THE GRAVITY AND MAGNETIC FIELDS FROM ELLIPSOIDAL BODIES IN THE WAVENUMBER DOMAIN*

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.     

LINEAR TRANSFORMATIONS OF GRAVITY AND MAGNETIC FIELDS*

The spectral representation of gravity and magnetic fields shows that the mathematical expressions describing these fields are the result of convolution of factors which depend on the geometry of the causative body, the physical properties of the body and the type of field being observed. If a field is known, it is possible to remove or alter these factors to map other fields or physical parameters which are linearly related to the observed field. The transformations possible are: continuation, reduction to the pole, converting between gravity and magnetic fields, converting between components of measurement, calculation of derivatives, and mapping magnetization and density distribution, relief on interfaces, and vertical thicknesses of layers.     

三维重磁交互解释及区域与局部异常的分离

探讨了三维重磁交互解释的方法技术,提出构制任意形状三维物体模型的用于计算机的橡皮膜技术．针对通常分离区域与局部异常所存在的问题,提出在异常交互解释过程中同时进行异常分离的方法,并提出相应的算法．最后介绍一个应用实例．     

APPROXIMATION OF THE GRAVITATIONAL ATTRACTION OF GEOLOGICAL BODIES*

The gravitational attraction produced by a geological body of irregular shape can be easily determined by dividing it into cubes of small size. The exact expression of the effect of a cube is very complicated, but it can be calculated by using an electronic computer. 4851 values of the gravitational attraction were determined for different positions of a cube with the side of 2l and the center in M(x₀, y₀, z₀), for x₀∈[0;20], y₀∈[0;20] and z₀∈[0;10]. Using these values, templates in double logarithmic representation were drawn, with x₀ and z₀ as parameters and y₀ as abscissa, and with x₀ and y₀ as parameters and z₀ in abscissa; this double set of templates permits a good interpolation for all cubes in the considered domain of M. The use of templates was tested to approximate the effect produced by a theoretical model of spherical shape and in a real case of a three-dimensional salt body of known shape based on a large number of boreholes. In both cases very good results were obtained.     

三维重磁交互解释及区域与局部异常的分离

探讨了三维重磁交互解释的方法技术，提出构制任意形状三维物体模型的用于计算机的橡皮膜技术．针对通常分离区域与局部异常所存在的问题，提出在异常交互解释过程中同时进行异常分离的方法，并提出相应的算法．最后介绍一个应用实例．     

A FAST FOURIER TRANSFORM METHOD FOR RAPID COMPUTATION OF GRAVITY AND MAGNETIC ANOMALIES DUE TO ARBITRARY BODIES*

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.     

SIMULTANEOUS INTERACTIVE MAGNETIC AND GRAVITY INVERSION*

Gravity and magnetic anomalies may have the same source but it is always difficult to achieve correlated solutions if interpretations are carried out separately. Therefore it is useful to invert both anomalous gravity and magnetic profiles at the same time, so that the solution mav be more constrained. Existing inversion techniques do not lend themselves equally to this kind of extension, since the parameters—such as density and susceptibility contrasts—should not be related. The algorithms more easily adaptable to simultaneous inversion are those which look for the shape and the position of anomalous bodies with constant unknown density or susceptibility contrasts. In this study, we use the generalized inverse method in the 2 1/2 D case. Parameters are the coordinates of the vertices of the polygonal cross-sections of the anomalous bodies and both density and susceptibility contrasts of each body. The two types of profile to be processed must be superposable but may have different lengths, spacings, and coordinate origins. Both synthetic examples and field data from geologically known areas have been processed, and it appears that the simultaneous inversion technique may provide an important additional degree of control in the interactive interpretation process.     

APPLICATION OF WIENER FILTERS TO TRANSFORMATIONS OF GRAVITY AND MAGNETIC FIELDS*

Consideration of the spectral representation of gravity and magnetic fields shows the field to be the result of the convolution of factors depending on the parameters of the field. Wiener filters, calculated using model transformations, provide an optimum method for altering these factors and hence effecting field transformations.     

二度体重、磁异常频谱与三度体重、磁异常频谱关系的理论问题

设F（x-ξ,y,z,η,ζ,α,…）为一在x方向无限薄的层的异常,若物体的异常可表示为: T（x,y）=integral from n=a to （a+2c）（F（x-ξ,y,z,η,ζ,α,…））dξ T（y）=integral from n=-∞ to ∞（F（x-ξ,y,z,η,ζ,α,…））dξ 则有 式中T（v）和T（u,v）分别为T（y）和T（x,y）的频谱,2c是物体在x方向的宽度,（x,y,z）和（ξ,η,ζ）分别为观测点和物体上任一点的坐标。 如果2c=dξ,则物体即为薄层本身;如果2c=dξ,薄层的面积无限缩小,则物体即为一质点;如果2c=dξ,薄层只有长度而无宽度,则物体即为一曲线。 对于形状复杂的物体,可以将物体分成若干简单而又符合所述条件的物体元,而有式中2cm为第m个物体元在x方向的宽度,Tm（v）为第m个物体元当其沿x方向的长度为无限长时的异常的频谱。     

二度体重、磁异常频谱与三度体重、磁异常频谱关系的理论问题

设F（x-ξ,y,z,η,ζ,α,…）为一在x方向无限薄的层的异常,若物体的异常可表示为: T（x,y）=integral from n=a to （a+2c）（F（x-ξ,y,z,η,ζ,α,…））dξ T（y）=integral from n=-∞ to ∞（F（x-ξ,y,z,η,ζ,α,…））dξ 则有 式中T（v）和T（u,v）分别为T（y）和T（x,y）的频谱,2c是物体在x方向的宽度,（x,y,z）和（ξ,η,ζ）分别为观测点和物体上任一点的坐标。 如果2c=dξ,则物体即为薄层本身;如果2c=dξ,薄层的面积无限缩小,则物体即为一质点;如果2c=dξ,薄层只有长度而无宽度,则物体即为一曲线。 对于形状复杂的物体,可以将物体分成若干简单而又符合所述条件的物体元,而有式中2c_m为第m个物体元在x方向的宽度,T_m（v）为第m个物体元当其沿x方向的长度为无限长时的异常的频谱。     

JOINT INTERPRETATION OF GRAVITY AND MAGNETIC DATA OVER AXIAL SYMMETRIC BODIES WITH APPLICATION TO THE DARNLEY BAY ANOMALY,NWT CANADA*

A method is presented for determining bounds of the properties of axial symmetric bodies from a finite number of gravity and magnetic observations based on Parker's theory of ideal bodies. Bounds on the density contrast and the intensity of magnetization are calculated as a function of depth to the top of the anomalous source, restricting the range of smallest possible solutions to fit the data. The model studied is approximated by an array of vertical annuli cylinders, each of uniform density and magnetization. Linear programming algorithms based on the ideal body theory were used to calculate the distribution of these parameters within the body. Simultaneous inversion of gravity and magnetic data is performed assuming a constant ratio between the density contrast and the intensity of magnetization and that a common body is responsible for both observed fields. The parameter k(|J|/δp) provides information about the rock type of the structure. Interpretation of gravity and aeromagnetic data from Darnley Bay, NWT, Canada, indicated the presence of a shallow ultrabasic intrusion.     

SOME COMMENTS ON THE CALCULATION OF THE GRAVITATIONAL AND MAGNETIC ATTRACTION OF A HOMOGENEOUS RECTANGULAR PRISM *

The use of arctangents rather than arcsines in the expression for the gravitational attraction of a homogeneous rectangular prism reduces computational difficulties. Once a subroutine is available to compute one component of attraction in a Cartesian coordinate system, the other components may be obtained by cyclic permutation of the field point and body coordinate parameters. This technique also readily provides derivatives of the gravitational attraction and hence forms a compact method for the calculation of a magnetic anomaly due to a homogeneous rectangular magnetic prism.     

RAPID NUMERICAL EVALUATION OF TWO-DIMENSIONAL GRAVITY PROFILES*

A precise method of gravity interpretation is described which uses tables for the attraction of rectangular prisms in all positions relative to a given point. Hence the attraction of any body or combination of bodies may be assessed without complicated mathematical procedures.     

A TWO-AND THREE-DIMENSIONAL GRAVITY DOT CHART*

This paper presents a dot chart for calculating gravity effects from two-dimensional bodies with the addition of a supplementary set of curves which can be used to determine the “end corrections” to modify the calculated effects when the length of the body is not relatively long with respect to the horizontal dimensions of the cross-section. The use of the chart is demonstrated by different examples. The results of theoretical cases amenable to mathematical treatment and computer results both compare favorably to those computed using the present chart. A fundamental incentive of constructing this chart, in a final form ready to use, is to make it available to gravity interpreters for utilization. Future effort in chart construction is thereby avoided.     

CROSS-CORRELATION AS AN AID IN SIMULTANEOUS GRAVITY AND MAGNETIC ANALYSIS *

The properties of cross-correlation functions for the case of gravity and magnetic total field anomalies produced by three geometric models of geological bodies (point-mass, horizontal line-mass, and vertical semi-infinite rectangular prism mass distributions) are studied. This study is carried out on four cases of cross-correlation: the whole curve of anomalies, a single branch of anomalies, the branch between the two inflexion points and the branch between the apex point and the inflexion point. In all cases, the crosscorrelation function can serve as a good indicator to discriminate the so-called genetically related anomalies from those produced by different geological bodies situated on the same vertical line; but the best results are obtained using the cross-correlation of the branches between the apex point and the inflexion point of the two geophysical anomalies. A practical procedure is developed in order to undertake such analysis. The tests in some cases of real gravity and magnetic anomalies mapped in Romania show the validity of this practical procedure.     

A CONTROL OF TWO-DIMENSIONAL MAGNETIC INTERPRETATION BY THREE-DIMENSIONAL MODEL BODY ANOMALIES*

In magnetic routine interpretation the comparison of two-dimensional model curves with measured magnetic anomalies is widely used for an approximate evaluation of the position and depth of magnetic models. Before starting an interpretation of a survey by means of two-dimensional models, it is very useful to have an idea of the shape of anomalies caused by extended but finite bodies, taking into account various strike directions: Three sets of anomalies of thin plates (horizontal length 19, downward length 9, width 1) dipping 30°, 60°, and 90° resp. for various strike directions and an inclination of 20° were computed. Some of these anomalies, e.g. those with nearly N-S strike direction look rather complicated, and at the first glance one would not expect that they are caused by such simple bodies. Several profiles crossing the computed anomalies perpendicularly were interpreted two-dimensionally. For less extended anomalies the depths determined for the top of the plates are 10-20% too small, the magnetization amounts to 50–75 % of the value of the finite bodies. The interpretation of the profiles covering more extended anomalies gave very accurately the same values for the position, depth and magnetization for the two-dimensional body as for the original three-dimensional model. Anomalies of vertical prisms with varying extensions in the y-direction were computed. Their differences in amplitude and in the distance maximum-minimum show that interpretation of short anomalies by two-dimensional methods yields depth errors of up to 20 percent. To see the possibilities of the separation of superimposed anomalies dike anomalies were added to the anomaly of a broad body in great depth and several attempts were made to interpret parts of the composite anomalies. The interpreted bodies lie too deep. In complicated cases the depth values can have large errors, but experienced interpreters should be able to keep the errors in the range of one third of the depth values.     

THREE-DIMENSIONAL INTERPRETATION OF GRAVITY DATA FROM SEDIMENTARY BASINS USING AN EXPONENTIAL DENSITY-DEPTH FUNCTION*

In mapping the topography of the basement of deep sedimentary basins by gravity modelling, the accuracy can be improved by incorporating an exponential increase in density with depth. For calculating the gravity effect of a three-dimensional (3D) structure with such an exponential density-depth relation a frequency-domain forward algorithm based on series expansion is presented, the numerical evaluation of which can be performed efficiently by fast Fourier transform. The algorithm can be applied in a recursive procedure to give the inverse solution in terms of basement relief. The inversion procedure is satisfactorily tested on a 2D synthetic example and a 3D field example of gravity data from the western margin of the Pannonian Basin in eastern Austria, where up to 2.2 km of Tertiary sediments overlie an igneous or metamorphic basement. The results are confirmed by basement intersections in several wells.     

黑水─泉州地学断面的重磁解释

讨论华南黑水─泉州地学断面的重磁解释。在解释中除应用常规的２．５维重磁异常反演外，还采用等密度线算法构制梯度层密度模型，用于分辨地壳密度细结构，以及计算自由地幔而深度用于分析上地幔密度的横向不均匀性。重磁模拟结果揭示了１０ｋｍ深度内的上地壳构造以及地壳与上地幔密度的分布。提出华南造山带以低密度上地幔为特征，它可能与上地幔的改造有关；四川盆地具有较高的上地幔密度，为未经改造的原始地幔。扬子克拉通与华南造山带的分界线与上地幔密度的分界线一致。根据布格异常、地表岩石密度和地形资料的综合分析，圈定出反映内生成矿作用深部标志的密度倒转区，可作为进一步找矿的远景区。     

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.