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.     

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.     

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.     

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。     

Euler homogeneity equation along ridges for a rapid estimation of potential field source properties

The Euler deconvolution is the most popular technique used to interpret potential field data in terms of simple sources characterized by the value of the degree of homogeneity. A more recent technique, the continuous wavelet transform, allows the same kind of interpretation. The Euler deconvolution is usually applied to data at a constant level while the continuous wavelet transform is usually applied to the points belonging to lines (ridges) connecting the m -order partial derivative modulus maxima of the upward-continued field at different altitudes in the harmonic region. In this paper a new method is proposed that unifies the two techniques. The method consists of the application of Euler's equation to the ridges so that the equation assumes a reduced form. Along each ridge the ratio among the m -order partial derivative of the field and its vertical partial derivative, for isolated source model, is a straight line whose slope and intercept allows the estimation of the source depth and degree of homogeneity. The method, strictly valid for single source model, has also been applied to the multisource case, where the presence of the interference among the field generated by each single source causes the path of the ratio to be no longer straight. The method in this case gives approximate solutions that are good estimations of the source depth and its degree of homogeneity only for a restricted range of altitudes, where the ratio is approximately linear and the source behaves as if it were isolated.     

An Improved Analytic Signal Technique for the Depth and Structural Index from 2D Magnetic Anomaly Data

This paper presents a new inversion method for the interpretation of 2D magnetic anomaly data, which uses the combination of the analytic signal and its total gradient to estimate the depth and the nature (structural index) of an isolated magnetic source. However, our proposed method is sensitive to noise. In order to lower the effect of noise, we apply upward continuation technique to smooth the anomaly. Tests on synthetic noise-free and noise corrupted magnetic data show that the new method can successfully estimate the depth and the nature of the causative source. The practical application of the technique is applied to measured magnetic anomaly data from Jurh area, northeast China, and the inversion results are in agreement with the inversion results from Euler deconvolution of the analytic signal.     

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.     

Visualizing and interpreting 3-D Euler solutions using enhanced computer graphics

The Euler 3-D deconvolution technique uses potential field data to calculate the position of discontinuities in physical properties without the need of any prior local geological information. Initial difficulties in presenting the Euler solutions in a manner which facilitated interpretation have been overcome using new techniques of image analysis, colour graphics and with real-time rotation of 3-D models. This has provided additional insight into the structure and tectonic history of part of eastern England.     

A derivative-based interpretation approach to estimating source parameters of simple 2D magnetic sources from Euler deconvolution, the analytic-signal method and analytical expressions of the anomalies

The major advantage of using either the analytic‐signal or the Euler‐deconvolution technique is that we can determine magnetic‐source locations and depths independently of the ambient earth magnetic parameters. In this study, we propose adopting a joint analysis of the analytic signal and Euler deconvolution to estimate the parameters of 2D magnetic sources. The results can avoid solution bias from an inappropriate magnetic datum level and can determine the horizontal locations, depths, structural types (indices), magnetization contrasts and/or structural dips. We have demonstrated the feasibility of the proposed method on 2D synthetic models, such as magnetic contacts (faults), thin dikes and cylinders. However, the method fails to solve the parameters of magnetic sources if there is severe interference between the anomalies of two adjacent magnetic sources.     

Interpretation case study of the Sahl El Qaa area, southern Sinai Peninsula, Egypt

The study area is located in the Sinai Peninsula, which is considered one of the most promising regions for oil resources. Three different tectonic forces affect the area in the triple junction structures associated with the opening of the Gulf of Suez and the strike slip movement along the Gulf of Aqaba. The main goal of this work is to model the structure of the basement rocks in the study area using magnetic methods. To achieve this, a high-resolution land magnetic survey was acquired and the results were combined with existing seismic reflection data. The magnetic interpretation was carried out using the analytical signal, horizontal gradient, Euler and Werner deconvolution and 3D magnetic modelling methods. We concluded that most of the deduced structures are trending in N–S, N35°–N45° west and E–W directions. The Aqaba trend (N15°–N25° east) is barely noticeable. The depth to the basement rocks ranges from 1 km to more than 2 km below sea level and these results are in agreement with the available well log data. In addition, interpretation of seismic reflection sections was carried out and compared with overlapping magnetic profiles interpreted using Euler deconvolution. They show that the sedimentary section was affected by the basement tectonics, with faults extending from the basement upwards through the sedimentary cover. These faults constitute good potential structural traps for oil accumulation.     

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.     

用Euler反褶积方法反演台湾海峡磁异常

用Ｅｕｌｅｒ反褶积方法反演台湾海峡磁异常，选取的窗口不能大，而决定取舍反褶积解的误差限又不能小，这势必对解的质量有较大影响．为此，不仅用了不同延深的单体模型，而且进行了多体模型实验．结果表明，复杂分布的磁性体，用Ｅｕｌｅｒ反褶积方法确定磁性体的轮廓可能是困难的，但可确定磁性体的水平位置和深度，从而降低了对资料精度和计算参数选取的要求．在磁异常比较复杂的地区．     

Microlevelling procedures applied to regional aeromagnetic data: an example from the Transantarctic Mountains (Antarctica)

The extensive application of digital enhancement and filtering as a powerful tool for aeromagnetic interpretation, not only of high resolution but also of regional data, requires an improved levelling. Two microlevelling techniques were thus compared in order to find an effective but relatively simple procedure to remove, or at least to reduce, residual magnetic errors remaining after standard levelling processes. This study was carried out on regional aeromagnetic data recently acquired at high magnetic latitudes along the Transantarctic Mountains in Antarctica, where it is particularly critical to remove time-dependent magnetic variations. Two-dimensional FFT filters applied to the gridded data, namely the Butterworth and a directional cosine filter, proved to be more effective than previously proposed one-dimensional space-domain filters in the reduction of the 'residual corrugation' not removed by statistical levelling. Tectonic interpretation of trends detected in the total field magnetic anomaly map and in the 3D analytic signal improved after application of frequency-domain microlevelling. However, we also show that when interpreting microlevelled data, two factors must be considered: (i) the possible presence of real geological trends aligned along the flight lines; (ii) modifications in the results yielded by depth estimates of magnetic sources due to the FFT filters applied during the microlevelling procedure. Such changes were seen both in the well-established 2D FFT method, based on the slope of the energy spectrum, and in the more recent 3D Euler deconvolution technique. Overall our results indicate that microlevelling could profitably be applied to older gridded aeromagnetic data sets in Antarctica, thus improving the accuracy and geological significance of future regional magnetic compilations, as already seen in other continents.     

Some aspects of interpretation of tensor gradiometry data

Aspects of the interpretation of measured data on the gravity gradient tensor (GGT) are examined. The problem is posed in relation to the great progress achieved in recent years in the development of instrumentation and the method of GGT measurements on mobile carriers. In our opinion, the new methods of measurement and the new data obtained with their help require the development of new methods of interpretation of potential fields. The paper addresses two methods taking advantage of simultaneous measurements of all components of the GGT and anomalies of the gravitaty field V _z. It is shown that the joint analysis of all GGT components can provide independent constraints on the noise level in various components. The method of tensor deconvolution proposed in the paper is a tensor analogue of the Euler method. The method is based on the calculation of invariants and is, therefore, stable with respect to the orientation uncertainties of the measuring system. The method provides means for estimating the structural index and, therefore, is particularly effective in the treatment of fields that contain isometric and/or elongated anomalies. The calculation of invariants and the tensor ratio can also be used for the development of procedures enabling automatic estimation of the axis strike azimuths of elongated anomaly-forming bodies.     

AUTOMATIC INTERPRETATION OF GRAVITY GRADIOMETRIC DATA IN TWO DIMENSIONS: VERTICAL GRADIENT1

The magnetic and gravity field produced by a given homogeneous source are related through Poisson's equation. Starting from this consideration, it is shown that some 2D interpretation tools, widely applied in the analysis of aeromagnetic data, can also be used for the interpretation of gravity gradiometric data (vertical gradient). This paper deals specifically with the Werner deconvolution, analytic signal and Euler's equation methods. After a short outline of the mathematical development, synthesized examples have been used to discuss the efficiency and limits of these interpretation methods. These tools could be applied directly to airborne gravity gradiometric data as well as ground gravity surveys after transformation of the Bouguer anomalies into vertical gradient anomalies. An example is given of the application of the Werner deconvolution and Euler's equation methods to a microgravity survey.     

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.     

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.

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SOME IMPROVEMENTS IN THE INTERPRETATION OF VERTICAL ELECTRIC SOUNDING CURVES*

The process of VES interpretation is discussed, including the following points. (a) Preliminary interpretation by means of master curves. It is shown that the positions of the auxiliary points K and Q depend on the resistivity of the substratum. The interpretation is improved if the auxiliary points are determined separately for each master curve. (b) The individual parts of the measured curves are shifted in overlapping MN electrode positions so that the total sum of squares of the shifts is minimal. (c) The ambiguity may be reduced by means of supplementary information or assumptions on the resistivities. Fixing the resistivities is not always possible because discrepancies may arise between the ground measurement and the well-logging data. The simultaneous interpretation of several VES curves is recommended assuming constant resistivities. This assumption may be subsequently verified by means of the F-test. (d) A nonlinear algorithm is proposed for the determination of confidence intervals. As the multi-dimensional confidence intervals are often very complicated, it is recommended to construct only one-dimensional confidence intervals for the estimable parametric functions. (e) A ‘double-least-squares’ optimization technique is presented. The optimization is performed on the estimable parametric functions, and the individual parameters are determined so that the solution remains near the initial guess. This technique is faster than the singular value decomposition.     

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.