井地电位测量系统的三维正演

针对井地电位测量系统的正演数值模拟问题,首先介绍了该系统的正演基本理论,然后采用"镜像"法,推导了垂直有限长线电流源在无限大半均匀介质中任意深度时的正常电位解析式,并在此基础上,推导了相应的装置系数和混合边界条件公式.结合有限差分法,实现了正演模型的求解计算.数值模拟中,采用不同的正常电位公式,混合边界条件公式和装置系数公式,分别建立了均匀介质模型、高阻异常体模型和低阻异常体模型,并对异常体正演响应的异常特征和分布规律进行了对比分析.结果表明,所推导的公式模拟异常体电位响应更准确,为后续更精确的反演和解释现场测量数据奠定了基础.     

磁偶极子构造法球形磁性矿体磁异常正演

球形磁性地质体是地质勘探中广泛遇到的基本地质体之一,选择合适的磁异常分析方法对其磁场进行准确正演计算具有重要的意义.为了准确快速地求解球形磁性地质体磁场特征,本文提出了一种基于磁偶极子构造原理求解球形磁性地质体磁场的数值计算方法,对球形磁性地质体按半径由小增大进行多层单元分割,再对每层单元进行块单元分割,将每个块单元视为"磁偶极子",利用基本磁偶极子公式计算了所有块单元在探测平面(线)的磁场大小,同时分析了球体磁场的剖面特征,并将数值解与解析解结果进行了对比和误差分析,最后建立双球组合模型结合Comsol多物理场仿真工具对其仿真对比,验证了磁偶极子构造法求解球体磁场的准确性.研究结果表明:数值解法结果与解析解结果磁异常波动趋势完全一致;不同测点处绝对误差有所差别,但磁异常值均在1 nT以下.本文提出的数值解法无需复杂的数学推导,计算结果稳定可靠.     

三维直流电场数值模拟的拟解析近似法:张量近似

拟解析近似方法是一种解决电磁场散射问题的快速求解积分方程的近似方法,它绕开了传统数值方法中的求解大型代数方程组或大型矩阵问题,适用于强散射和大扰动问题.本文应用孙建国提出的求解异常电场积分方程的张量拟解析近似理论公式,研究用其求解直流电场积分方程.利用接近实际的地电模型对异常电场进行模拟,研究了均匀场中异常球体的张量拟解析近似解;并对均匀场中的立方体异常体进行了数值计算.效果良好并具有很高的计算速度.研究结果为三维直流电场快速正反演模拟打下基础.     

基于单元细分H-自适应有限元全张量重力梯度正演

在重力梯度各阶张量正演中,难于求取复杂地质体的解析式,转而利用已知解析解的简单规则几何形体近似剖分地质体,其会引起较大的几何拟合误差.本文首先从全张量重力梯度解析公式出发结合等参变换基于有限元分析给出了满足地下复杂地质体正演的型函数,引入体积分的误差指示器用以估计全张量重力梯度误差,提出了适用于全张量重力梯度正演的自适应计算的策略及迭代算法,数值算例证明了本文方法的正确性和有效性.     

基于电场总场矢量有限元法的接地长导线源三维正演

电磁场数值模拟的背景场/异常场算法是三维正演的有效策略之一,优点为采用解析法计算电磁场背景场代替场源项、克服了场源奇异性,缺点为不适用于发射源布置于起伏地表或背景模型复杂的情形.总场算法是直接对电磁场总场开展数值模拟,其难点是有效加载场源、保证近区与过渡区数值解精度.本文以水平电偶源形式分段加载接地长导线源,并以电场总场Helmholtz方程为矢量有限元法控制方程,实现了基于非结构化四面体网格剖分的接地长导线源频率域电磁法三维正演.通过与均匀全空间中水平电偶源产生的电场解析解对比,验证了本文算法的正确性,并分析了四面体外接圆半径与其最短棱边的最大比值和四面体二面角最小值对数值解精度的影响规律.通过与块状高导体地电模型的积分方程法、有限体积法和基于磁矢量势Helmholtz方程的有限元法数值解对比,进一步验证了本文算法正确性,同时说明了非结构化四面体网格能够更加精细地剖分电性异常体,利于获得精确数值解.     

三度体（均质模型）位场波谱的正演计算

本文给出任意指向的均质直线段、多边形面和多面体的重磁异常谱的解析表达式。利用它们可以进行不规则三度体的重磁异常谱的数值计算。此外,尚导出任意指向的斜平行六面体重磁异常谱的解析表达式。这些表达式结构简单,易于计算,用作位场的正演反演计算都十分方便。     

基于MPICH环境下的复杂重力异常体并行正演模拟

任意多面体重力异常正演公式常用于解决复杂几何形态地质体的正演问题。 本文以均匀物性多面体重力异常正演公式为基础, 应用有限元技术中的网格离散化思想, 以任意四面体为基本单元, 通过并行计算技术在MPICH环境下实现了任意连续空间物性分布复杂异常体网格模型的重力异常正演模拟, 通过并行处理可以有效加速正演计算速度。 本文研究结果对于联合重力异常场正演建模和开展复杂模型网格的重力场计算有一定参考意义。     

二维激电中梯数据磁类比拟合反演研究

本文通过柯尔-柯尔模型研究地下不同几何参数的极化椭球体在地表的中梯双频激电响应,从而了解不同产状极化体产生极化率的特征,便于后期交互建模。由理论推导可知,任何具有激发极化特征的地质体在地表产生的激电异常都可近似为一个地下电偶极子的三维空间积分,这个积分形式与磁场正演计算公式一致,这意味着我们可以用解释磁法异常的方法解释激电异常,即磁类比法。只要根据先验信息实时改变模型,使激电正演得到的曲线与实测激电曲线逼近,当拟合差小于均方误差时,我们将最后一次模型近似看作是真实模型,由此获得地下极化体的顶埋深、倾角和几何大小等空间参数。实测数据测试表明,这种磁类比人机交互方法加入矿体的一些已知信息后,其拟合反演结果比自动反演更具合理性。      

地下铁质管线磁偶极子构造法正演单元划分策略

地下铁质管线经过地球磁场的磁化产生磁异常,对铁管磁异常进行数值分析可以帮助探测地下铁管的位置.为了在保证磁偶极子构造法地下铁质管线探测面磁异常正演准确性的前提下,尽量减少正演计算时间,在相对球坐标系中建立了管道单元划分策略分析模型,分析了测点空间位置对不同单元划分条件下磁异常计算差异的影响.研究结果表明:不同单元划分计算测点磁异常差异仅与距径比有关,相对误差与单元磁化率和外磁场特征无关;随着距径比增大,磁异常计算差异逐渐减小.在6.5倍距径比范围内,使用分块单元划分建立地下管线磁异常计算模型,在6.5倍距径比范围外,使用分节单元划分建立地下管线磁异常计算模型,不仅保证了管道磁异常正演计算的准确性,而且能够提高管道磁异常计算速率.     

基于Lorenz规范条件下磁矢势和标势耦合方程的频率域电磁法三维正演

为了克服空气层和地表耦合以及避免一次场计算,开发适合不同类型场源、不同应用范围的频率域三维正演模拟统一平台,本文从麦克斯韦基本方程出发,推导基于Lorenz规范条件的磁矢势和标势耦合方程;通过将不同类型场源分解成一系列短导线(电性)源组合,采用交错网格采样和有限体积技术对方程进行离散得到对称大型稀疏线性方程组,并采用Jacobi迭代预处理QMR(Quasi-Minimum-Residual,拟最小残差)算法进行求解,我们成功实现不同类型场源、不同应用范围的频率域电磁法三维正演模拟.通过层状模型下大地电磁法以及有限长接地导线和大回线磁性源激发下的电磁场响应模拟,并与一维解析解对比验证算法的有效性.进而,我们利用该算法平台的模拟结果对典型地电模型在不同场源激发下频率域电磁法响应特征进行对比分析.本文算法研究及实现为建立频率域电磁法三维正反演统一框架打下基础.     

非均匀物性条件下多尺度窗口修正法换算磁源重力异常及在寻找DSO的应用

DSO(Direct Shipping Ore,直运块矿)是一种高品位的富铁矿.在加拿大拉布拉多(Labrador)地槽谢菲尔威利(Schefferville)铁矿成矿带,含铁建造苏克曼(Sokoman)组地层全铁含量低、具高密度、强磁性,能够引起高重力异常与高磁异常;而风化淋滤后富集的赤铁矿和针铁矿等(也称DSO)全铁含量高,具高密度、无磁性,仅能够引起高重力异常.采用一般的滤波方法不能提取DSO的重、磁异常.本文采用基于泊松(Poisson)公式的磁场换算磁源重力异常(pseudo-gravity anomaly)方法,由磁场换算磁源重力异常,再与实测重力异常对比,得到纯粹由高密度、无磁性的DSO产生的剩余重力异常,对剩余重力异常采用密度成像与2.5D反演方法解释DSO.泊松公式虽然提出时间很长,但迄今为止仅仅用在资料解释中的定性分析,本文推导并实现了密度磁性非均匀条件下经典泊松公式的形式与实现过程,提出了多尺度窗口滑动线性回归修正的磁场换算磁源重力异常方法,使该公式的数学原理能够对重、磁异常的反演解释定量化.最后本文将多尺度窗口滑动线性回归修正的换算磁源重力异常方法用于加拿大拉布拉多地槽谢菲尔威利铁矿成矿带铁矿勘探,较好地解决了寻找高密度、无磁性DSO的问题.     

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.     

Gravity Anomalies of Arbitrary 3D Polyhedral Bodies with Horizontal and Vertical Mass Contrasts

During the last 15 years, more attention has been paid to derive analytic formulae for the gravitational potential and field of polyhedral mass bodies with complicated polynomial density contrasts, because such formulae can be more suitable to approximate the true mass density variations of the earth (e.g., sedimentary basins and bedrock topography) than methods that use finer volume discretization and constant density contrasts. In this study, we derive analytic formulae for gravity anomalies of arbitrary polyhedral bodies with complicated polynomial density contrasts in 3D space. The anomalous mass density is allowed to vary in both horizontal and vertical directions in a polynomial form of \(\lambda =ax^m+by^n+cz^t\), where m, n, t are nonnegative integers and a, b, c are coefficients of mass density. First, the singular volume integrals of the gravity anomalies are transformed to regular or weakly singular surface integrals over each polygon of the polyhedral body. Then, in terms of the derived singularity-free analytic formulae of these surface integrals, singularity-free analytic formulae for gravity anomalies of arbitrary polyhedral bodies with horizontal and vertical polynomial density contrasts are obtained. For an arbitrary polyhedron, we successfully derived analytic formulae of the gravity potential and the gravity field in the case of \(m\le 1\), \(n\le 1\), \(t\le 1\), and an analytic formula of the gravity potential in the case of \(m=n=t=2\). For a rectangular prism, we derive an analytic formula of the gravity potential for \(m\le 3\), \(n\le 3\) and \(t\le 3\) and closed forms of the gravity field are presented for \(m\le 1\), \(n\le 1\) and \(t\le 4\). Besides generalizing previously published closed-form solutions for cases of constant and linear mass density contrasts to higher polynomial order, to our best knowledge, this is the first time that closed-form solutions are presented for the gravitational potential of a general polyhedral body with quadratic density contrast in all spatial directions and for the vertical gravitational field of a prismatic body with quartic density contrast along the vertical direction. To verify our new analytic formulae, a prismatic model with depth-dependent polynomial density contrast and a polyhedral body in the form of a triangular prism with constant contrast are tested. Excellent agreements between results of published analytic formulae and our results are achieved. Our new analytic formulae are useful tools to compute gravity anomalies of complicated mass density contrasts in the earth, when the observation sites are close to the surface or within mass bodies.     

重力位谱分析及重力异常导数换算新方法--余弦变换

为了提高重力异常导数换算的精度,真实有效地反映地质体的异常特征,提出用余弦变换计算异常导数的新方法. 给出并证明了两个定理,利用它们推导出重力位余弦谱一般表达式以及重力异常各阶导数计算公式,建立了位场余弦谱分析理论. 模型实验中发现,用Fourier变换计算的一阶导数与理论导数偏差很大,而余弦变换计算的导数与理论异常导数拟合效果非常好,除边界几个数据因重力异常的有限截断产生的吉布斯效应残留使误差较大外,数据的计算精度均很高,误差为-009%～5%.     

大地电磁法三维交错采样有限差分数值模拟

系统地论述了大地电磁三维交错采样有限差分数值模拟算法实现过程中交错网格剖分、积分公式离散化、边界条件、方程组求解、三维张量阻抗的计算等内容. 由于提出了简洁的边界条件，采用了解大型系数矩阵方程组的双共轭梯度稳定解法，所实现的三维交错采样有限差分数值模拟算法具有迭代收敛稳定、计算精度高、速度快等特点. 通过两个理论模型的计算结果检验了算法的正确性和计算精度. 所实现的三维交错采样有限差分数值模拟算法为研究三维反演问题奠定了基础.     

A direct method of interpreting gravity and magnetic anomalies: The case of a horizontal cylinder

Summary A new method of interpreting the gravity and magnetic anomalies is introduced with special reference to the magnetic anomalies of a horizontal cylinder. The method consists of calculating the functions of the anomaly and its distance from an arbitrary point. These form a simple linear equation with coefficients related to the parameters defining the body. Since each observation forms a separate linear equation, the required normal equations are formed by the method of least squares and solved for the coefficients and hence for the various parameters defining the target. The discussion here is confined to the vertical magnetic anomalies. The application of the method to horizontal and total field anomalies of two dimensional bodies is also outlined.     

Interpretation of gravity and magnetic anomalies caused due to spherical bodies

Summary The method of continuation has been used to obtain the master curves for gravity and magnetic anomalies caused by spherical bodies. The procedure to calculate the depth of burial and radius of spherical bodies has been outlined.     

Detection of potential fields source boundaries by enhanced horizontal derivative method

A high‐resolution method to image the horizontal boundaries of gravity and magnetic sources is presented (the enhanced horizontal derivative (EHD) method). The EHD is formed by taking the horizontal derivative of a sum of vertical derivatives of increasing order. The location of EHD maxima is used to outline the source boundaries. While for gravity anomalies the method can be applied immediately, magnetic anomalies should be previously reduced to the pole. We found that working on reduced‐to‐the‐pole magnetic anomalies leads to better results than those obtainable by working on magnetic anomalies in dipolar form, even when the magnetization direction parameters are not well estimated. This is confirmed also for other popular methods used to estimate the horizontal location of potential fields source boundaries. The EHD method is highly flexible, and different conditions of signal‐to‐noise ratios and depths‐to‐source can be treated by an appropriate selection of the terms of the summation. A strategy to perform high‐order vertical derivatives is also suggested. This involves both frequency‐ and space‐domain transformations and gives more stable results than the usual Fourier method. The high resolution of the EHD method is demonstrated on a number of synthetic gravity and magnetic fields due to isolated as well as to interfering deep‐seated prismatic sources. The resolving power of this method was tested also by comparing the results with those obtained by another high‐resolution method based on the analytic signal. The success of the EHD method in the definition of the source boundary is due to the fact that it conveys efficiently all the different boundary information contained in any single term of the sum. Application to a magnetic data set of a volcanic area in southern Italy helped to define the probable boundaries of a calderic collapse, marked by a number of magmatic intrusions. Previous interpretations of gravity and magnetic fields suggested a subcircular shape for this caldera, the boundaries of which are imaged with better detail using the EHD method.     

High speed gravity and magnetic calculations of uniform cylindrical bodies of arbitrary cross-section and finite length

Summary A simple method is designed for programming the gravity and magnetic calculations of a right circular cylinder (vertical or horizontal) by treating it as a combination of thin rectangular slabs. It takes only a few seconds to compute a profile of each kind and the accuracy is comparable to that obtained by using exact expressions (involving complete elliptic integrals) instead. The method is also applicable to cylindrical bodies of arbitrary cross-section and could as well be used for rapid computation of derivatives of gravity and magnetic anomalies.     

Full gravity gradient tensors from vertical gravity by cosine transform

We present a method to calculate the full gravity gradient tensors from pre-existing vertical gravity data using the cosine transform technique and discuss the calculated tensor accuracy when the gravity anomalies are contaminated by noise. Gravity gradient tensors computation on 2D infinite horizontal cylinder and 3D ??Y?? type dyke models show that the results computed with the DCT technique are more accurate than the FFT technique regardless if the gravity anomalies are contaminated by noise or not. The DCT precision has increased 2 to 3 times from the standard deviation. In application, the gravity gradient tensors of the Hulin basin calculated by DCT and FFT show that the two results are consistent with each other. However, the DCT results are smoother than results computed with FFT. This shows that the proposed method is less affected by noise and can better reflect the fault distribution.