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1.
The direct gravity problem and its solution belong to the basis of the gravimetry. The solutions of this problem are well known for wide class of the source bodies with the constant density contrast. The non-uniform density approximation leads to the relatively complicated mathematical formalism. The analytical solutions for this type of sources are rare and currently these bodies are very useful in the gravimetrical modeling. The solution for the vertical component of the gravitational attraction vector for the 3D right rectangular prism is known in the geophysical literature for the density variations described by the 3-rd degree polynomial. We generalized this solution for an n-th degree, not only for the vertical component, but for the horizontal components, the second-order derivatives and the potential as well. The 2D modifications of all given formulae are presented, too. The presented general solutions, which involve a hypergeometric functions, can be used as they are, or as an auxiliary tool to derive desired solution for the given degree of the density polynomial as a sum of the elementary functions. The pros-and-cons of these approaches (the complexity of the programming codes, runtimes) are discussed, too.  相似文献   

2.
Variable-density sources have been paid more attention in gravity modeling. We conduct the computation of gravity gradient tensor of given mass sources with variable density in this paper. 3D rectangular prisms, as simple building blocks, can be used to approximate well 3D irregular-shaped sources. A polynomial function of depth can represent flexibly the complicated density variations in each prism. Hence, we derive the analytic expressions in closed form for computing all components of the gravity gradient tensor due to a 3D right rectangular prism with an arbitrary-order polynomial density function of depth. The singularity of the expressions is analyzed. The singular points distribute at the corners of the prism or on some of the lines through the edges of the prism in the lower semi-space containing the prism. The expressions are validated, and their numerical stability is also evaluated through numerical tests. The numerical examples with variable-density prism and basin models show that the expressions within their range of numerical stability are superior in computational accuracy and efficiency to the common solution that sums up the effects of a collection of uniform subprisms, and provide an effective method for computing gravity gradient tensor of 3D irregular-shaped sources with complicated density variation. In addition, the tensor computed with variable density is different in magnitude from that with constant density. It demonstrates the importance of the gravity gradient tensor modeling with variable density.  相似文献   

3.
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.  相似文献   

4.
Magnetic anomalies of complicated 3D sources can be calculated by using a combination of analytical and numerical integration. Two surfaces and the magnetization parameters (the amplitudes of the induced and remanent components and the direction cosines) of the source can be defined by arbitrary functions or by discrete data points in a plane. When combined with a polynomial magnetization function in the direction of the third axis, 3D magnetization distribution can also be modelled. The method gives very general equations for anomaly calculation. It can be used for direct modelling of sources interpreted by seismic or other methods and also for interactive interpretation with fast computers. It is possible to calculate anomalies of, for example, intrusives or folded sedimentary beds whose surfaces are functions of horizontal coordinates and which have polynomial magnetization variations in the vertical direction due to gravitational differentiation and arbitrarily varying magnetization in the horizontal direction due to regional metamorphosis. If the distribution of magnetization parameters in the vertical direction cannot be described satisfactorily by polynomials, models can be used whose surfaces are functions of the vertical coordinate and which can then have any arbitrary magnetization distribution in the vertical direction.  相似文献   

5.
本文将均质的任意二维、三维物体位场的波谱解析表达式的研究成果推广到变密度、变磁化强度的更一般的情形。对密度差随深度呈指数函数衰减或线性变化的模型,获得了任意倾斜多边形质量面、斜平行六面体以及一般的多面体等形体的重力谱的解析表达式。它们的结构与均质体相应表达式一样简单,易于计算。以上结果表明,在很一般的条件下,位场波谱具有指数函数和的形式。  相似文献   

6.
Wavenumber domain expressions for bodies with elliptical cross-section and of ellipsoidal shape have been developed both for homogeneous bodies and for certain bodies of density/magnetization varying linearly with depth or, more generally, according to a polynomial with depth. The simple expressions thus obtained lend themselves to an easy analysis, especially for long and short wavelengths. At the long-wavelength end of the spectra their decay is governed by an exponential with a decay “depth” equal to the depth to the center of mass. At the short-wavelength end this depth is replaced by the depth to the upper focus of the ellipsoid (or the elliptic cross-section). For vertically inhomogeneous ellipsoids the decay rate is also dependent on the product of the vertical gradient of density/magnetization and their semi-axes.  相似文献   

7.
Summary The solution of the direct magnetic problem is derived for three-dimensional bodies under the assumption that the components of the magnetization vector are analytical functions of the co-ordinates within the body being considered. The solution to the problem is given in the form of Green-type integrals and, much like with gravitational fields of inhomogeneous bodies, it may serve the purpose of solving the problem of the analytical continuation of the external field into the body.  相似文献   

8.
Eigenvalues of the curvature gravity gradient tensor (CGGT) have been utilized to detect and outline edges of geologic bodies. In this paper, we analyzed and discussed the application scope and the limitation of CGGT using a complex synthetic model. We found that the large eigenvalue only can be used to outline edges of positive density bodies, and the small eigenvalue only can be used to delineate edges of negative density bodies. However, in the actual geological situation, the positive and negative density bodies usually coexist simultaneously. Therefore, it has certain limitations to detect edges using this method directly. In view of the limitations of the method, we combined the gravity anomaly with large eigenvalues, and proposed a new improved method for the eigenvalues of CGGT. The improved method can be utilized to outline edges of causative sources in any case. It was tested on synthetic model data and real field data. All of the results have shown that the new improved method is effective for edge detection.  相似文献   

9.
The ground magnetic response of deep ore bodies in the Daye iron-ore deposit is relatively weak, and sometimes concealed by the strong magnetic background of shallower sources. Apart from the low-quality ground magnetic data, another critical problem for reconstructing the deep skarn-type ore bodies is developing a versatile inversion scheme that can simultaneously resolve 3D sources with arbitrary shapes. In this case, we resort to interactive 3D forward modeling solution with the joint use of two data sets-total field surface and three-component borehole magnetic data. Joint inversion of the two data sets is expected to help resolve the ambiguity associated with either data set and greatly reduces the nonuniqueness of the magnetic inversion. Such nonuniqueness is especially severe when a 3-D distribution of magnetic susceptibility, instead of a simple body, is sought from the inversion.In this paper, we calculate the magnetic field on the surface and in the borehole caused by 3D arbitrarily-shaped bodies with the triple integral method. The complex 3D magnetic sources having arbitrary shapes are constructed with cross-sections, termination points and facets in our visualization technology. We specify, interactively and in a user-friendly environment, the outline of the sources in terms of geometric elements and their magnetic parameters. The method automatically fits the observations within a prescribed precision. If dissatisfied, the user can redefine the model parameters and proceed to a new inversion. The method's ability to interpret a complicated 3D geologic environment is demonstrated on synthetic models and real data profiles in the Daye iron-ore deposit in central China. The interactive forward modeling results in all tests demonstrate a good correlation of estimated magnetic sources with corresponding known geologic features.  相似文献   

10.
A generic gravity source moment is an integral, over the source volume, of the product of the density distribution by a polynomia in the Cartesian coordinates of a point belonging to this volume. We obtained a formal expression for a generic moment in terms of integrals involving the gravity anomaly and the gravity potential. By analyzing the conditions under which this expression is valid, we conclude that, without usinga priori information regarding the sources, it is possible to determine, from the gravity anomaly, any moment or linear combination of moments whose associated polynomial has null Laplacian and depends only on the coordinates defining the measurement plane. Additionally, no moment whose associated polynomial has a nonnull laplacian can be determined without usinga priori information of the source.  相似文献   

11.
本文以常用的几种粘弹性体模拟地球介质的粘弹性,建立了该种介质中的波动方程组;在运用合理的近似处理方法基础上,得到了基本解;提出了校正其它衰减因素的功率谱切比雪夫拟合方法,由此给出了粘性Q值的概念和算法,最后讨论了该方法的可行性。  相似文献   

12.
Summary In this paper the solution of the direct magnetic problem for two-dimensional bodies, founded on the application of Green's theorem is derived. This solution is derived under the assumption that the components of the magnetization vector have continuous derivatives with respect to the coordinates and that they are continuous within the body. The problem is solved in terms of Green-type integrals for the scalar and vector potential of the magnetostatic field and it may serve the purpose of solving the problem of the analytical continuation of the external field into the body.  相似文献   

13.
全球大气电模式是指大气和地表构成的闭合回路,其中考虑雷暴为主要充电电源,并将雷暴作为一个垂直电偶极子来考虑.本文从电流连续方程出发,采用球谐函数展开法,边界条件采用下边界为地表大气电势,上边界为电离层底部电势,非齐次电流源函数项采用垂直偶极子假设.通过所做的非线性函数变换,得出了考虑雷暴活动的全球电模式的近地稳态解析解,并证明其能展开为有限多项式级数之和.所得结论与过去的理论分析结果和实验观测是一致的.该文的工作是对以前相关工作仅能得到近似解的一种改进,同时本文所得公式可用于今后对大气电参量进行相关的数值计算及模拟之中.  相似文献   

14.
埋藏在6000m以下的地震反射异常体是礁体还是火成岩体？或者是海底滑塌体？在高质量重磁电资采集的基础上，通过对研究区域内的地层岩石物性细致研究，利用CEMP资料界面约束二维反演、重磁力异常剥离技术和LCT重磁力地震联合反演解释技术细致研究了异常体的物性参数，从密度、磁化率、电阻率、速度等多种特征推测判断了深层异常体地质属性，为钻井部署提供了间接依据。  相似文献   

15.
Most amplitude versus offset (AVO) analysis and inversion techniques are based on the Zoeppritz equations for plane‐wave reflection coefficients or their approximations. Real seismic surveys use localized sources that produce spherical waves, rather than plane waves. In the far‐field, the AVO response for a spherical wave reflected from a plane interface can be well approximated by a plane‐wave response. However this approximation breaks down in the vicinity of the critical angle. Conventional AVO analysis ignores this problem and always utilizes the plane‐wave response. This approach is sufficiently accurate as long as the angles of incidence are much smaller than the critical angle. Such moderate angles are more than sufficient for the standard estimation of the AVO intercept and gradient. However, when independent estimation of the formation density is required, it may be important to use large incidence angles close to the critical angle, where spherical wave effects become important. For the amplitude of a spherical wave reflected from a plane fluid‐fluid interface, an analytical approximation is known, which provides a correction to the plane‐wave reflection coefficients for all angles. For the amplitude of a spherical wave reflected from a solid/solid interface, we propose a formula that combines this analytical approximation with the linearized plane‐wave AVO equation. The proposed approximation shows reasonable agreement with numerical simulations for a range of frequencies. Using this solution, we constructed a two‐layer three‐parameter least‐squares inversion algorithm. Application of this algorithm to synthetic data for a single plane interface shows an improvement compared to the use of plane‐wave reflection coefficients.  相似文献   

16.
An inverse gravity problem is solved for a geological model consisting of bodies of Sretenskii’s class. The position of the middle plane is fixed for each body. It is required to determine the upper and lower boundaries of a body, which are described by analytical functions and are parameterized. The solution of the problem is illustrated by an example.  相似文献   

17.
The spectrum of a magnetic or a gravity anomaly due to a body of a given shape with either homogeneous magnetization or uniform density distribution can be expressed as a product of the Fourier transforms of the source geometry and the Green's function. The transform of the source geometry for any irregularly-shaped body can be accurately determined by representing the body as closely as possible by a number of prismatic bodies. The Green's function is not dependent upon the source geometry. So the analytical expression for its transform remains the same for all causative bodies. It is, therefore, not difficult to obtain the spectrum of an anomaly by multiplying the transform of the source geometry by that of the Green's function. Then the inverse of this spectrum, which yields the anomaly in the space domain, is calculated by using the Fast Fourier Transform algorithm. Many examples show the reliability and accuracy of the method for calculating potential field anomalies.  相似文献   

18.
重力勘探中复杂条件下的三维正演计算量大存储要求高,使得这种条件下重力勘探高效、精细正反演变得困难.针对这一问题,提出一种空间-波数混合域数值模拟方法,该方法将空间域引力位积分进行水平方向二维傅里叶变换,将三维空间域卷积问题转换为多个不同波数之间相互独立的空间垂向一维积分问题,一维积分垂向可离散为多个单元积分之和,每个单元采用二次形函数表征密度变化,可得出单元积分的解析表达式.该方法计算量和存储需求少,算法高度并行;保留垂向为空间域,优势之一在于可根据实际情况合理调整单元疏密程度,准确模拟任意复杂地形和密度异常体的重力异常,兼顾计算精度与计算效率;优势之二在于用形函数拟合求得积分的解析解,计算精度和效率高;充分利用一维形函数积分的高效和高精度,不同波数之间一维积分高度并行性及快速傅里叶变换的高效性,实现重力异常场三维数值模拟.设计棱柱体模型,通过数值解和解析解对比验证了该方法的正确性、适用性和高效性.针对任意复杂地形条件下的重力场及其张量的模拟问题,提出一种快速算法,对其有效性进行了验证.探究标准FFT法的截断效应对计算精度的影响,对比分析Gauss-FFT法和标准FFT扩边法两种方法的计算精度和效率,总结了二者的选取策略,结果表明选用标准FFT扩边法计算效率更高.实际地形的数值模拟表明本文算法适用于任意复杂地形的高效计算.  相似文献   

19.
多辐射源地空瞬变电磁响应三维数值模拟研究   总被引：1，自引：0，他引：1  
地空瞬变电磁法结合地面和航空电磁法的优点,可实现探测深度和工作效率的平衡.当前地空瞬变电磁法采用单一线源激发电磁辐射场,仅能从一个侧面与地质体耦合,难以获得地质体的全息影像.采用多辐射源是解决这一问题的途径.本文采用三维矢量有限元法对两个不同地质体多个辐射源情况下的地空瞬变电磁响应开展了模拟研究,分析了多辐射源在不同辐射方向、不同飞行高度电磁响应的分布特征.研究表明,由多辐射场源作为地空电磁法的发射源,通过分散布设的线源,可以在地下激发与地质体多方位耦合的电磁场,能够获得地下地质体多方位不同高度情况下的耦合信息.同时,多辐射场源能够增强源电磁场的辐射强度,减少单一线源体积效应影响,飞行高度较低时可获得较强的响应幅值,研究结果为多辐射地空瞬变电磁法深部精细探测提供理论依据.  相似文献   

20.
Monitoring of the vadose zone is a potentially complex, time-consuming, and expensive problem. The location of monitoring points and selection of monitoring instruments can be optimized by using computer models. Numerical models developed for this purpose in the past have often been expensive and difficult to use. This paper describes a fast, three-dimensional, approximate analytical solution to the moisture content in the unsaturated zone. An analytical solution is available for steady-state drainage, whereas an approximate analytical solution is available for the transient case. The model will handle an arbitrary distribution of fluid sources, as well as vertical and horizontal impermeable boundaries.
The model may be applied to predict the incursion of fluid from accidental leakage or infiltration over large areas from unlined ponds and land treatment sites. The model is quite useful as an aid in designing monitoring or premonitoring programs near hazardous waste sites. Examples are presented to demonstrate the model's utility in estimating the maximum spread of a contaminant, the extent to which the fluid may spread with depth, the regions of high and low capillary pressure, and the non-linear behavior of the saturation when drainage from several sources in considered. These results are useful for the placement of monitoring locations and the selection of appropriate instruments, and as a tool in working with regulatory agencies to design monitoring programs. A glimpse of the future is necessary for today's planning. Computer models are some of the most useful crystal balls we have available.  相似文献   

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