3D ELECTROMAGNETIC INVERSION USING INTEGRAL EQUATIONS1

Utilizing electromagnetic data in geophysical exploration work is difficult when measured responses are complicated by the effects of 3D structures. 1D and 2D models may not be capable of accurately simulating the physical processes that contribute to a measured response. 3D conductive-host modelling is difficult, costly and time-consuming. Using a 3D inverse procedure it is possible to automate the interpretation of controlled-source electromagnetic data. This procedure uses an inverse formulation based on frequency-domain, volume integral equations and a pulse-basis representation for the internal electrical field and anomalous conductivity. Beginning with an initial model composed of a 3D inhomogeneous region residing in a laterally homogeneous (layered-earth) geoelectrical section, iterative least-squares algorithms are used to refine the geometry and the conductivity of the inhomogeneity. This novel approach for 3D electromagnetic interpretation yields a reliable and stable inverse solution provided constraints on how much the variable can change at each iteration are incorporated. Integral-equation-based inverse formulations that do not correctly address the non-linearity of this inverse problem may have poor convergence properties, particularly when dealing with the high conductivity contrasts that are typical of many exploration problems. While problems associated with contamination of the data by random noise and non-uniqueness of solutions do not usually influence the inverse solution in an adverse manner, problems associated with model inadequacy and errors in an assumed background conductivity structure can produce undesirable effects.     

FIXED LOOP SOURCE EM MODELLING RESULTS USING 2D FINITE ELEMENTS1

Among electromagnetic sounding techniques, the Mélos method possesses the specific feature of including an apparent resistivity computation. This acts as a normalizing scheme so that 2D modelling results can be obtained without accounting for a true 3D source. However, in order to get reliable numerical modelling results for a 2D magnetic dipole source, improved algorithms are required in order to apply the standard finite-element technique: quadratic basis functions must be used in place of linear basis functions, and a more sophisticated method than conventional ones is necessary for properly solving the resulting system of linear equations. Such modelling results have been used to study theoretical responses for the Mélos method in the search for conductive bodies in mineral exploration. Two sets of models are presented and discussed. They show that the typical Mélos response to a conductive target is a bipolar anomaly on the apparent resistivity pseudo-section, with a conductive pole at low frequency which is centred above the target.     

APPROXIMATING RAINFALL-RUNOFF MODELLING RESPONSE USING A STOCHASTIC INTEGRAL EQUATION

Rainfall-runoff modelling uncertainty can be analysed by the use of a stochastic integral formulation. The stochastic integral equation can be based on the rainfall–runoff model input of model rainfall or model rainfall excess. Similarly, the stochastic integral equation can be based on the rainfall–runoff model output of the modelled runoff hydrograph. The residual between actual measured runoff data and modelled runoff (from the rainfall–runoff model) is analysed here by the use of a stochastic integral equation. This approach is used to develop a set of convolution integral transfer function realizations that represent the chosen rainfall–runoff modelling error. The resulting stochastic integral component is a distribution of possible residual outcomes that may be directly added to the rainfall–runoff model's deterministic outcome, to develop a distribution of probable runoff hydrograph realizations from the chosen rainfall–runoff model.     

POTENTIAL FIELD OF A STATIONARY ELECTRIC CURRENT USING FREDHOLM'S INTEGRAL EQUATIONS OF THE SECOND KIND*

An integral equation method is described for solving the potential problem of a stationary electric current in a medium that is linear, isotropic and piecewise homogeneous in terms of electrical conductivity. The integral equations are Fredholm's equations of the ‘second kind’ developed for the potential of the electric field. In this method the discontinuity-surfaces of electrical conductivity are divided into ‘sub-areas’ that are so small that the value of their potential can be regarded as constant. The equations are applied to 3-D galvanic modeling. In the numerical examples the convergence is examined. The results are also compared with solutions derived with other integral equations. Examples are given of anomalies of apparent resistivity and mise-a-la-masse methods, assuming finite conductivity contrast. We show that the numerical solutions converge more rapidly than compared to solutions published earlier for the electric field. This results from the fact that the potential (as a function of the location coordinate) behaves more regularly than the electric field. The equations are applicable to all cases where conductivity contrast is finite.     

TRANSIENT EM ANALOGUE MODELLING FOR KOREAN TREASURE HUNTING WITH THE SIROTEM SYSTEM1

The inductive transient electromagnetic method (TEM) shows good potential for the detection of metallic relics of historical interest, such as buried antique bells, Buddhist idols, or precious metal nuggets. The effectiveness of the method was investigated with analogue models, using transmitter-receiver loop configurations with sizes or receiver spacings slightly different from those usually applied in earth resources exploration with the Sirotem system. The analogue modelling results show that the location and depth of the buried treasure may be obtained from the Sirotem data. A solid metallic object such as an antique bell could be detected to a depth about 12 times greater than its size, and treasure consisting of separate metallic objects, such as gold nuggets, could be detected to a depth 5 times its linear dimensions with a transmitter current of ? 20 A. In scaling down the dimensions of a target, its conductivity should be increased in order to preserve the same TEM conditions found in the field. However, since the buried treasure consists of gold or copper objects, it is not possible to properly scale the conductivity. Hence, in the field, the depth detection limit is expected to be greater than that derived from analogue modelling.     

双频电磁波电导率层析成像反演

介绍一个新的跨井电磁波层析成像方法,该方法利用两个频率的透射电磁波的电场强度之比值重建研究区域的电导率分布图像,无需计算发射天线的初始辐射场强与辐射方向因子.详细讨论了双频电磁波电导率层析成像的方法与其适用条件,并给出了应用实例.双频电磁波电导率层析成像适用于良导介质(＞0.01 S/m)和1-100MHz工作频率情况.该方法应用于实际数据,获得了较传统吸收成像更精细的层析图.     

电阻率测井响应的积分方程解法

从电流守恒条件出发,得到了两个积分方程.其中第一个和在1964年提出的相同,它的未知函数是边界面上的隐电流源分布密度;第二个积分方程的未知函数则是电位函数本身. 在的文章以及本文作者的另一篇文章中,供电电极和测量电极都被当成点电极处理.用这种方法来处理电位电极系和梯度电极系已不够准确,用来处理三侧向或双侧向等具有大电极的电极系则完全无能为力. 测井中所用的电极系是包着金属外皮的绝缘心棒,金属外皮就是电极.本文提出了处理这种电极系的方法.结果表明,第二个积分方程在处理这种电极系时有明显优点.     

电阻率测井响应的积分方程解法

从电流守恒条件出发,得到了两个积分方程.其中第一个和在1964年提出的相同,它的未知函数是边界面上的隐电流源分布密度;第二个积分方程的未知函数则是电位函数本身. 在的文章以及本文作者的另一篇文章中,供电电极和测量电极都被当成点电极处理.用这种方法来处理电位电极系和梯度电极系已不够准确,用来处理三侧向或双侧向等具有大电极的电极系则完全无能为力. 测井中所用的电极系是包着金属外皮的绝缘心棒,金属外皮就是电极.本文提出了处理这种电极系的方法.结果表明,第二个积分方程在处理这种电极系时有明显优点.     

基于边界积分方程的"准三极"法电阻率层析成像

基于边界积分方程,提出了依据二极采样数据、但利用电位差的某种线性组合的"准三极"电阻率层析成像方法.研究表明,该方法不仅较好地解决了对高阻异常体的成像问题,而且使低阻异常体的成像质量获得了明显改善和提高,因此增强了电阻率层析成像技术的实用性.     

SEISMIC VIBRATOR MODELLING1

The wavefield in, and at the surface of, a homogeneous, isotropic, perfectly elastic half-space, excited by a traction distribution at the surface of the medium is investigated. The emitted wavefield is a spatial convolution of the surface tractions and the spatial impulse response. The properties of the wavefield in the far-field of the medium are derived and it is shown that the far-field particle velocity is essentially equal to a weighted sum of the time derivative of the integrated surface tractions, that is, of the components of the ‘ground force’. The theory is valid for an arbitrary geometry and orientation of the surface tractions, and is independent of the boundary conditions at the surface of the medium. The surface tractions are related to a source that consists of a mass distribution with an arbitrary force distribution imposed upon it. A boundary condition is introduced that accounts for the mass load and the forces applied to it but neglects vibrations within the mass. The boundary condition follows from the equation of motion of the surface mass load. The theory is applied to the Vibroseis configuration, using a P-wave vibrator model with a uniformly distributed force imposed on top of the baseplate, and assuming that horizontal surface traction components are absent. The distribution of displacement and stress directly underneath the baseplate of a single vibrator and an array of vibrators is investigated. Three different boundary conditions are used: (1) assuming uniform pressure, (2) assuming uniform displacement, (3) using the equation of motion of the baseplate as a boundary condition. The calculations of the distribution of stress and displacement over the plate for different elastic media and several frequencies of operation show that only the results obtained with the mixed boundary condition agree with measurements made in the field. The accuracy of three different phase-feedback signals is compared using synthetic data. Baseplate velocity phase-feedback leads to huge deviations in the determination of the far-field wavelet; reaction mass acceleration phase-feedback looks stable but neglects the differentiating earth filter; and phase-feedback to a weighted sum of baseplate and reaction mass accelerations becomes unstable with increasing frequency. The instability can be overcome using measurements over the whole baseplate. The model can be extended to a lossy layered earth.     

AREAL RAINFALL EVALUATION USING TWO SURFACE FITTING TECHNIQUES

ABSTRACT

Two trend surface fitting techniques, the use of bi-cubic splines and multiquadric analysis, have been applied to the determination of areal rainfall. On two mathematically defined test surfaces, the methods were applied using gridded data; the bi-cubic splines gave the better results. For non-gridded data, selected and random points, the multiquadric analysis gives satisfactory values. With real catchment and storm data, multiquadric analysis is shown to be a potentially very useful and flexible method for determining areal rainfall.     

地幔对流的数值模拟及其与表面观测的关系

本文从基本的热对流方程出发,并结合地幔对流特点,特别考虑到自重及非线性影响,探讨地幔对流及其与表面观测的关系,发展了相应的数值方法.结果表明,计算得到的长波大地水准面、地表地形、板块速度场水平散度与观测值符合程度较好.上、下地幔的非绝热温度异常与由地震层析得到的地震波速异常显示一定的相关性.地幔内部的流动呈现复杂形态,反映了高瑞利数对流的特征.     

地幔对流的数值模拟及其与表面观测的关系

本文从基本的热对流方程出发,并结合地幔对流特点,特别考虑到自重及非线性影响,探讨地幔对流及其与表面观测的关系,发展了相应的数值方法.结果表明,计算得到的长波大地水准面、地表地形、板块速度场水平散度与观测值符合程度较好.上、下地幔的非绝热温度异常与由地震层析得到的地震波速异常显示一定的相关性.地幔内部的流动呈现复杂形态,反映了高瑞利数对流的特征.     

边界积分方程用于电阻率Zohdy反演的初步研究

研究利用边界积分方程进行电阻率Ｚｏｈｄｙ反演的有关技术,结果表明在探测区域划分为数千个单元的情况下,为节省正、反演过程中所需要的计算机内存和 ＣＰＵ时间,可采取以下措施：（１）假定电位分块线性变化,可以使节点数目显著减少,这里每一块都包含若干个单元;（２）利用边界积分方程的特点,可事先计算并存储所有必要的边界积分值,供每一步迭代中调用;（３）利用边界积分方程所特有的数值延拓功能,可以进一步减少节点数目．为了说明这些手段的有效性,先给出了一个数值模拟成像结果,它清楚地表明基于边界积分方程的电阻率 Ｚｏｈｄｙ反演是快速而有效的,然后给出一个实测剖面的成像实例,成像结果经钻孔检验正确．     

边界积分方程用于电阻率Zohdy反演的初步研究

研究利用边界积分方程进行电阻率Ｚｏｈｄｙ反演的有关技术,结果表明在探测区域划分为数千个单元的情况下,为节省正、反演过程中所需要的计算机内存和 ＣＰＵ时间,可采取以下措施：（１）假定电位分块线性变化,可以使节点数目显著减少,这里每一块都包含若干个单元;（２）利用边界积分方程的特点,可事先计算并存储所有必要的边界积分值,供每一步迭代中调用;（３）利用边界积分方程所特有的数值延拓功能,可以进一步减少节点数目．为了说明这些手段的有效性,先给出了一个数值模拟成像结果,它清楚地表明基于边界积分方程的电阻率 Ｚｏｈｄｙ反演是快速而有效的,然后给出一个实测剖面的成像实例,成像结果经钻孔检验正确．     

EM target location1

We consider the problem of computing the most probable location of a target based on radar measurements of the subsurface. Our algorithm makes use of the maximum likelihood estimator (MLE), which represents a correlation between the measured data and synthetic data generated for the object of interest at different locations. Previous studies assume a plane-wave acquisition geometry and target object(s) embedded in a uniform background. In this paper, a generalization of the MLE method is presented which is valid for discrete point sources (and receivers) and a 2D model (i.e. a 2.5D acquisition geometry). Within this formulation the treatment of a non-uniform background model is also possible. We concentrate on geotechnical ground investigations and assume that the characteristic dimensions of the target object are in the range 1–2λ, (λ being the wavelength). The potential of the method is demonstrated employing cross-hole radar data acquired in a controlled field experiment. The MLE result is also compared with the image obtained employing a full reconstruction method such as diffraction tomography.     

HYDRODYNAMIC MODELLING OF ESTUARINE CIRCULATION USING A SPLINE METHOD OF FRACTIONAL STEPS

I. INTRODUCTIONNumerical models of estuaries incorporating simulation of the density induced circulation are consideredimPOrtant tools in the assessment of their dispersive characteristics. Most models to date have used finite difference approximations to the governing equations (see for example thericke and Hogan, 1977, Blumberg,1 977, Perrels and Karelse, 1978). A numeriCal model of estuarine circulation using a SADI (Spline Alternating Direction implicit) procedure together with a…     

沿球面大圆的积分逆变换及其应用

本文给出了从沿球面上大圆的积分值直接反推球面上函数分布的关系式,即得到了球面上大圆的积分变换和反变换对。如利用的是整个大圆的积分,得到的是球面上任何点及其对蹠点上函数的平均值,或者说得到的是球面上函数的偶对称部分。如果进行积分的球面域小于半球面,反演将是唯一的。所得结果可应用于从地震表面波资料反演三维地球内部构造的地球物理问题。     

沿球面大圆的积分逆变换及其应用

本文给出了从沿球面上大圆的积分值直接反推球面上函数分布的关系式,即得到了球面上大圆的积分变换和反变换对。如利用的是整个大圆的积分,得到的是球面上任何点及其对蹠点上函数的平均值,或者说得到的是球面上函数的偶对称部分。如果进行积分的球面域小于半球面,反演将是唯一的。所得结果可应用于从地震表面波资料反演三维地球内部构造的地球物理问题。