A New Algorithm for Depth Determination from Total Magnetic Anomalies due to Spheres

We have developed an automatic method to determine the depth of a buried sphere from numerical second horizontal derivative anomalies obtained from total field magnetic data. The method is based on using a relationship between the depth and a combination of observations at symmetric points with respect to the coordinate of the projection of the center of the source in the plane of the measurement points with a free parameter (graticule spacing). The problem of depth determination has been transformed into the problem of finding a solution of a nonlinear equation of f(z) = 0. Procedures are also formulated to determine the magnetic moment and the effective angle of magnetization. The method is applied to synthetic examples with and without random errors and tested on a field example from Senegal. In all cases, the depth solutions are in good agreement with the actual ones.     

Self-potential data interpretation using standard deviations of depths computed from moving-average residual anomalies

We have developed a least‐squares minimization approach to determine simultaneously the shape (shape factor) and the depth of a buried structure from self‐potential (SP) data. The method is based on computing the standard deviation of the depths determined from all moving‐average residual anomalies obtained from SP data, using filters of successive window lengths for each shape factor. The standard deviation may generally be considered a criterion for determining the correct depth and shape factor of the buried structure. When the correct shape factor is used, the standard deviation of the depths is less than the standard deviations computed using incorrect shape factors. This method is applied to synthetic data with and without random errors, complicated regionals and interference from neighbouring sources, and is tested on a known field example from Turkey. In all cases, the shape and depth solutions obtained are in a good agreement with the actual values.     

A simple formula for shape and depth determination from residual gravity anomalies

This paper presents a simple method for shape and depth determination of a buried structure from residual gravity anomalies along profile. The method utilizes the anomaly values of the origin and characteristic points of the profile to construct a relationship between the shape factor and depth of the causative source. For fixed points, the depth is determined for each shape factor. The computed depths are then plotted against the shape factor representing a continuous monotonically increasing curve. The solution for the shape and depth of the buried structure is then read at the common intersection point of the depth curves. This method is applied to synthetic data with and without random errors. Finally, the validity of the method is tested on two field examples from the USA.     

Detection of potential fields source boundaries by enhanced horizontal derivative method

A high‐resolution method to image the horizontal boundaries of gravity and magnetic sources is presented (the enhanced horizontal derivative (EHD) method). The EHD is formed by taking the horizontal derivative of a sum of vertical derivatives of increasing order. The location of EHD maxima is used to outline the source boundaries. While for gravity anomalies the method can be applied immediately, magnetic anomalies should be previously reduced to the pole. We found that working on reduced‐to‐the‐pole magnetic anomalies leads to better results than those obtainable by working on magnetic anomalies in dipolar form, even when the magnetization direction parameters are not well estimated. This is confirmed also for other popular methods used to estimate the horizontal location of potential fields source boundaries. The EHD method is highly flexible, and different conditions of signal‐to‐noise ratios and depths‐to‐source can be treated by an appropriate selection of the terms of the summation. A strategy to perform high‐order vertical derivatives is also suggested. This involves both frequency‐ and space‐domain transformations and gives more stable results than the usual Fourier method. The high resolution of the EHD method is demonstrated on a number of synthetic gravity and magnetic fields due to isolated as well as to interfering deep‐seated prismatic sources. The resolving power of this method was tested also by comparing the results with those obtained by another high‐resolution method based on the analytic signal. The success of the EHD method in the definition of the source boundary is due to the fact that it conveys efficiently all the different boundary information contained in any single term of the sum. Application to a magnetic data set of a volcanic area in southern Italy helped to define the probable boundaries of a calderic collapse, marked by a number of magmatic intrusions. Previous interpretations of gravity and magnetic fields suggested a subcircular shape for this caldera, the boundaries of which are imaged with better detail using the EHD method.     

Analytic signal approach and its applicability in environmental magnetic investigations

We investigate the analytic signal method and its applicability in obtaining source locations of compact environmental magnetic objects. Previous investigations have shown that, for two-dimensional magnetic sources, the shape and location of the maxima of the amplitude of the analytic signal (AAS) are independent of the magnetization direction. In this study, we show that the shape of the AAS over magnetic dipole or sphere source is dependent on the direction of magnetization and, consequently, the maxima of the AAS are not always located directly over the dipolar sources. Maximum shift in the horizontal location is obtained for magnetic inclination of 30°. The shifts of the maxima are a function of the source-to-observation distance and they can be up to 30% of the distance. We also present a method of estimating the depths of compact magnetic objects based on the ratio of the AAS of the magnetic anomaly to the AAS of the vertical gradient of the magnetic anomaly. The estimated depths are independent of the magnetization direction. With the help of magnetic anomalies over environmental targets of buried steel drums, we show that the depths can be reliably estimated in most cases. Therefore, the analytic signal approach can be useful in estimating source locations of compact magnetic objects. However, horizontal locations of the targets derived from the maximum values of the AAS must be verified using other techniques.     

Continuous wavelet transform,theoretical aspects and application to aeromagnetic data at the Huanghua Depression,Dagang Oilfield,China

We use the continuous wavelet transform based on complex Morlet wavelets, which has been developed to estimate the source distribution of potential fields. For magnetic anomalies of adjacent sources, they always superimpose upon each other in space and wavenumber, making the identification of magnetic sources problematic. Therefore, a scale normalization factor, a^?n, is introduced on the wavelet coefficients to improve resolution in the scalogram. By theoretical modelling, we set up an approximate linear relationship between the pseudo‐wavenumber and source depth. The influences of background field, random noise and magnetization inclination on the continuous wavelet transform of magnetic anomalies are also discussed and compared with the short‐time Fourier transform results. Synthetic examples indicate that the regional trend has little effect on our method, while the influence of random noise is mainly imposed on shallower sources with higher wavenumbers. The source horizontal position will be affected by the change of magnetization direction, whereas the source depth remains unchanged. After discussing the performance of our method by showing the results of various synthetic tests, we use this method on the aeromagnetic data of the Huanghua depression in central China to define the distribution of volcanic rocks. The spectrum slices in different scales are used to determine horizontal positions of volcanic rocks and their source depths are estimated from the modulus maxima of complex coefficients, which is in good accordance with drilling results.     

Toward a full multiscale approach to interpret potential fields

The way potential fields convey source information depends on the scale at which the field is analysed. In this sense a multiscale analysis is a useful method to study potential fields particularly when the main field contributions are caused by sources with different depths and extents. Our multiscale approach is built with a stable transformation, such as depth from extreme points. Its stability results from mixing, in a single operator, the wavenumber low‐pass behaviour of the upward continuation transformation of the field with the enhancement high‐pass properties of n‐order derivative transformations. So, the complex reciprocal interference of several field components may be efficiently faced at several scales of the analysis and the depth to the sources may be estimated together with the homogeneity degrees of the field. In order to estimate the source boundaries we use another multiscale method, the multiscale derivative analysis, which utilizes a generalized concept of horizontal derivative and produces a set of boundary maps at different scales. We show through synthetic examples and application to the gravity field of Southern Italy that this multiscale behaviour makes this technique quite different from other source boundary estimators. The main result obtained by integrating multiscale derivative analysis with depth from extreme points is the retrieval of rather effective information of the field sources (horizontal boundaries, depth, structural index). This interpretative approach has been used along a specific transect for the analysis of the Bouguer anomaly field of Southern Apennines. It was set at such scales, so to emphasize either regional or local features along the transect. Two different classes of sources were individuated. The first one includes a broad, deep source with lateral size of 45∼50 km, at a depth of 13 km and having a 0.5 structural index. The second class includes several narrower sources located at shallowest depths, ranging from 3–6 km, with lateral size not larger than 5 km and structural indexes ranging from 1–1.5. Within a large‐scale geological framework, these results could help to outline the mean structural features at crustal depths.     

A Least-Squares Standard Deviation Method to Interpret Gravity Data due to Finite Vertical Cylinders and Sheets

We have developed a least-squares approach to determine simultaneously the depth to both the top and base of a buried finite vertical cylinder (vertical line element approximation) and a 2-D vertical thin sheet from moving average residual anomaly profiles obtained from gravity data using filters of successive window lengths. The method involves using a relationship between the depth to the top, and base of the source and a combination of windowed observations. The method is based on computing the standard deviation of the depths to the top, determined from all moving average residual anomalies for each value of the depth to the base. The standard deviation may generally be considered a criterion for determining the correct depth to the top and base of the buried structure. When the correct depth to the base value is used, the standard deviation of the depths to the top is less than the standard deviation using incorrect values of the depth to the base. This method can be applied to residuals as well as to the observed gravity data. The method is applied to synthetic examples with and without random errors and tested on two field examples from the USA and Canada.     

Quantitative Interpretation of Self-Potential Anomalies of Some Simple Geometric Bodies

We have developed a new numerical method to determine the shape (shape factor), depth, polarization angle, and electric dipole moment of a buried structure from residual self-potential (SP) anomalies. The method is based on defining the anomaly value at the origin and four characteristic points and their corresponding distances on the anomaly profile. The problem of shape determination from residual SP anomaly has been transformed into the problem of finding a solution to a nonlinear equation of the form q = f (q). Knowing the shape, the depth, polarization angle and the electric dipole moment are determined individually using three linear equations. Formulas have been derived for spheres and cylinders. By using all possible combinations of the four characteristic points and their corresponding distances, a procedure is developed for automated determination of the best-fit-model parameters of the buried structure from SP anomalies. The method was applied to synthetic data with 5% random errors and tested on a field example from Colorado. In both cases, the model parameters obtained by the present method, particularly the shape and depth of the buried structures are found in good agreement with the actual ones. The present method has the capability of avoiding highly noisy data points and enforcing the incorporation of points of the least random errors to enhance the interpretation results.     

A gradient‐ratio method for the semi‐automatic interpretation of gravity map data sets

The application of semi‐automatic interpretation techniques to potential field data can be of significant assistance to a geophysicist. This paper generalizes the magnetic vertical contact model tilt‐depth method to gravity data using a vertical cylinder and buried sphere models. The method computes the ratio of the vertical to the total horizontal derivative of data and then identifies circular contours within it. Given the radius of the contour and the contour value itself, the depth to the source can be determined. The method is applied both to synthetic and gravity data from South Africa. The Matlab source code can be obtained from the author upon request.     

位场数据解释的Theta-Depth法

Theta图是利用位场(重磁)数据识别边界的常用方法,其表达式为重磁异常水平变化与垂直变化的比值函数.该方法计算浅源地质体边界的效果较好,而由于深源位场数据在换算过程中会产生趋同效应,在深源地质体识别应用中计算结果不准确,为此,本文提出Theta-Depth法并进行地质体埋深的计算.首先给出直接利用Theta图像进行场源体深度估算的方法,然后推导出基于Theta导数的线性方程来自动估算场源位置参数,本文方法可有效地利用Theta图像的特征为约束条件来提高反演结果的精度.理论模型试验证明本文提出的Theta-Depth法能有效地计算出场源体位置和深度.将该方法应用于满都拉地区实测磁数据的解释,帮助圈定了矿脉的分布.     

Refraction traveltime and amplitude corrections for very near- surface inhomogeneities

The performance of refraction inversion methods that employ the principle of refraction migration, whereby traveltimes are laterally migrated by the offset distance (which is the horizontal separation between the point of refraction and the point of detection on the surface), can be adversely affected by very near‐surface inhomogeneities. Even inhomogeneities at single receivers can limit the lateral resolution of detailed seismic velocities in the refractor. The generalized reciprocal method ‘statics’ smoothing method (GRM SSM) is a smoothing rather than a deterministic method for correcting very near‐surface inhomogeneities of limited lateral extent. It is based on the observation that there are only relatively minor differences in the time‐depths to the target refractor computed for a range of XY distances, which is the separation between the reverse and forward traveltimes used to compute the time‐depth. However, any traveltime anomalies, which originate in the near‐surface, migrate laterally with increasing XY distance. Therefore, an average of the time‐depths over a range of XY values preserves the architecture of the refractor, but significantly minimizes the traveltime anomalies originating in the near‐surface. The GRM statics smoothing corrections are obtained by subtracting the average time‐depth values from those computed with a zero XY value. In turn, the corrections are subtracted from the traveltimes, and the GRM algorithms are then re‐applied to the corrected data. Although a single application is generally adequate for most sets of field data, model studies have indicated that several applications of the GRM SSM can be required with severe topographic features, such as escarpments. In addition, very near‐surface inhomogeneities produce anomalous head‐wave amplitudes. An analogous process, using geometric means, can largely correct amplitude anomalies. Furthermore, the coincidence of traveltime and amplitude anomalies indicates that variations in the near‐surface geology, rather than variations in the coupling of the receivers, are a more likely source of the anomalies. The application of the GRM SSM, together with the averaging of the refractor velocity analysis function over a range of XY values, significantly minimizes the generation of artefacts, and facilitates the computation of detailed seismic velocities in the refractor at each receiver. These detailed seismic velocities, together with the GRM SSM‐corrected amplitude products, can facilitate the computation of the ratio of the density in the bedrock to that in the weathered layer. The accuracy of the computed density ratio improves where lateral variations in the seismic velocities in the weathered layer are known.     

Determination of depths to centroids of three-dimensional sources of potential-field anomalies with examples from environmental and geologic applications

A method is developed for determining the depth to the centroid (the geometric center) of ‘semi-compact' sources. The method, called the anomaly attenuation rate (AAR) method, involves computing radial averages of AARs with increasing distances from a range of assumed source centers. For well-isolated magnetic anomalies from ‘semi-compact' sources, the theoretical AARs range from 2 (close to the sources) to 3 (in the far-field region); the corresponding theoretical range of AARs for gravity anomalies is 1 to 2. When the estimated source centroid is incorrect, the AARs either exceed or fall short of the theoretical values. The levelling-off of the far-field AARs near their theoretical maximum values indicates the upper (deeper) bound of the centroid location. Similarly, near-field AARs lower than the theoretical minimum indicate the lower (shallower) bound of the centroid location. It is not always possible to determine usable upper and lower bounds of the centroids because the method depends on characteristics of sources/anomalies and the noise level of the data. For the environmental magnetic examples considered in this study, the determined deeper bounds were within 4% of the true centroid-to-observation distance. For the case of the gravity anomaly from the Bloomfield Pluton, Missouri, USA, determination of only the shallower bound of the centroid location (7 km) was possible. This estimate agrees closely with the centroid of a previously determined three-dimensional model of the Bloomfield Pluton. For satellite magnetic anomalies, the method is appropriate only for high-amplitude, near-circular anomalies due to the inherent low signal-to-noise ratio of satellite magnetic anomalies. Model studies indicate that the AAR method is able to place depths within ±20–30 km of actual center locations from a 400-km observation altitude. Thus, the method may be able to discriminate between upper crustal, lower crustal, and mantle magnetic sources. The results from the prominent Kentucky anomaly are relatively well-resolved (centroid depth 30 km below the Earth's surface). For the Kiruna Magsat anomaly, the deleterious effects from neighboring anomalies make a determination difficult (possible depth could be between 20 and 30 km). The centroid depths are deeper for the Kursk anomaly (40–50 km). These depths may indicate that magnetic anomalies from the near-surface Kursk iron formations (a known contributor) and deep crustal magnetic sources could combine to form the Kursk Magsat anomaly.     

Mapping basement structures in the northwestern offshore of Abu Dhabi from high‐resolution aeromagnetic data

This paper presents a case study of mapping basement structures in the northwestern offshore of Abu Dhabi using high‐resolution aeromagnetic data. Lineament analysis was carried out on the derivatives of the reduced‐to‐the‐pole magnetic data, along with supporting information from published geologic data. The lineament analysis suggests three well‐defined basement trends in the north–south, northeast–southwest, and northwest–southeast directions. The reduced‐to‐the‐pole magnetic data reveal high positive magnetic anomalies hypothesized to be related to intra‐basement bodies in the deep seated Arabian Shield. Depth to basement was estimated using spectral analysis and Source Parameter Imaging techniques. The spectral analysis suggests that the intruded basement blocks are at the same average depth level (around 8.5 km). The estimated Source Parameter Imaging depths from gridded reduced‐to‐the‐pole data are ranged between 4 km and 12 km with a large depth variation within small distances. These estimated depths prevent a reliable interpretation of the nature of the basement relief. However, low‐pass filtering of the horizontal local wavenumber data across two profiles shows that the basement terrain is characterized by a basin‐like structure trending in the northeast–southwest direction with a maximum depth of 10 km. Two‐dimensional forward magnetic modelling across the two profiles suggests that the high positive magnetic anomalies over the basin could be produced by intrusion of mafic igneous rocks with high susceptibility values (0.008 to 0.016 SI.     

A CONTROL OF TWO-DIMENSIONAL MAGNETIC INTERPRETATION BY THREE-DIMENSIONAL MODEL BODY ANOMALIES*

In magnetic routine interpretation the comparison of two-dimensional model curves with measured magnetic anomalies is widely used for an approximate evaluation of the position and depth of magnetic models. Before starting an interpretation of a survey by means of two-dimensional models, it is very useful to have an idea of the shape of anomalies caused by extended but finite bodies, taking into account various strike directions: Three sets of anomalies of thin plates (horizontal length 19, downward length 9, width 1) dipping 30°, 60°, and 90° resp. for various strike directions and an inclination of 20° were computed. Some of these anomalies, e.g. those with nearly N-S strike direction look rather complicated, and at the first glance one would not expect that they are caused by such simple bodies. Several profiles crossing the computed anomalies perpendicularly were interpreted two-dimensionally. For less extended anomalies the depths determined for the top of the plates are 10-20% too small, the magnetization amounts to 50–75 % of the value of the finite bodies. The interpretation of the profiles covering more extended anomalies gave very accurately the same values for the position, depth and magnetization for the two-dimensional body as for the original three-dimensional model. Anomalies of vertical prisms with varying extensions in the y-direction were computed. Their differences in amplitude and in the distance maximum-minimum show that interpretation of short anomalies by two-dimensional methods yields depth errors of up to 20 percent. To see the possibilities of the separation of superimposed anomalies dike anomalies were added to the anomaly of a broad body in great depth and several attempts were made to interpret parts of the composite anomalies. The interpreted bodies lie too deep. In complicated cases the depth values can have large errors, but experienced interpreters should be able to keep the errors in the range of one third of the depth values.     

Gravity and magnetic field characteristics and hydrocarbon prospects of the Tobago Basin

The Tobago Basin, which is located offshore northern Venezuela with a southern margin close to Trinidad and Tobago, has an area of approximately 59,600 km². The Tobago Basin has relatively favourable hydrocarbon prospects, and to date, exploration work has mainly concentrated on small areas of the southwestern portion of the basin. To conduct a comprehensive study of the structural framework of the basin and the characteristics of the basement in order to identify prospective zones for hydrocarbon exploration, shipborne‐measured and satellite‐measured gravity data, shipborne‐measured magnetic data, and aeromagnetic survey data were analysed. A regularisation filtering method was used to separate and obtain regional and residual gravity and magnetic anomalies. Directional gradients of gravity and magnetic anomalies and the total horizontal gradient and vertical second derivative of gravity anomalies were employed to extract information about fault structures. Regression analysis methods were used to determine the basement depth. The geological significance of the gravity and magnetic fields was examined, the structural framework of the basin was assessed, the basement depth was estimated, and favourable hydrocarbon exploration prospects within the basin were identified. The results show that the Tobago Basin contains complex structures consisting mainly of two groups of faults trending in northeasterly and northwesterly directions and that the major northeasterly trending faults control the main structural configuration and depositional system within the basin. The basement of the Tobago Basin has deep rises and falls. It can be divided into the following four secondary tectonic units: the western sub‐basin, the central uplift area, the southern sub‐basin, and the northeastern sub‐basin. The central uplift area and northeastern sub‐basin are most likely to have developed hydrocarbon accumulations and should be targeted for further exploration.     

A Least-squares Approach to Depth Determination from Numerical Horizontal Self-potential Gradients

We have developed a least-squares minimization approach to depth determination of a buried ore deposit from numerical horizontal gradients obtained from self-potential (SP) data using filters of successive window lengths (graticule spacings). The problem of depth determination from SP gradients has been transformed into the problem of finding a solution to a nonlinear equation of the form f(z)=0. Formulas have been derived for vertical and horizontal cylinders and spheres. Procedures are also formulated to estimate the electrical dipole moment and the polarization angle. The method is applied to synthetic data with and without random noise. Finally, the validity of the method is tested on two field examples. In both cases, the depth obtained is found to be in a very good agreement with that obtained from drilling information.     

A Least-squares Window Curves Method for Interpretation of Magnetic Anomalies Caused by Dipping Dikes

We have developed a least-squares method to determine simultaneously the depth and the width of a buried thick dipping dike from residualized magnetic data using filters of successive window lengths. The method involves using a relationship between the depth and the half-width of the source and a combination of windowed observations. The relationship represents a family of curves (window curves). For a fixed window length, the depth is determined for each half-width value by solving one nonlinear equation of the form f (z) = 0 using the least-squares method. The computed depths are plotted against the width values representing a continuous curve. The solution for the depth and the width of the buried dike is read at the common intersection of the window curves. The method involves using a dike model convolved with the same moving average filter as applied to the observed data. As a result, this method can be applied to residuals as well as to measured magnetic data. Procedures are also formulated to estimate the amplitude coefficient and the index parameter. The method is applied to theoretical data with and without random errors. The validity of the method is tested on airborne magnetic data from Canada and on a vertical component magnetic anomaly from Turkey. In all cases examined, the model parameters obtained are in good agreement with the actual ones and with those given in the published literature.     

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.     

A New Best-Estimate Methodology for Determining Magnetic Parameters Related to Field Anomalies Produced by Buried Thin Dikes and Horizontal Cylinder-like Structures

A new best estimate methodology is proposed and oriented towards the determination of parameters related to a magnetic field anomaly produced by a simple geometric-shaped model or body such as a thin dike and horizontal cylinder. This approach is mainly based on solving a system of algebraic linear equations for estimating the three model parameters, e.g., the depth to the top (center) of the body (z), the index parameter or the effective magnetization angle (θ) and the amplitude coefficient or the effective magnetization intensity (k). The utility and validity of this method is demonstrated by analyzing two synthetic magnetic anomalies, using simulated data generated from a known model with different random errors components and a known statistical distribution. This approach was also examined and applied to two real field magnetic anomalies from the United States and Brazil. The agreement between the results obtained by the proposed method and those obtained by other interpretation methods is good and comparable. Moreover, the depth obtained by such an approach is found to be in high accordance with that obtained from drilling information. The advantages of such a proposed method over other existing interpretative techniques are clarified, where it can be generalized to be automatically applicable for interpreting other geological structures described by mathematical formulations.