On the computation of the electrical potential inside a horizontally‐layered half‐space

A new formulation is proposed for the electrical potential developed inside a horizontally‐layered half‐space for a direct current point‐source at the surface. The recursion formula for the kernel coefficient in the potential integral is simpler than the generally used two‐coefficient recursion. The numerical difficulties that may occur during the computation of the integrals and near the source axis are examined and solutions are proposed. The set of equations permits a stable and accurate computation of the tabular potential everywhere in the medium.     

TEM Response of a Large Loop Source over a Homogeneous Earth Model: A Generalized Expression for Arbitrary Source-Receiver Offsets

In the present research, an attempt is made to derive the generalized expressions for the transient electromagnetic response of a large loop source over the surface of a homogeneous earth model for arbitrary receiver positions inside or outside the source loop. Expressions are derived for the impulse as well as step excitations of the source loop. As a cross check for validity of expressions, the step response expression is obtained from the impulse response expression and vice versa. Computations are performed for the TEM response over a homogeneous earth model for source-receiver offset (r = 0) pertinent to the central loop configuration and the results are compared with the published results for the central loop TEM responses. The results are in well coincidence with each other and thus provide the check for the authenticity of the expressions. To exemplify the nature of TEM response at various source receiver offsets, results are presented for the TEM response at source receiver offsets r = 0, r = a/2, r = a and r = 2a, 6a, 12a pertaining to the central loop, in-loop, on-loop and offset loop configurations, respectively. The results depict their characteristic variations. At receiver positions inside the loop source, both the impulse as well as step responses are of same sign, whereas at the receiver positions outside the loop source, both the curves exhibit a change of sign that shifts towards the later times with increase in the offset distances. The change of sign in impulse response occurs at a relatively later time than that in the step response. This is the initial presentation of TEM response expressions for the large loop source over a homogeneous earth model for arbitrary receiver position inside and/ or outside the loop source except for the case of receiver at the center of the loop and at the coincident loop point. This research would be of immense use in the development and use of the large loop TEM method in its various configurations and thus would enhance the applicability and cost effectiveness of the large loop source TEM method.     

位于导电介质中三维导体的电磁响应

本文给出一种用积分方程法配合有限差分法计算位于导电介质中三维导体的电磁响应的新途径,在二层大地条件下给出了具体计算方法和结果。详细讨论了围岩介质的导电性对导体异常的影响。这对于开展低阻覆盖层地区的电磁法工作具有一定的指导意义。     

On calculating transient electromagnetic fields of a small current-carrying loop over a homogeneous earth

Summary The basic theory of airborne EM surveying, in the time domain, is considered. Rather than resorting to tediousdouble numerical integration, a more direct approach is adopted. This method, valid in the quasi-static regime, is illustrated for a homogeneous flat earth. The results exhibit a number of clear-cut features that are relevant to remote sensing. For example, a vertical co-axial loop system has a desirable transient response from the standpoint of yielding conductivity data without requiring accurate information on the height of the device above the ground.The research reported here was supported by AFCRL Contract No. PRO-Y-71-872.     

Two field applications of the rotating current EM method1

The rotating current EM method has been applied to the delineation of two conductive orebodies, Elura near Cobar, NSW, and Thalanga near Charter's Towers, Queensland. The field data were collected in the form of observations of the vertical magnetic field strength ratio and phase difference using a Turam-style receiver with twin vertical coils. By reconstituting this data back to the ring source field and phase, i.e. the observed H_z, phasor, it is possible to present contoured maps of the EM field. Anomaly phasors are obtained by subtracting theoretical phasors from the observed phasors in the complex plane of the H_z phasor. The theoretical phasors for the finite source are based on horizontally layered, half-space earth models, computed at each point of the survey grids, then normalized to selected points of the observed fields. Use is made of the intrinsic circular symmetry of the method in X–Y plots of field versus source-receiver distance to ascertain geoelectric parameters for the earth models. A steel picket fence at Thalanga is modelled by a line source grounded at each end and its H_z, phasor is removed by the same process. A considerable improvement in anomaly delineation is gained over previous Turam-style anomalies and the two survey examples illustrate the limitations of the method in the presence of a conductive overburden (Elura) and its abilities in the absence of a conductive overburden (Thalanga).     

Zone of main contribution to the measured signal for a circular current loop source and receiver on the surface of a conductive half‐space

The electric and magnetic fields generated by an individual horizontal current ring induced inside a homogeneous conductive half‐space, originating from an external large circular loop source of current in the presence of a flat half‐space, are deduced. A check of self‐consistency for these expressions led to the known general functions for these fields due to the same external source in the presence of that medium. The current rings’ mutual coupling related to the magnetic field's radial component is thoroughly analysed and its specific members are presented. The existence of a relatively small zone inside the half‐space responsible for the main contribution for the signal measured at the observation point, with the source and receiver on the ground surface, is made evident. For increasing values of frequency, at a given transmitter‐receiver (T–R) configuration, this zone shrinks and its central point moves away from a maximum depth of about 30% and horizontal distance of nearly 85%, of the T–R separation, to a point very close to the receiver position. The coordinates of the central point of this zone of main contribution are provided as approximated functions in terms of the induction number .     

ELASTIC EXTRAPOLATION OF PRIMARY SEISMIC P- AND S-WAVES1

The elastic Kirchhoff-Helmholtz integral expresses the components of the monochromatic displacement vector at any point A in terms of the displacement field and the stress field at any closed surface surrounding A. By introducing Green's functions for P- and S-waves, the elastic Kirchhoff-Helmholtz integral is modified such that it expresses either the P-wave or the S-wave at A in terms of the elastic wavefield at the closed surface. This modified elastic Kirchhoff-Helmholtz integral is transformed into one-way elastic Rayleigh-type integrals for forward extrapolation of downgoing and upgoing P- and S-waves. We also derive one-way elastic Rayleigh-type integrals for inverse extrapolation of downgoing and upgoing P- and S-waves. The one-way elastic extrapolation operators derived in this paper are the basis for a new prestack migration scheme for elastic data.     

A RECTANGULAR LOOP SOURCE OF CURRENT ON A TWO-LAYERED EARTH*

Using the concept of reciprocity and the known solution for the electric field of a vertical oscillating magnetic dipole source placed over a two-layered halfspace, an integral expression for the vertical magnetic field produced by a horizontal rectangular loop, carrying an oscillating current and placed on the surface of the same halfspace, is deduced. This expression is such that it can be evaluated by a combination of straightforward numerical integration and digital linear filter techniques. Displacement currents everywhere in space are neglected. Fields both inside and outside the loop are presented as sounding curves. A little over a decade of frequencies (100–3000 Hz) appears to be the minimum requirement and two decades (100–10000 Hz) are sufficient for most situations. Parametric and geometric sounding curves both show good resolution of subsurface layering. Phase shows better resolution than amplitude. Similarly, resolution is higher when conductivity increases with depth than when it decreases. Besides being useful for the electromagnetic depth sounding for layered earth structures, such computations can be applied to determine normal corrections to Turam observations, whenever a rectangular loop of finite size is used as a transmitter.     

THE USE OF TWO-ELECTRODE AND SCHLUMBERGER FILTERS FOR COMPUTING RESISTIVITY AND EM SOUNDING CURVES*

Different sets of filter coefficients for the linear filter technique for the computations of resistivity and EM sounding curves are evaluated for several electrode and coil configurations. Instead of this procedure, the two-electrode filter can be used for computations of Wenner, Schlumberger, and dipole—dipole apparent resistivity model curves by defining convolutional expressions which contain the new input functions in terms of the resistivity transform function. Similarly, the Schlumberger filter performs the computations of dipole—dipole apparent resistivity model curves. The Wenner, Schlumberger, and dipole—dipole filter functions are defined in terms of the two-electrode filter using the new convolutional expressions. A relationship between the Schlumberger and dipole—dipole filter functions is given. The above arguments are adopted for the computations of EM sounding curves. It is shown that the EM filter for the horizontal coplanar loop system (which is identical to the two-electrode filter) performs the computations of the mutual coupling ratios for perpendicular, vertical coplanar, and vertical coaxial loop systems. In the same way, the Schlumberger filter can be used to compute vertical coaxial sounding curves. The corresponding input functions are defined in terms of the EM kernel for all convolutional expressions presented. After these considerations, integral expressions of the mutual coupling ratios involving zero-order Bessel function are derived. The mutual coupling ratio for the vertical coaxial loop system is given in the same form as the mutual coupling ratio for the vertical coplanar loop system.     

On two-dimensional elastic dislocations in a multilayered half-space

The two-dimensional problem of a long displacement dislocation in a multilayered half-space is studied. Both plane strain and antiplane strain cases are considered. The Thomson-Haskell matrix method is used to obtain the static field caused by the dislocation. The source is represented in terms of the jumps at the source level in the depth-dependent factors in the Fourier integral representation of the displacements and stresses due to the source in an infinite medium. Explicit expressions for the surface displacements due to dip-slip and strike-slip faults of arbitrary dip are obtained. The formulation developed is quite convenient for numerical computation.     

声电效应测井的有限差分模拟

本文研究声电效应测井波场的有限差分模拟算法.忽略井外地层中诱导电磁场对孔隙弹性波的影响,将求解动电耦合波方程组的问题解耦,先计算孔隙弹性波,再计算其诱导电磁场.基于轴对称柱坐标系下的速度-应力交错网格,采用时域有限差分计算井孔流体声波和井外地层孔隙弹性波.将电磁场近似看作似稳场,基于轴对称柱坐标系下的5点式有限差分网格,求解不同时刻的电位Poisson方程,计算诱导电场.结果表明:本文算法可准确模拟频率6.0 kHz的声电效应测井全波;在声波测井频率范围内,电导率、动电耦合系数和动态渗透率的低频近似对伴随电磁场的计算影响不大;地层水平界面导致伴随反射斯通利波的电场和显著的界面电磁波,后者对于探测地层界面具有潜在的应用价值.     

ELECTROMAGNETIC SOUNDING NEAR A LARGE SQUARE LOOP SOURCE OF CURRENT*

An electromagnetic sounding experiment with a large square loop as source was carried out on the dried-up bed of a water reservoir near the town of Dharmavaram in Andhra Pradesh. The sounding was performed in both geometric and parametric modes, and involved measurements of phase as well as amplitude of the vertical magnetic field inside and outside the loop. The six-frequency EM system used for the experiment was found to be a workable system for electromagnetic sounding. The results of the experiment more or less confirm the conclusions from theoretical modeling. They also show that even though the earth is not always electrically horizontally layered over a sufficiently large lateral extent for the one-dimensional model to be strictly valid, it is still possible to apply such modeling to sounding curves taken one part at a time and obtain layer parameters which check qualitatively with the layer parameters obtained from direct current resistivity sounding.     

The disturbance due to a point source in a homogeneous liquid layer over a heterogeneous liquid half-space

Summary The problem of a periodic point source in a homogeneous liquid layer overlying a heterogeneous liquid half-space is discussed. After obtaining the formal solution, the path of integration for the displacement potential of the layer is transformed from the positive real axis to the positive imaginary axis and the Sommerfeld contour and the latter is further distorted to the modified Sommerfeld contour. The residues of the integrand at the poles contained within the Sommerfeld loop constitute the normal mode solution to the problem. The integrands in the expressions for the integrals along the imaginary axis are expanded in a series of negative powers of exponentials and then some of the terms in these expansions are evaluated approximately. This gives various waves reflected from the interface and the integral along the Sommerfeld loop vanishes. The frequency equation is obtained, also by the principle of constructive interference. An expression for the reflection coefficient at an interface of two liquid media, the upper medium being homogeneous and the lower one inhomogeneous, is obtained.     

伴随前兆蠕动和震后滑动的准静态形变——模型与观测实例

为了建立断层蠕动与其周围地壳形变的关系, 本文给出了线性粘弹性半空间中任意倾角的断层滑动引起的准静态位移场的完整的解析表达式.其中, 滑动时间函数的形式是根据经验拟定的, 而介质被视为广义开尔文体或麦克思韦体.通过数值计算讨论了介质的流变性对位移场的影响.根据1976年唐山地震前后在地震区取得的短期观测资料的分析, 论证了唐山主震前夕在极震区地壳上部曾发生过前兆蠕动, 而且主震后断层仍继续滑动, 但其速率呈指数型衰减.估算了符合观测数据的断层滑动参数, 并对这些结果的意义作了简要的讨论.      

FREQUENCY ELECTROMAGNETIC SOUNDING USING A VERTICAL MAGNETIC DIPOLE*

A horizontal transmitter loop (vertical magnetic dipole) is used for frequency electromagnetic (FEM) soundings. The frequency ranges from approximately 6 Hz to about 4000 Hz. The vertical and radial magnetic field components are measured for 20 frequencies per decade several hundred meters from the transmitter loop. A very small bandwidth is selected for amplification using a reference signal. An Apple computer is used for data acquisition. A computer program for Marquardt inversion optimizes the parameters for the n-layer case: the resistivities and thicknesses of individual beds and a correction factor for the primary magnetic field. Interpretation of each component individually yields practically the same parameters. Examples from the field are given with interpretation; comparison with dc resistivity measurements is provided. The ratio of vertical/radial magnetic field components vs. frequency can be transformed simply into apparent resistivity vs. apparent depth. This can be done in the field to obtain an estimation of the depth of the layer boundaries. FEM results are compared with Schlumberger d.c. sounding obtained at the same site.     

THE VERTICAL MAGNETIC FIELD OF AN ALTERNATING CURRENT DIPOLE FOR HORIZONTALLY STRATIFIED MEDIA *

For the computation of the vertical component H_z of the magnetic field of a horizontal A.C. dipole lying on the earth's surface, a recurrence formula is presented for a horizontally stratified half space, to obtain the (n+ 1)-layer case from the w-layer case. By means of several computed diagrams for the two-layer case, H_z can be determined for different ratios of conductivity of the subsoil and that of the overburden. Thereby the distance from the dipole as well as the layer thickness h are expressed in terms of the wave length A of a plain wave in the overburden. Assuming a sufficiently large conductivity difference, the results show that evidence about the subsurface conditions can be obtained if the distance between the measuring coil and the dipole is of the order of A/3, and if the thickness h of the layer varies within the range A/100 < h < A/6. As an example for the 3-layer case, a nonconducting intermediate layer is assumed.     

TRANSIENT FINITE INTEGRAL POINT SOURCE RESPONSE FROM A PLANAR ACOUSTIC INTERFACE: THE CAUSALITY APPROACH*

Point source reflection/transmission responses from planar interfaces based on the Sommerfeld-Weyl integral are causal and are given in the form of a solution integral with an infinite integration limit. They can be simply transformed to integrals with a finite limit by an approach that takes into account causality as well as time-symmetry aspects. To introduce the fundamental principles involved, the transformation is applied to the simple case of a point-source response from a planar acoustic interface. For a point source above a planar interface separating an upper and lower acoustic half-space of velocity c₁and c₂ the inhomogeneous plane waves in the Sommerfeld-Weyl source representation do not contribute to the final result if c₁≤c₂ and only a limited spectrum of inhomogeneous plane waves comes into play if c₁c₂.     

时间域航空电磁系统瞬变全时响应正演模拟

近年来很多专家学者致力于时间域航空系统正反演的研究.本文针对一维均匀层状介质和三维模型进行正演.不仅计算垂直方向上的电磁响应,还计算了同线方向上的电磁响应,为航空电磁多分量观测提供理论依据.通过比较航空电磁系统的脉冲响应和阶跃响应特征,发现脉冲响应在早期时间存在奇异性,造成计算不稳定.然而,阶跃响应在早期时间没有奇异性,因而利用系统的阶跃响应可得到一种计算时间域航空电磁系统全时响应的稳定算法.该算法具有较高的精度,并很好地保持了磁场强度B和磁感应dB/dt关系的一致性.该算法推广到三维地质体的时间域正演模拟亦取得很好的效果.     

长偶极大功率可控源电磁波响应特征研究

地球物理学中关于电磁波勘探研究通常采用的是地球半空间模型.然而,对于几十公里的有限长电缆源（长偶极源）,远距离电磁波场探测必须要考虑电离层的影响,它是一个全空间问题.关于包含电离层、空气层和地球介质（我们称“地-电离层”模式）的电磁波场特征的研究在国外较少,国内几乎是空白.本文采用全空间积分方程法首先对小尺度的可控源电磁波场特征进行了研究,由于此时电离层的影响可忽略,它应该和半空间成熟的CSAMT模拟结果一致,对比结果表明,二者是一致的,验证了全空间模拟方法的可靠性和有效性.随后进行了50 km长电缆电离层和空气层高度都为100 km的“地-电离层”模式大尺度电磁波场模拟,以探讨大尺度可控源电磁波场的特征.给定频率的“地-电离层”模式电磁场的衰减曲线表明长电缆远距离电磁波场由于受电离层的作用存在衰减逐渐变小的过渡场和衰减变小的波导场.为了探讨复杂介质“地-电离层”模式电磁波特征,对“地-电离层”模式的典型地盾和地台多层介质模型进行了数值模拟,得到了偶极源长度50 km、电流200A、收发距离远达1600 km和2500 km的合理的电磁场结果.最后,对一简单含油储层结构模型进行了长偶极、大功率、远距离电磁波场响应计算.储层横向不均匀复杂结构模拟的结果表明,考虑电离层和大气层的“地-电离层”模式大尺度深层复杂介质模拟时,电磁场对深部目标体仍有很好的异常响应.     

A Review Of Environmental Applications Of Quasi-Stationary Electromagnetic Techniques

Electromagnetic (EM) techniques are the most commonly used geophysical methods in mineral exploration. However, the use of EM measurements for environmental and engineering applications like the detection of contaminant plumes or the exploration of waste sites is relatively new.The reason for the success of the application of EM methods to environmental problems lies in the variation of conductivity caused by different geometry of pore fluids and clay contents in rocks, and by the presence of organic and inorganic contaminants.Many EM methods/instruments used for mapping near surface geology exist and nowadays they play a central role in environmental geophysics. In general, these methods can be classified in two blocks: EM methods using a plane wave source of excitation and EM methods using a controlled source like a magnetic dipole or a loop source. The Very Low Frequency (VLF, VLF-R) and Radiomagnetotelluric (RMT) methods are chosen as representative methods for plane wave techniques, while horizontal loop EM methods operating in low induction numbers (EM31, EM34) and Transient Electromagnetic methods (TEM) are chosen as representatives of magnetic dipole or loop source techniques. Basic principles, advantages and disadvantages of each technique as well as their connection to specific environmental problems will be discussed.Different successful applications of these methods are reported in the literature. However, this review will focus on three major subjects: waste site exploration, detection of contaminated earth layers, and groundwater exploration. Case histories are presented illustrating the suitability of EM methods for solving such problems.