Magnetic field transforms with low sensitivity to the direction of source magnetization and high centricity

Magnetic data interpretation faces difficulties due to the various shapes of magnetic anomalies and the positions of their extrema with respect to the causative bodies for different directions of the source magnetization. The well‐known transforms — reduction to the pole, pseudogravity field, and analytic signal (total gradient) — help in reducing the problem. Another way to achieve the required effect is the transformation of magnetic data, ΔT or Z, into values of the anomalous magnetic intensity T. In this respect, we have found some transforms based on differential operators such as the gradient of T and its modulus R = |?T|, the Laplacian L = ?²T, the product T ?²T and its square root Q, and the Laplacian ?²(T²) and its square root E, to be useful. They are slightly sensitive to the magnetization orientation and their extrema occur above the sources. For a 2D anomaly of a homogeneous causative body, the proposed transforms do not depend on the inclination of magnetization. In the 3D case, such independence does not exist even for the elementary field of a point dipole. The influence of the magnetization direction is estimated by an integral coefficient of sensitivity. This coefficient takes values of up to 2.0 for ΔT or Z anomalies, while their transforms T, R, E, Q and L have values of less than 0.28, 0.29, 0.24, 0.16 and 0.07, respectively, i.e. on average, 10 times less. The estimation of the centricity is carried out using the relative deviation of the principal extremum of the anomaly or its transforms from the epicentre of the model body at a depth equal to 100 units. For a ΔT anomaly this deviation is up to 67%; for the L transform it is less than 8%; for Q, E, R and T it is less than 10%, 15%, 20% and 25%, respectively. The proposed transforms take only non‐negative values. With respect to their shape, the peripheral magnetic extrema are removed, the anomalous configuration is simplified and the resolution of complicated interference patterns is improved. Their calculation does not require additional data for the direction of magnetization, which is an essential advantage over the reduction‐to‐the‐pole and pseudogravity‐field transforms. A joint analysis of the measured field and its transforms T, E and L offers possibilities for more confident separation of the anomalous effects and direct correlation to their sources. The model tests performed and the 3D field applications to real magnetic data confirm the useful properties of the transforms suggested here.     

SOME NEW MAGNETIC TRANSFORMATIONS*

All magnetic transformations are governed by one simple differential relation between the observed and the transformed quantities. A magnetic map for any component, at any location, and for any given direction of magnetization can be converted into one for which any one, two, or all three parameters differ. Three new magnetic transformations are introduced: (i) reduction to equator, (ii) orthogonal reduction, and (iii) elimination of remanence. The first eliminates (or minimizes) the asymmetry and the lateral shift of the measured total field anomalies, exactly as in Baranov's reduction to pole. The second produces perfect asymmetry so that a symmetrical target lies vertically below the zero anomaly point, midway between the maximum and minimum. When remanence is a contributing factor, the direction of resultant magnetization must be known a priori in all cases, except for transformation of one component into another in the same area. Explicit working formulae are presented for reduction to equator and pole, and orthogonal reduction.     

Reconstruction of the harmonic component of the magnetic field modulus anomalies

An algorithm is proposed for calculating a harmonic function equal to the projection of the anomalous magnetic field vector onto the normal field direction from in situ measurements of the anomalous magnetic field modulus (the scalar magnetic anomaly) ΔT, which is a nonharmonic function and is nonlinearly related to the magnetization of anomaly sources. It is shown that the inferred estimates tend to the desired harmonic function if the iterative algorithm converges. The convergence conditions and stability of the process are studied numerically in a wide range of amplitudes of the anomalous field. The results of the modeling simulation demonstrate the efficiency of the proposed algorithm in solving magnetic field interpretation problems often encountered in practice.     

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.     

Palaeomagnetism of the monzonite porphyry from Milton,New South Wales

Summary Directions of natural remanent magnetization aftermagnetic cleaning of specimens from monzonite porphyry at Milton are given. Thermal and alternating magnetic field stability tests indicate that the resultant direction is that of the earth's magnetic field at the time of cooling. Comprison of the pole position calculated from this result with other pole positions from rocks of known age confirms the probable Permian age of the intrusion and the wide divergence of Australian from European and North American pole positions of this age.     

Methods for reconstructing harmonic functions from the magnetic field ΔT and V.N. Strakhov’s function ΔS: A review

We give an overview of the methods designed for reconstructing close-to-harmonic functions from the magnetic field ΔT. The formula of Yu.P. Tafeev is refined. It is shown that this refined formula directly leads to the relation derived by V.M. Gordin and his colleagues that allows isolating the harmonic component in the function ΔT. V.N. Strakhov’s linearized representation of the function ΔT is immediately derived from the main approximate Tafeev formula for Q ΔT. The experience of using Strakhov’s ΔS function in the interpretation of the magnetic anomaly ΔT generated by the Krivoi Rog structure is described. It is noted that the problem of reconstructing the corresponding harmonic functions from the data of magnetic and gravity surveys has much in common. The specific features of measuring the magnetic field H and magnetic induction B in the material media are considered, and the physical interpretation of these fields is presented.     

A method to estimate the total magnetization direction from a distortion analysis of magnetic anomalies1

Knowledge of the declination and inclination of the total and induced magnetization vectors is normally required for the interpretation and analysis of magnetic anomalies. A new method of estimating the direction of the total magnetization vector of magnetized rocks from magnetic anomalies is proposed. The unknown declination and inclination (D*_T and I*_T) can be found by applying a reduction-to-the-pole operator to the measured anomalies for different couples of total magnetization direction parameters (D_T and I_T) and by observing the variation of the anomaly minimum as a function of both D_T and I*_T.and D*_T are estimated using the maximum of this function. Comparing our method to previous methods, one advantage is that our estimates are not zero-level dependent; furthermore, the method allows inclinations to be well estimated, with the same accuracy as declinations; finally declinations are not underestimated. Our method is applied to a real case and meaningful results are obtained; it is shown that the feasibility of the method is improved by removing the low-frequency components.     

Interpretation of Magnetic Anomalies and Estimation of Depth of Magnetic Crust in Slovakia

The magnetic map of Slovakia used in the paper was compiled as part of a project titled Atlas of Geophysical maps and profiles in 2001. The residual magnetic data were analyzed to produce Curie point estimates. To remove distortion of magnetic anomalies caused by the Earth’s magnetic field, reduction to pole transformation was applied to the magnetic anomalies using the magnetization angle of the induced magnetization. Anomalies reduced to the pole tend to be better correlated with tectonic structures. We applied a 3-km upward continuation to the residually compiled magnetic anomalies in order to remove effects of topography. The depth of magnetic dipoles was calculated by an azimuthally averaged power spectrum method for the entire area. Such estimates can be indicative of temperatures in the crust, since magnetic minerals lose their spontaneous magnetization according to Curie temperature of the dominant magnetic minerals in the rocks. The computed Curie point depths in the Slovakia region vary between 15.2 km and 20.9 km. Heat flow higher than 100 mWm⁻² occurs at the central volcanics and eastern part of Slovakia, where the Curie point depths values are shallow. The correlation between Curie point depths, heat flow and crust depth was investigated for two E-W cross sections. Heat flow and Curie point depth values are correlated with each other however, these values could not be correlated with crust depth. The Curie point isotherm, which separates magnetic and non-magnetic parts of the crust, is represented in two cross sections.     

EIN AUTOMATISCHES VERFAHREN ZUR INTERPRETATION MAGNETISCHER ANOMALIEN NACH DER METHODE DER KLEINSTEN QUADRATE*

The interpretation of magnetic anomalies on the basis of model bodies is preferably done by making use of “trial and error” methods. These manual methods are tedious and time consuming but they can be transferred to the computer by making the required adjustments by way of the method of least squares. The general principles of the method are described. Essential presumptions are the following:

• 1 the assumption of definite model bodies
• 2 the existence of approximation values of the unknown quantities (position, dip, magnetization, etc.)
• 3 a sufficiently large number of measuring values, so that the process of adjustment can be carried out.

The advantages of the method are the following:

• 1 substantial automatization and a quick procedure by using computers
• 2 determination of the errors of the unknown quantities.

The method was applied to the interpretation of two-dimensional ΔZ- and ΔT-anomalies. Three types of model bodies are taken as basis of the computer program, viz. the thin dyke, infinite resp. finite in its extension downward and the circular cylinder. Only the measuring values are given to the computer. The interpretation proceeds in the following steps:

• 1 calculation of approximation values
• 2 determination of the model body of best fit
• 3 iteration in the case of the model body of best fit.

The computer produces the end values of the unknown quantities, their mean errors, and the pertaining theoretical anomalies. These end results are given to a plotting machine, which draws the measured curve, the theoretical curve and the model bodies. Interpretation examples are given.     

A Least-squares Minimization Approach to Depth Determination from Magnetic Data

—We have developed a least-squares minimization approach to depth determination from magnetic data. By defining the anomaly value T(0) at the origin and the anomaly value T(N) at any other distance (N) on the profile, the problem of depth determination from magnetic data has been transformed into finding a solution to a nonlinear equation of the form f(z)=0. Formulas have been derived for a sphere, horizontal cylinder, dike, and for a geologic contact. Procedures are also formulated to estimate the effective magnetization intensity and the effective magnetization inclination. A scheme for analyzing the magnetic data has been formulated for determining the model parameters of the causative sources. The method is applied to synthetic data with and without random errors. Finally, the method is applied to two field examples from Canada and Arizona. In all cases examined, the estimated depths are found to be in goodagreement with actual values.     

基于离散余弦变换（DCT）的化磁极方法

针对提高磁异常化磁极的质量问题,提出基于离散余弦变换(DCT)的化磁极方法.以位场余弦变换谱分析为基础,从理论上推导了基于DCT的二度和三度体总场磁异常化磁极转换公式.在倾斜板状体模型实验中,化磁极误差小于0.001 nT,具有较高的精度;在单球体及多球体模型实验中,采用基于DCT的化磁极方法在5.倾斜磁化时就可以取得较好的化磁极结果,15°时化磁极的效果更加明显,其等值线的形态、幅值以及所反映的磁性体的水平位置都得到较好的恢复,这说明,采用本文方法进行化磁极时,可以取得较好的效果.     

THE MAGNETIZATION OF THE MAGNETITE OREBODY NEAR CORTEGANA,SPAIN*

The magnetic ground survey of ΔZ across the orebody near Cortegana suggests that the direction of the magnetization of the orebody deviates from the present earth field direction in that area. Magnetic measurements of more than 500 specimens of drilling cores of several vertical and one nearly horizontal drill holes showed that the magnetization of the orebody points essentially to the north in the direction of the inclination of the orebody and the banding of the ore. In the central part of the orebody with an average magnetite content of about 50 vol% the magnetization amounts to 0.35 Gauss, the remanent and induced component having the same order of magnitude. The outer parts of the orebody have a much smaller magnetization according to both the smaller magnetite content and greater inhomogeneity of the remanent magnetization, also partly due to their reversed magnetization which is brought about by the stray field of the central part of the orebody. As all drilling cores have been chemically analyzed with respect to their Fe content a logarithmic relationship could be established between the magnetite content, ranging from 25 to 80 vol%, and the susceptibility.     

Age of the source of the Jarrafa gravity and magnetic anomalies offshore Libya and its geodynamic implications

The interpretation of the Jarrafa magnetic and gravity highs, NW Libyan offshore, suggests that it may be caused by a body of high-density and high magnetization. Analysis of their power spectra indicates two groups of sources at: (1) 2.7 km depth, probably related to the igneous rocks, some of which were penetrated in the JA-1 borehole, (2) 5 km depth, corresponding to the top of the causative body and (3) 10 km depth, probably referring to the local basement depth. The boundary analysis derived from applied horizontal gradient to both gravity and magnetic data reveals lineaments many of which can be related to geological structures (grabens, horsts and faults).The poor correlation between pseudogravity fields for induced magnetization and observed gravity fields strongly suggests that the causative structure has a remanent magnetization (D = −16°, I = 23°) of Early Cretaceous age, fitting with the opening of the Neo Tethys 3 Ocean.Three-dimensional interpretation techniques indicate that the magnetic source of the Jarrafa magnetic anomaly has a magnetization intensity of 0.46 A/m, which is required to simulate the amplitude of the observed magnetic anomaly. The magnetic model shows that it has a base level at 15 km.The history of the area combined with the analysis and interpretation of the gravity and magnetic data suggests that: (1) the source of the Jarrafa anomaly is a mafic igneous rock and it may have formed during an Early Cretaceous extensional phase and (2) the Jarrafa basin was left-laterally sheared along the WNW Hercynian North Graben Fault Zone, during its reactivation in the Early Cretaceous.     

Short note: A simple nomogram for interpretation due to magnetic horizontal cylinders

The magnetic anomaly ΔF due to a long horizontal cylinder yields three points related to its extrema. A nomogram is presented for the determination of some source parameters of the causative body.     

Effect of Stress on the Magnetic Memory of Induced Magnetic Anisotropy of Rocks and Its Mathematical Model

When re-heated to temperatures below the Curie temperature and subsequently cooled in a constant magnetic field (H _T), rock samples which contain magnetic minerals can acquire an induced magnetic anisotropy (IMA). As the result of acquiring the IMA, a constriction develops in the hysteresis loop of the magnetization of these rocks at the values of the magnetizing field close or equal to the HT. Thus the IMA is capable of retaining the information on the palaeointensity of the geomagnetic field, i.e., if IMA was created in a rock in the geomagnetic field in a past geological epoch, it preserves the information on the intensity of that field. Investigations have shown, that when IMA is created in a rock under external stress, the stress has an impact on the magnetic memory. Here we also deal with the issue of how stress affects the magnetic memory of IMA. A mathematical model for the effect of stress on magnetic memory phenomena related to induced magnetic anisotropy in rocks containing multidomain magnetite and titanomagnetite grains is proposed herewith. The effect of temperature on the magnetic memory of rocks is discussed also.     

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.     

Estimating magnetic dike parameters using a non‐linear constrained inversion technique: an example from the Särna area,west central Sweden

In this paper, we describe a non‐linear constrained inversion technique for 2D interpretation of high resolution magnetic field data along flight lines using a simple dike model. We first estimate the strike direction of a quasi 2D structure based on the eigenvector corresponding to the minimum eigenvalue of the pseudogravity gradient tensor derived from gridded, low‐pass filtered magnetic field anomalies, assuming that the magnetization direction is known. Then the measured magnetic field can be transformed into the strike coordinate system and all magnetic dike parameters – horizontal position, depth to the top, dip angle, width and susceptibility contrast – can be estimated by non‐linear least squares inversion of the high resolution magnetic field data along the flight lines. We use the Levenberg‐Marquardt algorithm together with the trust‐region‐reflective method enabling users to define inequality constraints on model parameters such that the estimated parameters are always in a trust region. Assuming that the maximum of the calculated g_zz (vertical gradient of the pseudogravity field) is approximately located above the causative body, data points enclosed by a window, along the profile, centred at the maximum of g_zz are used in the inversion scheme for estimating the dike parameters. The size of the window is increased until it exceeds a predefined limit. Then the solution corresponding to the minimum data fit error is chosen as the most reliable one. Using synthetic data we study the effect of random noise and interfering sources on the estimated models and we apply our method to a new aeromagnetic data set from the Särna area, west central Sweden including constraints from laboratory measurements on rock samples from the area.     

Relative paleointensity of the geomagnetic field during the past 200 ka from the West Philippine Sea and its chronological significance

Our rock magnetic analysis of core Ph05 from the West Philippine Sea demonstrates that the core preserves a strong, stable remanent magnetization and meets the magnetic mineral criteria for relative paleointensity (RPI) analyses. The magnetic minerals in the sequence are dominated by pseudosingle-domain magnetite, and the concentration of magnetic minerals is at the same scale. Both the conventional normalizing method and the pseudo-Thellier method were used in conjunction with the examination of the rock magnetic properties and natural remanent magnetization. Susceptibility (χ), anhysteretic remnant magnetization (ARM) and saturation isothermal remnant magnetization (SIRM) were used as the natural remanent magnetization normalizer. However, coherence analysis indicated that only ARM is more suitable for paleointensity reconstruction. The age model of core is established based on oxygen isotope data and AMS¹⁴C data, which is consistent with the age model estimated from RPI records. The relative paleointensity data provide a continuous record of the intensity variation during the last 200 ka, which correlates well with the global references RPI stacks. Several prominent low paleointensity values are identified and are correlated to the main RPI minima in the SINT-200 record, suggesting that the sediments have recorded the real changes of geomagnetic field. Supported by National Natural Science Foundation of China (Grant No. 90411014) and Pilot Project of the Knowledge Innovation Program of Chinese Academy of Sciences (No. KZCX2-YW-211)     

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.     

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。