Palaeomagnetic distortion modelling and possible recovery by inversion

A robust finite-element technique is presented for computation of both the internal demagnetization effects and magnetic terrain effects in bodies with arbitrary shape and arbitrary susceptibility distribution. This method facilitates a flexible analysis of the palaeomagnetic deflection problem. Tests on geologically realistic settings of highly magnetic rocks demonstrate that deflections of several degrees may occur even for relatively simple two-dimensional models. Similarly, the magnetic intensity may well be biased by 5-15% by demagnetization effects. The present paper focuses on deflections and intensity variations inside the magnetized body, where we find a systematic shallowing of inclination for bodies with a horizontal elongation. Because the bodies sampled at a typical palaeomagnetic site will have a dominant direction of elongation, the magnetic deflection effect will tend to impose a systematic bias which doesn’t average out. An inversion-based procedure for elimination of the deflection effect is presented. It requires that the magnetic body is quite homogeneous and that its surface geometry is known, as may be the case for historical lava flows. Tests demonstrate that in order to recover both ambient palaeofield direction and the effective susceptibility at blocking temperature it is necessary to sample near strong topographic elements in the magnetic body. Since the surface geometry rarely is known it is proposed as an alternative to inversion that an effective susceptibility is assessed and a horizontal slab correction is applied for samples taken far from topographical features. When shape geometry is unknown and no correction applied, palaeomagnetic conclusions must take into account the possible bias from internal demagnetization and magnetic terrain effects.     

空间波数混合域磁异常场积分解三维数值模拟

本文提出一种空间波数混合域磁异常场三维数值模拟方法.该方法利用磁位三维空间域积分为卷积的特点,沿水平方向进行二维傅里叶变换,把空间域磁位满足的三维积分问题转化为不同波数之间相互独立的垂向一维积分问题.保留垂向为空间域,优势之一在于便于浅层单元剖分可适当加密,随着深度增加,单元剖分适当稀疏,可以准确模拟任意复杂地形和磁性体的磁异常,兼顾了计算精度与计算效率;优势之二在于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磁化强度,可得出单元积分的解析表达式,计算精度高、效率高.该方法充分利用一维形函数积分的高效和高精度、快速傅里叶变换的高效性及算法高度并行性,实现了磁异常场高效、高精度的数值模拟.设计棱柱体模型,将模型解析解与空间波数混合域法的数值解对比,结果表明该方法计算精度高、效率高.设计了组合棱柱体复杂模型,对比分析了标准FFT扩边法与Gauss-FFT法的计算精度与计算效率,总结了标准FFT的扩边系数选取策略.针对任意复杂地形条件下的磁异常模拟问题,本文提出一种适用于起伏地形条件下的磁异常场快速计算方法,并对其有效性进行了验证.     

强磁性三度体内部磁化强度的分布

本文讨论稳定磁化场中强磁性三度体内部磁化强度的数值解法。展示了由于退磁作用而造成的不均匀性,并以长方体内一些典型截面为例,较详细地讨论了磁化强度的特征,作了分布图;推导了较均匀磁化体磁化强度分布的近似公式,并计算了一组长方体的视退磁系数。     

Modelling magnetic fields due to steel drum accumulations

Modelling the magnetic fields produced by accumulations of steel drums is a problem that is relevant to the detection and evaluation of disposal sites containing materials that are potentially hazardous to the environment. Accurate modelling is possible with existing integral equation techniques but these are numerically intensive due to the need to solve very large systems of linear equations. Use of an approximate iterative technique for the solution of the equations (system iteration) allows the integral equation technique to be extended to modelling the magnetic effect of substantially large accumulations, comprising up to several hundred drums, on very moderate computing facilities. However, even this process remains time‐consuming and suggests the use of more rapid, if less accurate, modifications. Several are available. Surprisingly, quite reasonable results can also be achieved with a very basic approximation that represents each drum by a discrete dipole located at its centroid. The dipole moments are found from the magnetic behaviour of single drums exposed to a uniform inducing field, which can be conveniently defined by a dyadic drum apparent susceptibility. The basic discrete dipole model for drum accumulations can be substantially improved by using a first‐order accommodation of the depolarizing effect produced by the shape of the accumulation. All of the above modelling techniques require details of individual drum locations and orientation. This information is generally unavailable to geophysical practitioners involved in environmental surveys and so prompts the idea of models that represent drum accumulations as a continuous distribution of magnetization. The convenience of neglecting details of drum location and orientation comes at the cost of some loss in accuracy of the modelled responses. However, for accumulations buried sufficiently deep and in which the drums are uniformly distributed, the total field magnetic anomaly is found to be reasonably approximated by the effect of a continuous magnetization, expressible in terms of an effective isotropic susceptibility. Again, the basic model can be improved by the accommodation of demagnetization effects due to the shape of the accumulation.     

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.     

Computing three-dimensional gravitational fields with equivalent sources

Existing techniques for computing the gravitational field due to a homogeneous polyhedron all transform the required volume integral, expressing the field due to a volume distribution of mass, into a surface integral, expressing the potential due to a surface mass distribution over the boundary of the source body. An alternative representation is also possible and results in a surface integral expressing the potential due to a variable-strength double layer located on the polyhedral source boundary. Manipulation of this integral ultimately allows the gravitational field component in an arbitrary direction to be expressed as a weighted sum of the potentials due to two basic source distributions. These are a uniform-strength double layer located on all faces and a uniform-strength line source located along all edges. The derivatives of the gravitational field components can also be expressed in a similar form as can the magnetic field components due to a homogeneous magnetic polyhedron. It follows that the present approach can be used to generate a universal program capable of modelling all the commonly used potential field responses due to 3D bodies of arbitrary shape.     

Estimation of depth and magnetization of magnetic sources from magnetic data: The linearized continuous inverse problem for 21/2D structures

The estimation of the depth to the top and bottom of a magnetic source from magnetic data defines a nonlinear inverse problem, while the evaluation of the distribution of magnetization determines a linear inverse problem. In this paper, these interpretation problems are resolved in the continuous case of 2_1/2D magnetized bodies with lateral magnetization variations. A formulation of the magnetic problem accounting for different directions of remanent and total magnetization vectors and including a more general definition of apparent susceptibility is presented. Differences between 2D and 2_1/2D formulations are stressed, as regards the anomaly amplitude, shape and zero-level.In order to utilize well-known continuous linear inverse methods, Fréchet derivatives of the data functionals with respect to the depth of the source top and bottom, are analytically described. Thus, using the spectral expansion inverse method (Parker, 1977) and linearizing the problem at several steps of an iterative process, the source depth is obtained within a few iterations, although the starting model is distant from the final solution. The interpretation of an anomaly in the Italian region shows the usefulness of the method.     

Singularity-free expression of magnetic field of cuboid under undulating terrain

Most of the current computing methods used to determine the magnetic field of a uniformly magnetized cuboid assume that the observation point is located in the upper half space without a source. However, such methods may generate analytical singularities for conditions of undulating terrain. Based on basic geomagnetic field theories, in this study an improved magnetic field expression is derived using an integration method of variable substitution, and all singularity problems for the entire space without a source are discussed and solved. This integration process is simpler than that of previous methods, and final integral results with a more uniform form. ΔT at all points in the source-free space can be calculated without requiring coordinate transformation; thus forward modeling is also simplified. Corresponding model tests indicate that the new magnetic field expression is more correct because there is no analytical singularity and can be used with undulating terrain.     

Origin of a magnetic lineation on Kyushu Island, Japan

Abstract Several linear magnetic anomalies over continental crust have been identified in and around the Japanese Islands. The anomalies are probably related to island arc tectonic structures, but identifying specific sources has been difficult. Several deep holes were drilled in and around Aso caldera, where a linear anomaly occurs along an active fault. One drillhole located on the linear anomaly encountered a zone of highly magnetized and altered basement rocks at least 100 m thick at a depth of ∼1000 m. The other hole was located away from the anomaly and did not encounter any high-magnetic zones. Rocks from the zone have exceptionally strong remanent magnetization (several tens of A/m) sub-parallel to the present field. AF demagnetization experiments indicated that the magnetization is hard and stable. Magnetic modeling indicates that the linear anomaly is caused mainly by this layer. Microscopic examination of core samples shows that the highly magnetized zone includes secondary magnetic minerals and abundant hydrothermal alterations. Temperatures determined by fluid inclusions and down-hole temperatures show that the temperature of the highly magnetized zone was elevated in the past relative to surrounding rocks. The high temperature could destroy primary magnetic minerals and replace them with secondary magnetic minerals. Thus, the past hydrothermal system may have enhanced thermo-chemical remanent magnetization. The results can produce a model indicating that there was a past hydrothermal system related to the tectonic structure.     

Evolution of the Tharsis region of Mars: insights from magnetic field observations

Mars Global Surveyor (MGS) observations of crustal magnetic fields over Tharsis provide new constraints on models for the thermal and magmatic evolution of this region. We analyze the distribution of magnetic field anomalies over Tharsis surface units of Noachian, Hesperian and Amazonian age. These data suggest that early Noachian crust underlies the Tharsis province, and formed contemporaneously with the existence of a martian dynamo. This crust either pre-dates the formation of Tharsis, or formed during the earlier phases of Tharsis volcanism. The preservation of strong magnetic field anomalies over some of the earliest Noachian and topographically high units, together with the observation of magnetic field anomalies over Hesperian- and Amazonian-age surface units, indicate that a large fraction of the magnetized crust has remained cool (below the blocking temperature of the magnetic carrier) throughout the construction of Tharsis. Moreover, the distributions of magnetic anomaly amplitudes over Noachian, Hesperian, and Amazonian surface units suggest that the youngest units overlie sites of prolonged intrusion and have undergone a greater extent of thermal demagnetization. The absence of magnetic anomalies around the Tharsis Montes and Olympus Mons argues for strong, localized heating, as would be expected at volcanic centers. We show that end-member models for progressive thermal demagnetization of a Noachian magnetized crustal layer are consistent with the anomaly amplitude distributions. We integrate the magnetic field observations with constraints from tectonics, gravity, and topography, and present a revised scenario for the evolution of the Tharsis region.     

复杂地下异常体的可控源电磁法积分方程正演

可控源电磁法具有分辨率高及抗干扰能力强等特点,是一种重要的地电磁勘探方法.目前,可控源电磁法的高精度正演计算一直是其核心研究问题之一.传统积分方程法一般采用近似积分公式、简单矩形网格和近似的奇异性体积分计算技术,制约了体积分方程法处理复杂地下异常体的能力,降低了计算精度.针对上述问题,本文基于完全积分公式、四面体非结构化网格和奇异体积分的精确解析解来高精度求解复杂可控源电磁模型的正演响应.首先,从电场积分公式出发,推导了可控源电磁问题满足的积分方程;其次,借助于非结构化四面体网格离散技术,实现了地下复杂异常体的有效模拟.最后,利用散度定理把强奇异值体积分转换为一系列弱奇异性的面积分公式,并通过推导获得了这些弱奇异性的面积分公式的解析解,从而最终实现三维可控源电磁问题的高精度积分求解.以块状低阻体地电模型为测试模型,采用本文提出的积分方程方法获得的数值解与其他公开数值算法解进行对比分析,其对比结果具有高度的吻合性,验证了算法的正确性;同时,设计了球状及复杂地电模型进行算法收敛性测试,进一步验证算法的正确性以及能够处理地下复杂模型的能力.     

考虑自退磁影响的三维复杂形体磁场正反演研究进展与展望

忽视自退磁效应的影响,会导致磁异常反演结果的误差,影响地质解释的效果,这使得人们在实际应用中不得不考虑退磁的影响.本文对考虑自退磁影响的三维复杂形体磁场正演和反演方法进行了系统总结,论述了各种方法的特点和实际应用中的困难,提出了考虑自退磁影响的三维复杂形体磁场正反演研究的发展方向.     

磁赤道处化极方法

化向地磁极(化极)是最基本的磁测资料处理方法之一,化极能消除或减少斜磁化影响,提高对磁测资料的认识程度和解释水平,对研究地壳产生的磁异常具有重要意义.但低纬度地区特别是磁赤道处,化极处理很不稳定甚至奇异,一直是位场研究的难点.针对地磁纬度较低特别是磁赤道地区磁异常化极的困难,利用从磁北极处垂直磁化向低纬度地区水平磁化方向转换稳定的特点,提出"狭义化赤"概念,并将其与低纬度磁异常"倒相"解释方法结合,提出专门用于磁赤道处化极的方法.该方法扩展了现有的化极理论,实现了磁赤道处的稳定化极.区别于目前任何方法,专门用于(近)水平磁化条件下的化极计算,具有原理简单,实现方便,收敛速度快等特点.对理论模型和实际资料计算表明这种针对磁赤道地区磁异常的化极处理方法是稳定、可靠的.     

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.     

CURRENT CHANNELLING IN SQUARE PLATES WITH APPLICATIONS TO MAGNETOMETRIC RESISTIVITY1

The magnetometric resistivity (MMR) method uses a sensitive magnetometer to measure the low-level, low-frequency magnetic fields associated with the galvanic current flow between a pair of electrodes. While the MMR anomalies of simple structures such as dikes and vertical contacts have been determined analytically, there is a lack of systematic information on the expected responses from simple three-dimensional bodies. We determine the characteristic anomalies associated with square, plate-like conductors, which are excellent models of many base metal mineral deposits. The anomalies of plates of finite size are determined numerically using an integral equation method. A plate is subdivided into many sections and the current flow within each section is solved by equating the electrical field within each section to the tangential electrical field just outside it. When the plate size is small in relation to either the depth or the transmitter spacing, the shape and amplitude of the anomaly produced is closely approximated by a current dipole model of the same length and depth. At the other extreme, a large plate is represented by a half-plane. The dipole and half-plane models are used to bracket the behaviour of plates of finite size. The form of a plate anomaly is principally dependent on the shape, depth and orientation of the plate. A large, dipping plate near the surface produces a skewed anomaly highly indicative of its dip, but the amount of skew rapidly diminishes with increased depth or decreased size. Changes in plate conductivity affect the amplitude of the anomaly, but have little effect on anomaly shape. A current channelling parameter, determined from the conductivity contrast, can thus be used to scale the amplitude of an anomaly whose basic shape has been determined from geometrical considerations. The separation into geometrical and electrical factors greatly simplifies both the interpretation and modelling of MMR anomalies, particularly in situations with multiple plates. An empirical formula, using this separation, predicts the anomaly of two or more parallel plates with different conductances. In addition, the relation between the resolution of two vertical, parallel plates of equal conductance and their separation is determined. The ability of the integral equation method to model plate-like structures is demonstrated with the interpretation of an MMR anomaly in a survey conducted at Cork Tree Well in Western Australia. The buried conductor, a mineralized graphitic zone, is modelled with a vertical, bent plate. The depth to the top of the plate, and the plate conductance, is adjusted to fit the anomaly amplitude as closely as possible. From the modelling it would appear that this zone is not solely responsible for the observed anomaly.     

同时考虑退磁和剩磁的有限体积法正演模拟

为分析同时考虑退磁和剩磁对磁测数据解释的影响,探讨了利用有限体积法求静磁场数值解的方法.从静磁场中的麦克斯韦方程出发,导出了有限体积法控制方程的离散表达式,对边界条件近似处理后求解方程组得到磁异常.通过与退磁改正计算结果对比,验证了方法的正确性,并分析得到忽略剩磁的相对误差与科尼斯布格比（Q）相关;利用有限体积法计算长方体模型在无地磁场情况下的磁异常和内部磁化强度,从数值模拟上说明剩磁也需要进行退磁改正,并表明退磁作用对剩磁的影响不仅与磁化率相关,而且与剩磁的方向和磁性体的形态相关;组合模型的计算结果对比表明,退磁作用对剩磁的影响还会因为临近强磁性体的作用发生改变.在青海灶火河西工区的应用说明,开展同时考虑退磁和剩磁的解释方法对准确识别强磁性岩体具有实用价值.     

The applicability of two computer techniques to separate the remanent magnetization (RM) components acquired in laboratory fields

Thermal remanent magnetization (TRM) and anhysteretic remanent magnetization (ARM) components were imposed on natural rock samples. The artificial laboratory components had different directions and the blocking temperature and/or coercivity spectra were overlapping. Two methods, principal component analysis (PCA) by Kirschvink and analytical modelling of demagnetization data (by Stupavsky and Symons, S&S) were used to resolve these components. The PCA technique calculated lines fitted to the demagnetization path with ASD = 10° (angular standard deviation), and the S&S method used four types of intensity decay curves for calculated components.

Both methods (PCA and S&S) resolved perfectly the one-component case. The two- or three-component case results strongly depended on spectra overlapping, and on the angles between component directions and magnetic minerals in samples. Principal component analysis gave more reliable results for separated spectra of TRM and thermally cleaned samples, whereas the S&S technique was more efficient for the case of strong spectra overlapping of ARM components and the alternative current field (AF) demagnetization method. Remarkable anisotropy of RM was observed which influences the results for the haematite-bearing samples.     

Magnetotelluric responses of three-dimensional conductive and magnetic anisotropic anomalies

A magnetotelluric finite-element modelling algorithm is developed, which is capable of handling three-dimensional conductive and magnetic anisotropic anomalies. Different from earlier three-dimensional magnetotelluric anisotropic modelling methods, the algorithm we presented has taken the magnetic anisotropy into consideration. The variational equations are produced by the Galerkin method and the governing equations are solved using a hexahedral vector edge finite-element method. The accuracy of this algorithm is firstly validated by comparing its solutions with the results of finite-difference method for a three-dimensional conductive arbitrary anisotropic model, and then validated by comparing with analytical solutions for a one-dimensional magnetic model. The responses of four kinds of models under different conditions are studied, and some conclusions are obtained. It shows that for materials with a high magnetic permeability, its influence on magnetotelluric responses cannot be ignored in some circumstances. Especially, if the magnetic susceptibility is exceptionally high, it may really distort the apparent resistivities of lower resistive anomalies. These conclusions are also beneficial for magnetotelluric survey.     

On deducing the magnetization of the lunar surface from orbital surveys

A combination of orbital photographic, selenochemical and magnetic surveys may elucidate the mechanism by which the lunar surface became magnetized and possibly yield an estimate of the intensity of the ancient magnetizing field and its time variation. The determination of the size and shape of the magnetized regions requires the measurement of the altitude dependence of field, especially at low altitudes (< 100 km) and with a high enough sampling rate to resolve the profile at the edges of magnetized bodies. The planned Lunar Polar Orbiter may well provide the necessary data.     

各向异性标准化方差计算重磁源边界

在重磁源边界定位方法中,传统的梯度方法易受干扰的影响使计算的边界混乱,而且在弱异常处由于叠加异常的影响很难识别场源边界.本文首先利用坐标旋转构造了各向异性高斯函数,提出了各向异性标准化方差计算重磁源边界的方法.理论分析与模型实验详细阐明了该方法的数学含义,并通过干扰分析验证了方法的稳定性与有效性,结合中扬子地区航磁异常...