两斜交柱坐标系间谐波函数坐标变换的辅助平面方法.

针对利用波函数展开法进行三维地震响应研究中的坐标变换问题,提出了在两斜交柱坐标系间谐波函数空间坐标变换的辅助平面方法．通过建立与空间点对应的一系列辅助平面,将地震波三维散射研究中的谐波函数表达式从柱坐标系变换至辅助平面内的极坐标系,然后在辅助平面内运用Graf加法公式,将柱坐标系下的波函数表达式变换至与该柱坐标系斜交的另一柱坐标系下,从而将三维问题转换成二维问题进行处理,最终得到了斜交柱坐标系下内域问题和外域问题的坐标变换公式．      

Gravity and magnetic field interpretation based on transformations of horizontal gradients in the VECTOR system

A method of potential field processing based on the transformation of vectors of the total horizontal gradient in windows of various sizes is considered. The gradients are calculated at the centers of triangles, whose vertices are points of observations, as a rule, of gravity and magnetic fields. Averaging of horizontal gradients of the field rather than initial values of the field is the main distinction of this approach from the known methods. This procedure, referred to in this paper as vector scanning of the field, makes it possible to obtain layer distributions of field sources in a 3-D diagram that is a quasi-density model of the study medium within the framework of certain model concepts. The paper presents a model example demonstrating the possibility of separating the fields produced by two sources located on a vertical line and an example illustrating the application of this method to the interpretation of the gravity field in the zone of the geodynamic influence of the Urals.     

三维TTI介质高效射线追踪方法

TTI介质是石油地震勘探领域最常用的各向异性介质,快速计算TTI介质射线路径和走时信息有重要的研究意义.TTI介质传统运动学射线追踪方法一般基于任意弹性介质射线方程,利用Bond变换或者四阶张量变换来处理复杂的21个弹性参数,因而非常耗时.实际野外对称轴统一的TTI介质模型,一般可以看成VTI介质模型旋转一定角度获得.为此,本文推导了三维VTI介质射线追踪方程,提出先在本构坐标系中进行VTI介质射线追踪,再通过坐标旋转将射线路径旋转至观测坐标系中,获得TTI介质射线路径.数值模型计算表明该方法高效和精确,较传统方法效率提高了近4倍.在强各向异性等特殊情况下,体波波前面都与理论群速度面一致.     

薄层状沉积旋迴介质中垂直地震剖面法反射波勘探基础及解释方法的讨论

本文从地震模型实验入手,讨论和测试了垂直地震剖面法在多个薄层粘合的二维固体模型中地震直达波、反射波的运动规律,及其各向异性现象。利用坐标变换方法,将多层的各向异性介质转换成均匀各向同性介质,从而推导出地震反射波理论时-深方程。通过模型的测定,介绍了各向异性系数的求取及地质剖面的构制方法。比较了各向异性理论及均匀同性理论解释的地质剖面,肯定了各向异性理论在垂直地震剖面法解释中的可靠性。     

Magnetic fields of 3-D anisotropic bodies: Theory and practice of calculations

The general theory of the distribution of the volume and surface magnetic mass within 3-D anisotropic bodies and solving the forward problem is given in this paper. An algorithm for calculating the magnetic fields of monoclines of complex shape and folded structures with uniform anisotropy is constructed. The algorithm is based on the regularities in the relationship between the magnetic susceptibility of anisotropy, tectonic structure, and the anomalous magnetic field established experimentally by Zavoisky. These regularities not only simplify the solution of the problem, but significantly facilitate the preparation of original field data necessary for solving it. The latter circumstance is of especial importance. The algorithm is designed for wide practical application in the construction of 3-D magnetic models of local and regional geological structures.We draw attention to the fact that the use of a curvilinear coordinate system is reasonable in cases when the distribution of the magnetic mass density in anisotropic geological formations is studied.The features of the relationship between the intensity and induction of a magnetic field in different unit systems are pointed out in their application to magnetology problems.     

Singularity-free expression of magnetic field of cuboid under undulating terrain

Most of the current computing methods used to determine the magnetic field of a uniformly magnetized cuboid assume that the observation point is located in the upper half space without a source. However, such methods may generate analytical singularities for conditions of undulating terrain. Based on basic geomagnetic field theories, in this study an improved magnetic field expression is derived using an integration method of variable substitution, and all singularity problems for the entire space without a source are discussed and solved. This integration process is simpler than that of previous methods, and final integral results with a more uniform form. ΔT at all points in the source-free space can be calculated without requiring coordinate transformation; thus forward modeling is also simplified. Corresponding model tests indicate that the new magnetic field expression is more correct because there is no analytical singularity and can be used with undulating terrain.     

层状介质中三维大地电磁模拟

本文发展了积分方程法用于层状介质中三维不均匀体的大地电磁模拟算法（简称MT）.分为二个步骤:第一步,异常体用等效的散射电流代替,通过层状介质中的格林函数,建立以散射电流为未知参数的积分方程;第二步,把求得的散射电流乘上相应的格林函数,即得地面上的二次电磁场,由此而计算出各种MT响应. 文中采用了数值滤波与插值、群变换以及格林矩阵带状化三个方面的数值处理方法,提高了计算效率.通过与已发表的三维MT计算结果对比及格林函数互易性检验,表明了该算法的正确性.在此基础上,进行了数值模拟,初步讨论了三维MT曲线的畸变特点.     

地磁学中的球面各向同性随机矢量场

从地磁场随机模拟的需要出发，研究球面各向同性随机矢量场．首先指出，与三维空间中的随机矢量场不同，在球上均匀性蕴含于各向同性；三维均匀各向同性随机矢量场一定是球上各向同性随机矢量场，反之不真．其次，以地核磁场的随机模型为例，给出了球面各向同性模型的空间两点关联张量的并矢表示，并将其在不变坐标系分解；利用核幔边界地球主磁场的球谐功率谱的幂级数拟合模型，估算空间两点关联张量．因此，即使根据地球主磁场观测认定地核场具有球面各向同性，亦不能推定地核内部磁流体运动是三维均匀各向同性的．     

球坐标系下谱元法三维地震波场模拟

谱元法已成为区域性乃至大陆性尺度地震波场模拟的重要工具.对于区域或大陆尺度层析成像而言,地球曲率不可忽略,此时模拟地震波传播采用球坐标系更为合适.本文从球坐标系下弹性波动方程弱形式出发,基于球坐标系变分原理给出了球坐标系下求解三维地震波方程的谱元法.另一方面,计算Fréchet敏感核是进行全波形反演的关键,本文借助伴随原理,推导了全波走时层析成像三维Fréchet敏感核表达式.为了验证球坐标系下谱元法的精度,我们将数值模拟结果与normal mode方法得到的解析解在1-D PREM模型下进行了对比.同时,我们将此方法应用到华北克拉通区域,以期获得地球内部结构精确成像.基于3-D全球径向各向异性地幔模型S362ANI和3-D地壳模型Crust1.0,我们建立了华北克拉通初始3-D背景模型,并将数值模拟结果与实际观测台站记录波形资料进行对比分析,利用互相关方法提取走时残差,最后给出了Fréchet敏感核在3-D空间中的分布,这些工作为下一步开展球坐标系下三维大尺度全波形反演奠定了基础.     

1-D, 2-D, and 3-D analytical solutions of unsaturated flow in groundwater

An excellent tool for checking numerical models of unsaturated flow in groundwater is analytical solutions. However, because of the highly nonlinear nature of the governing partial differential equation, only a limited number of analytical solutions are available. This paper first gives some simple 1-D solutions. Next, by use of a transformation, the nonlinear partial differential equation is converted to a linear one for a specific form of the moisture content vs. pressure head and relative hydraulic conductivity vs. pressure head curves. This allows both 2-D and 3-D solutions to be derived, which is done in this paper. Finally, computations from a finite element computer program are compared with results from one of the analytical solutions to illustrate the use of the derived equations.     

PRACTICAL ASPECTS OF THE DETERMINATION OF 3-D STACKING VELOCITIES*

Proper stacking of three-dimensional seismic CDP-data generally requires the knowledge of normal moveout velocities in all source-receiver directions contributing to a CDP-gather. The azimuthal variation of the stacking velocities mainly depends on the dip of the seismic interfaces. For a single dipping plane a simple relation exists between the dip and the azimuthal variation of NMO-velocity. Varying strike and dip of subsequent reflectors, however, result in a complex dependency of the seismic parameters. Reliable information on the spatial distribution of the normal moveout (NMO)-velocity can be derived from a wavefront curvature estimation using a 3-D ray-tracing technique. These procedures require additional information, e.g. reflection time gradients or depth maps to show interval velocities between leading interfaces. Moreover, their application to an extended 3-D data volume is restricted by high costs. The need for a routine 3-D procedure resulted in a special data selection to create pseudo 2-D profiles and to apply existing velocity estimation routines to these profiles. At least three estimates in different directions are necessary to derive the full azimuthal velocity variation, characterized by the large and the small main axis and the orientation of the velocity ellipse. Errors are estimated by means of computer models. Stacking velocities obtained by mathematical routines (least-squares fit) and by seismic standard routines (NMO-correction and correlation) are compared. Finally, a general 3-D velocity procedure using cross-correlation of preliminarily NMO-corrected traces is proposed.     

Curie point depth from spectral analysis of aeromagnetic data from Cerro Prieto geothermal area,Baja California,México

Using aeromagnetic data acquired in the area from the Cerro Prieto geothermal field, we estimated the depth to the Curie point isotherm, interpreted as the base of the magnetic sources, following statistical spectral-based techniques. According to our results the Curie point isotherm is located at a depths ranging from 14 to 17 km. Our result is somewhat deeper than that obtained previously based only in 2-D and 3-D forward modeling of previous low-quality data. However, our results are supported by independent information comprising geothermal gradients, seismicity distribution in the crust, and gravity determined crustal thickness. Our results imply a high thermal gradient (ranging between 33 and 38 °C/km) and high heat flow (of about 100 mW/m²) for the study area. The thermal regime for the area is inferred to be similar to that from the Salton trough.     

有限长圆柱体磁异常场全空间正演方法

在经典位场理论中,许多简单形体位场异常难以通过积分得到全空间的解析式.圆柱体是一类很重要的理论模型体,常用于模拟圆柱状地质体或非地质体(如管线),但目前还不能用解析公式正演有限长圆柱体在三维空间里的磁异常,而多是采用近似简化为有限长磁偶极子或线模型代替.对于有限长圆柱体,特别是半径相对于上顶埋深较大时,这种近似的误差不可忽略.本文利用共轭复数变量替换法,推导出有限长圆柱体在全空间的引力位一阶、二阶导数,利用Poisson关系得到磁异常正演公式,进而利用有限长圆柱体磁异常正演公式求解管状体的磁异常,得到不同磁化方向、不同大小的管线产生的磁场的特征,并将其推广到截面为椭圆的情况.最后通过模拟计算定量给出了将圆柱体近似为线模型的条件.     

C-RESPONSE OF GEOMAGNETIC DEPTH SOUNDING ON A 1D THIN SHELL MODEL

This paper tries to formulate the C-response of geomagnetic depth sounding(GDS)on an Earth model with finite electrical conductivity. The computation is performed in a spherical coordinate system. The Earth is divided into a series of thin spherical shells. The source is approximated by a single spherical harmonic P₁⁰ due to the spatial structure of electrical currents in the magnetosphere. The whole solution space is separated into inner and external parts by the Earth surface. Omitting displacement current, the magnetic field in the external space obeys Laplacian equation, while in the inner part, due to the finite conductivity, the electromagnetic fields obey Helmholtz equation. To connect the magnetic fields in the inner and external space, the continuity condition of magnetic fields is used on the Earth surface. The external magnetic fields are expressed by the inner and external source coefficients, from which a new parameter called C-response is computed from the inner coefficient divided by the external coefficient, thus normalizing the actual source strength. The inner magnetic fields in each layer can be recursively derived by the continuity boundary condition of both normal and tangential components of the magnetic field from the initial boundary condition at core-mantle-boundary. The consistency of our C-responses with that from a typical 1-D global model validates the accuracy of the proposed algorithm. Numerical results also show that the C-response estimated from the geomagnetic transfer function method will deviate exceeding 5%from the actual response at longer periods than about 10⁶s, which means that ignoring the curvature of the Earth at extreme long periods will make inversion result unreliable. Therefore, an accurate C-response should be computed in order to lay a solid foundation for reliable inversion.     

3-D Global Induction in the Oceans and Solid Earth: Recent Progress in Modeling Magnetic and Electric Fields from Sources of Magnetospheric,Ionospheric and Oceanic Origin

Electromagnetic induction in the Earth’s interior is an important contributor to the near-Earth magnetic and electric fields. The oceans play a special role in this induction due to their relatively high conductivity which leads to large lateral variability in surface conductance. Electric currents that generate secondary fields are induced in the oceans by two different processes: (a) by time varying external magnetic fields, and (b) by the motion of the conducting ocean water through the Earth’s main magnetic field. Significant progress in accurate and detailed predictions of the electric and magnetic fields induced by these sources has been achieved during the last few years, via realistic three-dimensional (3-D) conductivity models of the oceans, crust and mantle along with realistic source models. In this review a summary is given of the results of recent 3-D modeling studies in which estimates are obtained for the magnetic and electric signals at both the ground and satellite altitudes induced by a variety of natural current sources. 3-D induction effects due to magnetospheric currents (magnetic storms), ionospheric currents (Sq, polar and equatorial electrojets), ocean tides, global ocean circulation and tsunami are considered. These modeling studies demonstrate that the 3-D induction (ocean) effect and motionally-induced signals from the oceans contribute significantly (in the range from a few to tens nanotesla) to the near-Earth magnetic field. A 3-D numerical solution based on an integral equation approach is shown to predict these induction effects with the accuracy and spatial detail required to explain observations both on the ground and at satellite altitudes. On leave from Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation, Russian Academy of Sciences, 142190 Troitsk, Moscow region, Russia.     

利用曲线网格有限差分方法研究三维倾斜断层的破裂动力学

利用曲线网格有限差分方法,研究了三维倾斜断层的破裂传播过程.基于断层面生成贴体曲线网格,并通过坐标变换将含曲线网格的物理空间转换到含均匀直角网格的计算空间,实现了有限差分方法对复杂界面的处理.通过模拟地震断层的自相似破裂和自发破裂,并与已有发表的结果对比,发现拟合程度较高,验证了本方法的有效性和精确性.重点研究了不同倾角的倾斜断层破裂,最后展望了今后用本方法对非均匀介质中和任意起伏地表下的任意非平面断层破裂动力学的进一步研究.     

Modelling converted seismic waveforms in isotropic and anisotropic 1-D gradients: discontinuous versus continuous gradient representations

Over the past decade, there have been numerous receiver function studies directed at imaging the lithosphere-asthenosphere boundary (LAB). Although it is generally accepted that receiver function phases observed in these studies are derived from physical mode conversions at depth within the lithosphere-asthenosphere transition, it is still debatable as to whether these phases are directly indicative of the LAB. This is because interpretation of receiver function LAB signals relies on understanding the elastic characteristics of the Earth??s outer thermal boundary layer. The main issues for receiver function imaging are the sharpness of the elastic material property transition and, more importantly, what specifically are the material gradients. To test the various transition models, a forward modelling approach is required that allows accurate waveform synthetics for a range of discontinuous and continuous gradients in anisotropic, elastic media. We present a derivation of the reflection and transmission response for continuous one-dimensional (1-D) gradients in generally anisotropic elastic media. We evaluate the influence of 1-D isotropic and anisotropic elastic gradients on the seismic waveform by comparing numerical results of models for discontinuous and continuous transitions. The results indicate that discontinuous representations using layers each with uniform parameters and with thicknesses on the order of approximately 1/3 to 1/8 of the dominant seismic wavelength can be used to accurately model P-to-S and S-to-P mode conversions due to continuous transitions of both isotropic and anisotropic elastic properties. From a practical point of view, when comparing synthetic modelling with observation, this constraint can be relaxed further. The presence of signal noise and/or the result of receiver function stacking techniques will likely obscure these subtle waveform e ff ects. Hence this study suggests that accurate synthetic waveforms for LAB transitions can be modelled with discontinuous gradient representations using a reasonable number of discrete transition layers with layer thicknesses no greater than 1/2 to 1/3 the dominant seismic wavelength.     

Interpretation of aeromagnetic anomalies over part of Southeastern Nigeria using three-dimensional Hilbert transformation

We present the results of the application of three-dimensional Hilbert transformation to the analysis of airborne total field magnetic anomalies over part of Southeastern Nigeria. This study not only substantiates the usefulness of 3-D Hilbert transforms in the interpretation of magnetic anomaly maps but also more clearly delineates the structural pattern of the area, of study. Results from the previous study are discussed in relation to the results of previous geological and geophysical studies of the area.     

Exact solution for two-dimensional flow to a well in an anisotropic domain

Although most current applications of the analytic element method are formulated for isotropic hydraulic conductivity, anisotropic domains can be modeled with analytic elements using the well-known coordinate transformation where one coordinate axis is scaled by the square root of the anisotropy ratio. If the standard analytic solution for steady radial flow to a well is used with this coordinate transformation, the resulting solution correctly models the far field but it does not meet the constant head boundary condition at the well radius. This could be a significant shortcoming if you are interested in the flow field close to the well or want to estimate the head at the pumping well. A new solution for two-dimensional steady flow to a well in an anisotropic domain is presented. This solution satisfies the governing equations exactly and meets the constant head boundary condition at the well radius exactly. It was derived using a conformal mapping.     

Forward computation of the magnetic field of a 3D body with arbitrary boundary and continually varying magnetization

The forward computation of the gravitational and magnetic fields due to a 3D body with an arbitrary boundary and continually varying density or magnetization is an important problem in gravitational and magnetic prospecting. In order to solve the inverse problem for the arbitrary components of the gravitational and magnetic anomalies due to an arbitrary 3D body under complex conditions, including an uneven observation surface, the existence of background anomalies and very little or no a priori information, we used a spherical coordinate system to systematically investigate forward methods for such anomalies and developed a series of universal spherical harmonic expansions of gravitational and magnetic fields. For the case of a 3D body with an arbitrary boundary and continually varying magnetization, we have also given the surface integral expressions for the common spherical harmonic coefficients in the expansion of the magnetic field due to the body, and a very precise numerical integral algorithm to calculate them. Thus a simple and effective method of solving the forward problem for magnetic fields due to 3D bodies of this kind has been found, and in this way a foundation is laid for solving the inverse problem of these magnetic fields. In addition, by replacing the parameters and unit vectors in the spherical harmonic expansion of a magnetic field by gravitational parameters and a downward unit vector, we have also derived a forward method for the gravitational field (similar to that for the magnetic case) of a 3D body with an arbitrary boundary and continually varying density.