Automatic Interpretation of Magnetic Data Using Euler Deconvolution with Nonlinear Background

The voluminous gravity and magnetic data sets demand automatic interpretation techniques like Naudy, Euler and Werner deconvolution. Of these techniques, the Euler deconvolution has become a popular choice because the method assumes no particular geological model. However, the conventional approach to solving Euler equation requires tentative values of the structural index preventing it from being fully automatic and assumes a constant background that can be easily violated if the singular points are close to each other. We propose a possible solution to these problems by simultaneously estimating the source location, depth and structural index assuming nonlinear background. The Euler equation is solved in a nonlinear fashion using the optimization technique like conjugate gradient. This technique is applied to a published synthetic data set where the magnetic anomalies were modeled for a complex assemblage of simple magnetic bodies. The results for close by singular points are superior to those obtained by assuming linear background. We also applied the technique to a magnetic data set collected along the western continental margin of India. The results are in agreement with the regional magnetic interpretation and the bathymetric expressions.     

TSVD analysis of Euler deconvolution to improve estimating magnetic source parameters: An example from the Åsele area,Sweden

In this paper, I introduce a new approach based on truncated singular value decomposition (TSVD) analysis for improving implementation of grid-based Euler deconvolution with constraints of quasi 2D magnetic sources. I will show that by using TSVD analysis of the gradient matrix of magnetic field anomaly (reduced to pole) for data points located within a square window centered at the maximum of the analytic signal amplitude, we are able to estimate the strike direction and dip angle of 2D structures from the acquired eigenvectors. It is also shown that implementation of the standard grid-based Euler deconvolution can be considerably improved by solving the Euler's homogeneity equation for source location and structural index, simultaneously, using the TSVD method. The dimensionality of the magnetic anomalies can be indicated from the ratio between the smallest and intermediate eigenvalues acquired from the TSVD analysis of the gradient matrix. For 2D magnetic sources, the uncertainty of the estimated source location and structural index is significantly reduced by truncating the smallest eigenvalue.Application of the method is demonstrated on an aeromagnetic data set from the Åsele area in Sweden. The geology of this area is dominated by several dike swarms. For these dolerite dikes, the introduced method has provided useful information of strike directions and dip angles in addition to the estimated source location and structural index.     

Euler deconvolution of the analytic signal and its application to magnetic interpretation

Euler deconvolution and the analytic signal are both used for semi‐automatic interpretation of magnetic data. They are used mostly to delineate contacts and obtain rapid source depth estimates. For Euler deconvolution, the quality of the depth estimation depends mainly on the choice of the proper structural index, which is a function of the geometry of the causative bodies. Euler deconvolution applies only to functions that are homogeneous. This is the case for the magnetic field due to contacts, thin dikes and poles. Fortunately, many complex geological structures can be approximated by these simple geometries. In practice, the Euler equation is also solved for a background regional field. For the analytic signal, the model used is generally a contact, although other models, such as a thin dike, can be considered. It can be shown that if a function is homogeneous, its analytic signal is also homogeneous. Deconvolution of the analytic signal is then equivalent to Euler deconvolution of the magnetic field with a background field. However, computation of the analytic signal effectively removes the background field from the data. Consequently, it is possible to solve for both the source location and structural index. Once these parameters are determined, the local dip and the susceptibility contrast can be determined from relationships between the analytic signal and the orthogonal gradients of the magnetic field. The major advantage of this technique is that it allows the automatic identification of the type of source. Implementation of this approach is demonstrated for recent high‐resolution survey data from an Archean granite‐greenstone terrane in northern Ontario, Canada.     

3D Gravity Inversion using Tikhonov Regularization

Subsalt exploration for oil and gas is attractive in regions where 3D seismic depth-migration to recover the geometry of a salt base is difficult. Additional information to reduce the ambiguity in seismic images would be beneficial. Gravity data often serve these purposes in the petroleum industry. In this paper, the authors present an algorithm for a gravity inversion based on Tikhonov regularization and an automatically regularized solution process. They examined the 3D Euler deconvolution to extract the best anomaly source depth as a priori information to invert the gravity data and provided a synthetic example. Finally, they applied the gravity inversion to recently obtained gravity data from the Bandar Charak (Hormozgan, Iran) to identify its subsurface density structure. Their model showed the 3D shape of salt dome in this region.     

Avoidable Euler Errors – the use and abuse of Euler deconvolution applied to potential fields

Window‐based Euler deconvolution is commonly applied to magnetic and sometimes to gravity interpretation problems. For the deconvolution to be geologically meaningful, care must be taken to choose parameters properly. The following proposed process design rules are based partly on mathematical analysis and partly on experience.

相似文献   

Inversion of potential field data using the structural index as weighting function rate decay

Nonparametric inverse methods provide a general framework for solving potential‐field problems. The use of weighted norms leads to a general regularization problem of Tikhonov form. We present an alternative procedure to estimate the source susceptibility distribution from potential field measurements exploiting inversion methods by means of a flexible depth‐weighting function in the Tikhonov formulation. Our approach improves the formulation proposed by Li and Oldenburg (1996, 1998) , differing significantly in the definition of the depth‐weighting function. In our formalism the depth weighting function is associated not to the field decay of a single block (which can be representative of just a part of the source) but to the field decay of the whole source, thus implying that the data inversion is independent on the cell shape. So, in our procedure, the depth‐weighting function is not given with a fixed exponent but with the structural index N of the source as the exponent. Differently than previous methods, our choice gives a substantial objectivity to the form of the depth‐weighting function and to the consequent solutions. The allowed values for the exponent of the depth‐weighting function depend on the range of N for sources: 0 ≤N≤ 3 (magnetic case). The analysis regarding the cases of simple sources such as dipoles, dipole lines, dykes or contacts, validate our hypothesis. The study of a complex synthetic case also proves that the depth‐weighting decay cannot be necessarily assumed as equal to 3. Moreover it should not be kept constant for multi‐source models but should instead depend on the structural indices of the different sources. In this way we are able to successfully invert the magnetic data of the Vulture area, Southern Italy. An original aspect of the proposed inversion scheme is that it brings an explicit link between two widely used types of interpretation methods, namely those assuming homogeneous fields, such as Euler deconvolution or depth from extreme points transformation and the inversion under the Tikhonov‐form including a depth‐weighting function. The availability of further constraints, from drillings or known geology, will definitely improve the quality of the solution.     

Can Euler deconvolution outline three-dimensional magnetic sources?

Severe limitations of the standard Euler deconvolution to outline source shapes have been pointed out. However, Euler deconvolution has been widely employed on field data to outline interfaces, as faults and thrust zones. We investigate the limitations of the 3D Euler deconvolution–derived estimates of source dip and volume with the use of reduced-to-the-pole synthetic and field anomalies. The synthetic anomalies are generated by two types of source bodies: (1) uniformly magnetized prisms, presenting either smooth or rough interfaces, and (2) bodies presenting smooth delimiting interfaces but strong internal variation of magnetization intensity. The dip of the first type of body might be estimated from the Euler deconvolution solution cluster if the ratio between the depth to the top and vertical extent is relatively high (>1/4). For the second type of body, besides dip, the source volume can be approximately delimited from the solution cluster envelope, regardless of the referred ratio. We apply Euler deconvolution to two field anomalies which are caused by a curved-shape thrust zone and by a banded iron formation. These anomalies are chosen because they share characteristics with the two types of synthetic bodies. For the thrust zone, the obtained Euler deconvolution solutions show spatial distribution allowing to estimate a source dip that is consistent with the surface geology data, even if the above-mentioned ratio is much less than 1/4. Thus, there are other factors, such as a heterogeneous magnetization, which might be controlling the vertical spreading of the Euler deconvolution solutions in the thrust zone. On the other hand, for the iron-ore formation, the solution cluster spreads out occupying a volume, in accordance with the results obtained with the synthetic sources having internal variation of magnetization intensity. As conclusion, although Euler deconvolution–derived solutions cannot offer accurate estimates of source shapes, they might provide a sufficient degree of reliability in the initial estimates of the source dip and volume, which may be useful in a later phase of more accurate modelling.     

Imaging magnetic sources using Euler's equation

The conventional Euler deconvolution method has the advantage of being independent of magnetization parameters in locating magnetic sources and estimating their corresponding depths. However, this method has the disadvantage that a suitable structural index must be chosen, which may cause spatial diffusion of the Euler solutions and bias in the estimation of depths to the magnetic sources. This problem becomes more serious when interfering anomalies exist. The interpretation of the Euler depth solutions is effectively related to the model adopted, and different models may have different structural indices. Therefore, I suggest a combined inversion for the structural index and the source location from the Euler deconvolution, by using only the derivatives of the magnetic anomalies. This approach considerably reduces the diffusion problem of the location and depth solutions. Consequently, by averaging the clustered solutions satisfying a given criterion for the solutions, we can image the depths and attributes (or types) of the causative magnetic sources. Magnetic anomalies acquired offshore northern Taiwan are used to test the applicability of the proposed method.     

Interpretation of Gravity Data using 3D Euler Deconvolution,Tilt Angle,Horizontal Tilt Angle and Source Edge Approximation of the North-West Himalaya

The collision of the Indian plate and the Eurasian plate created shortening and imbrications with thrusting and faulting which influences northward tectonic movement. This plate movement has divided the Himalaya into four parts, viz. Outer Himalaya, Lesser Himalaya, Greater Himalaya, and Tethys Himalaya. The crystalline basement rock plays an imperative role for structural and tectonic association. The study has been carried out near Rishikesh-Badrinath neighborhood in the northwestern part of the Himalayan girdle with multifarious tectonic set up with thrusted and faulted geological setting. In this study area, 3D Euler deconvolution, horizontal gradient analysis, tilt angle (TILT) and horizontal tilt angle (TDX) analysis have been carried out using gravity data to delineate the subsurface geology and heterogeneity in the northwestern part of Himalaya. The Euler depth solutions suggest the source depth of about 12 km and various derivative analyses suggest the trend of the delineation thrust-fault boundaries along with the dip and strike direction in the study area.     

A deep geophysical study in the Baikal region

Experimental data show that in East Siberia resistivity curves, irrespective of their trends, are affected by galvanic (local) distortions. The preliminary step of the magnetotelluric data processing is to obtain a steady shape of resistivity curves reflecting a true deep section. For this purpose statistical averaging and different criteria of impedance rejecting were used. The available MTS curves were normalized by level to the global magnetovariation curves. Two-dimensional modelling was performed from several sublatitudinal profiles crossing the Baikal rift zone. Three-dimensional models based on two-dimensional modelling and on induction vector distribution have been computed via programs of M. N. Yudin. Following other researchers, two conductive layers are distinguished: i) the mid- and low crustal and ii) the mantle one, with the layer surface uplifted from 100–110 km depth in the southern Baikal rift zone to 60–70 km northeastwards along the eastern Baikal coast. The top of this layer seems to correspond to the asthenospheric roof. The asthenosphere deepening in southern BRZ is likely to be related to a decrease in the asthenospheric bulge width and an increase in the rate of lithospheric thickening with mantle degasing. The origin and evolution of the Baikal rift is considered, proceeding from the model of passive rifting with regard to a long-existing lithospheric inhomogeneity between the Siberian platform and the Sayan-Baikal folded area.     

基于垂向一阶导数与解析信号比值的欧拉反演方法

常规欧拉反褶积法中构造指数的选取以及分散解存在较多的问题,提出了基于联立垂向一阶导数与解析信号的欧拉齐次方程的RDAS-Euler反演方法。该方法可以更为精确的估计场源的范围及埋深,且不需考虑构造指数N的影响,避免了因构造指数不当而引起的反演误差。通过对单一地质体及组合地质体模型的实验证明本文方法能有效地完成目标体的反演工作,反演结果与理论值之间的误差小于10%,且相对于常规欧拉反褶积法更加稳定准确,能够更好的得到地质体边界及深度信息。将RDAS-Euler法应用于黑龙江省虎林盆地实测布格重力异常数据,获得了丰富的断裂信息,说明RDASEuler法增强了对断裂平面位置的识别能力。     

Estimation of the Source Time Function Based on Blind Deconvolution with Gaussian Mixtures

It is demonstrated that the blind deconvolution method is fully capable of recovering the unknown Greens function and of estimating the source time functions from observed seismic data of small earthquakes. Based on the assumption of the Gaussian-mixture model of the Greens function, the newly-formulated algorithm is evaluated using synthetic seismic data along with those of the May 8, 1996 Mexico earthquake (M_c = 4.6). Since the estimated results closely match the theoretical input very well, the method is then employed to analyze the source time functions of the July 7, 1995 Pu-Li, Taiwan earthquake (M_L = 5.3). The stations triggered by this event were azimuthally well covered. Using the estimated source time functions, information pertaining to the directivity effect is readily obtained, and the actual fault plane of this event is identified, thus clearly indicating that this method provides a most efficient way to estimate the source time function of a small earthquake.Acknowledgment The authors would like to express their thanks to two anonymous reviewers for their valuable suggestions and Dr. I. Santamarias courteous assistance. They also appreciate the efforts of Drs. H.C. Chiu and R.J. Rau, who provided the seismic data and the fault plane solutions. The National Science Council, Taiwan, has supported this research (NSC 91-2119-M-194-011).     

The theory of the continuous wavelet transform in the interpretation of potential fields: a review

We consider the use of the continuous wavelet transform in the interpretation of potential field data. We report its development since the publication of the first paper by Moreau et al . in 1997. Basically, it consists in the interpretation in the upward continued domain since dilation of the wavelet transform is the upward continuation altitude. Thus within a range of altitudes, the wavelet transform of the noise is decreased faster than the wavelet transform of the potential field caused by underground sources; this means that the signal-to-noise ratio is much better than those involved in other enhancing methods (e.g., Euler deconvolution, gradient analysis, or the analytic signals). Similarly to the Euler deconvolution, its first target parameters were the source positions and shape. The method has then been developed to estimate size and directions of extended sources (e.g., faults and dikes of finite dimensions) and also the magnetization direction in the case of magnetic data. Latest developments show that when combined with a Radon transform, the continuous wavelet transform can help in the automatic detection of elongated structures in 3D, simultaneously to the estimation of their strike direction, shape and depth. Several applications to real case studies have been shown before; however for clarity's sake in the present paper, only synthetic cases have been reproduced to clearly sum up the development of the methodology.     

Multiridge Euler deconvolution

Potential field interpretation can be carried out using multiscale methods. This class of methods analyses a multiscale data set, which is built by upward continuation of the original data to a number of altitudes conveniently chosen. Euler deconvolution can be cast into this multiscale environment by analysing data along ridges of potential fields, e.g., at those points along lines across scales where the field or its horizontal or vertical derivative respectively is zero. Previous work has shown that Euler equations are notably simplified along any of these ridges. Since a given anomaly may generate one or more ridges we describe in this paper how Euler deconvolution may be used to jointly invert data along all of them, so performing a multiridge Euler deconvolution. The method enjoys the stable and high‐resolution properties of multiscale methods, due to the composite upward continuation/vertical differentiation filter used. Such a physically‐based field transformation can have a positive effect on reducing both high‐wavenumber noise and interference or regional field effects. Multiridge Euler deconvolution can also be applied to the modulus of an analytic signal, gravity/magnetic gradient tensor components or Hilbert transform components. The advantages of using multiridge Euler deconvolution compared to single ridge Euler deconvolution include improved solution clustering, increased number of solutions, improvement of accuracy of the results obtainable from some types of ridges and greater ease in the selection of ridges to invert. The multiscale approach is particularly well suited to deal with non‐ideal sources. In these cases, our strategy is to find the optimal combination of upward continuation altitude range and data differentiation order, such that the field could be sensed as approximately homogeneous and then characterized by a structural index close to an integer value. This allows us to estimate depths related to the top or the centre of the structure.     

Mean Field Dynamos with Algebraic and Dynamic α-Quenchings

Calculations for mean field dynamo models (in both full spheres and spherical shells), with both algebraic and dynamic -quenchings, show qualitative as well as quantitative differences and similarities in the dynamical behaviour of these models. We summarise and enhance recent results with extra examples. Overall, the effect of using a dynamic appears to be complicated and is affected by the region of parameter space examined.     

辽宁地区时变重力场源变化特征分析

地震重力分析通过研究时变重力场变化获取地球内部介质物性变化信息.采用贝叶斯重力网平差方法对辽宁地区2011-2014年共计四年7期的流动重力观测资料进行处理,对研究区重力观测网总体监测能力做出分析,选用欧拉反褶积对研究区的重力变化场源深度及空间分布特征进行反演和解释.通过反演计算发现,在2013年灯塔Ms5.1地震前沈...     

Integral Equations for Kinematic Dynamo Models

A new technique for the treatment of the kinematic dynamo problem is presented. The method is applicable when the dynamo is surrounded by a medium of finite conductivity and is based on a reformulation of the induction equation and boundary conditions at infinity into an integral equation. We show that the integral operator involved here is compact in the case of homogeneous conductivity, which is important for both mathematical and numerical treatment. A lower bound for the norm of then yields a necessary condition for the generation of magnetic fields by kinematic dynamos. Numerical results are presented for some simple ²-dynamo models. The far-field asymptotics for stationary and time-dependent field modes are discussed.     

Interpretation of gravity and magnetic anomalies when observation plane is inclined

Summary Not infrequently, in mining geophysics, measurements have to be made on the slopes of a hill that contain mineralisation. In this paper, procedures are evolved for interpreting gravity and vertical component magnetic data collected on such slopes and caused by geological bodies that can be approximated by infinite line pole, point pole, infinite line dipole and point dipole. From sets of theoretical magnetic anomaly curves (not reproduced), graphs have been constructed using characteristic points and are reproduced in Figures 5 to 10 and 12 to 17. These can be used for direct determination of depths, offsets, etc., by using information from suitably chosen field profiles. In the case of gravity, the mass can be computed as usual by a surface integration, provided a correction factor (1/cos²) is used, being the angle of the slope.     

不同高度数据联合欧拉反褶积法研究

基于不同测量高度重力场及其梯度数据可同时对应同一场源并用于反演场源位置的分析原理,拓展不同高度场数据在欧拉反褶积法中的应用范围.首先,立足于对欧拉反褶积方法的理论研究基础,提出不同高度数据融合联合欧拉反演公式.其次,在理论模型上对多种高度数据联合反演做了测试分析计算,验证了不同高度场数据融合联合欧拉反褶积法能够改善位场解释中单一观测面数据计算带来的解的发散问题,收敛过程由此改善.最后,将本文方法应用于龙门山地区实际重力数据的解释,获得了研究区断裂分布特征.     

重力全张量数据联合欧拉反褶积法研究及应用

全张量测量技术是在空中或海上用加载了多个加速度计的移动平台技术测量位场的五个独立分量.各张量分量包含不同方向的地下地质体信息,水平张量分量T_(xx)、T_(yy)、T_(xy)、T_(xz)、T_(yz)通常用于识别和映射与地质构造或地层变化有关的测量区域中的目标,垂直张量分量Tzz用于估计深度.然而,这些分量传统上是彼此分开解释,经常遇到错失关键信息的风险.本文所用全张量欧拉反褶积是在单独z方向的欧拉反演基础上发展而来的,它融合了重力异常垂直分量以及其三个方向导数、水平分量以及其三个方向导数.全张量数据信息得以有效应用的同时,欧拉反褶积结果也比常规欧拉反褶积结果更加收敛.最后,结合美国墨西哥湾地区实测航空FTG数据,用重力梯度张量数据进行联合欧拉三维反演研究,有效的识别岩盖的边界信息,划分岩盖范围,为进一步研究盖层底下深部复杂地质情况提供可靠的解释结果.