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Relations between potential fields and some equivalent sources
Abstract
The concept of equivalent sources is described from first principles. Two types of sources that produce the same magnetic field are studied in detail: thin sheets and uniaxially magnetized half-spaces. Simple analytic solutions show that their magnetization variations are essentially given by the pseudogravity field and the magnetic field, respectively. While the former varies slowly to account for the slow variation in the vertically integrated magnetization, the latter varies rapidly to account for the spatial variation in rock magnetization close to the observation points. An extension of these simple distributions is the sandwich distribution. Using statistical arguments, we construct sandwich distributions. The depths and mean magnetizations of each layer are found from the decay of the azimuthally averaged power spectrum. The equivalent layer magnetizations closely resemble the magnetic field anomaly when upward-continued to a height equal to the mean depth of the layer.
... Few authors in the geophysical exploration literature have addressed the use of the equivalent-layer technique for interpreting geologic sources. Pedersen (1991), for example, discusses the relationship between the potential field and the equivalent source. Medeiros and Silva (1996) and Silva et al. (2010) estimate an apparent-magnetization map on a layer by using Tikhonov and entropic regularizations, respectively. ...
... Hence, pðx 0 0 ; y 0 0 ; z c Þ is strictly positive at all points on the equivalent layer as well. This relation is similar to that presented by Pedersen (1991) and . They determine, in the wavenumber domain, the magnetic-moment distribution within a continuous equivalent layer vertically magnetized by induction. ...
... They also consider a planar equivalent layer located below and parallel to a horizontal plane containing the observed total-field anomaly data. Under these assumptions, Pedersen (1991) and conclude that the magnetic-moment distribution within the continuous equivalent layer is all positive and proportional to the pseudogravity anomaly produced by the source on the plane of the equivalent layer. Here, we do not follow the same wavenumber-domain reasoning used by those authors. ...
... "Sandwich model" method Pedersen (1991) uses Wiener filters to estimate magnetization distributions within a layered or "sandwich" model introduced by Jacobsen (1987). The sandwich model consists of a series of layers in which magnetization varies only laterally, producing a magnetic field y with Fourier transform given by ...
... Here, h i is the depth to the top of layer i, d is the constant layer thickness, x i is the i th layer magnetization, and VðkÞ is defined in equation 4. Assuming no correlation between the magnetizations in the different layers, Pedersen (1991) constructs a Wiener filter to determine the magnetization in a given layer i as ...
... The equivalence between the two different approaches is not surprising, considering the close connection between Wiener filter theory and inverse theory (Treitel and Lines, 1982). Pedersen (1991) extends his approach to allow varying with depth the expected value of magnetization power in the layers. Equation 8 then becomes, again assuming uncorrelated magnetizations, ...
Several noniterative, imaging methods for potential field data have been proposed that provide an estimate of the 3D magnetization/density distribution within the subsurface or that produce images of quantities related or proportional to such distributions. They have been derived in various ways, using generalized linear inversion, Wiener filtering, wavelet and depth from extreme points (DEXP) transformations, crosscorrelation, and migration. We demonstrated that the resulting images from each of these approaches are equivalent to an upward continuation of the data, weighted by a (possibly) depth-dependent function. Source distributions or related quantities imaged by all of these methods are smeared, diffuse versions of the true distributions; but owing to the stability of upward continuation, resolution may be substantially increased by coupling derivative and upward continuation operators. These imaging techniques appeared most effective in the case of isolated, compact, and depth-limited sources. Because all the approaches were noniterative, computationally fast, and in some cases, produced a fit to the data, they did provide a quick, but approximate picture of physical property distributions. We have found that inherent or explicit depth-weighting is necessary to image sources at their correct depths, and that the best scaling law or weighting function has to be physically based, for instance, using the theory of homogeneous fields. A major advantage of these techniques was their speed, efficiently providing a basis for further detailed, follow-up modelling.
... Poucos autores tem direcionado o uso da camada equivalente para a interpretação de fontes geológicas. PEDERSEN (1991) Nesta tese, apresentamos dois desenvolvimentos teóricos e suas aplicações na interpretação de dados magnéticos usando a técnica da camada equivalente. O primeiro desenvolvimento teóricoé fundamentado nas leis de Gauss para campo magnéticos e de Ampère. ...
... Consequentemente, a função p(x , y , z c ) tambémé positiva em todos os pontos sobre a camada. Esta relaçãó e similaràquela apresentada por PEDERSEN (1991) e LI et al. (2014. Aqueles autores determinaram, por meio de uma abordagem desenvolvida no domínio de Fourier, uma distribuição de intensidades de momento magnético positiva sobre uma camada equivalente contínua e com magnetização induzida na direção vertical. ...
In this thesis, I present two theoretical results and the applications of the equivalent layer for processing and interpreting magnetic data.
In the first one, I show that there is a unique planar and continuous layer of dipoles, with a given uniform magnetization direction, that is able to reproduce, simultaneously, the three components of the magnetic induction field produced by an arbitrary set of sources. This property holds true regardless of whether the magnetization direction of the layer is equal to the that of the sources or not. From this theoretical result, I show that it is possible to use a planar layer of dipoles with uniform and arbitrary magnetization direction to estimate the three components of the magnetic induction field produced by an arbitrary set of sources via linear inversion of single component data.
Results with synthetic data produced by numerical simulations and real data obtained on a rock sample from the Vredefort impact crater, South Africa, show the utility of the method in the processing of magnetic microscopy data and identification of regions with largest
concentrations of magnetic minerals. In the second theoretical development presented in this thesis, I show that the magnetic
moment intensity distribution on a planar and continuous layer of dipoles is all positive if the uniform magnetization direction of the layer is equal to that of the true sources. Using this positivity property, I present an iterative method for estimating the uniform magnetization direction of a set of 3D sources by inverting total-field anomaly data.
At each iteration, the method solves a linear inverse problem to estimate a positive magnetic moment intensity distribution and a non-linear inverse problem to estimate the magnetization direction on a planar layer of dipoles. At the end of the iterative process, the uniform
magnetization direction of the equivalent sources approximates that of the true sources. Tests with data produced by models simulating different geological scenarios show that the method can be a powerful tool for estimating the uniform magnetization direction of a set of geological sources. Applications to airborne data over the Montes Claros de Goiás complex, located in the Goiás Alkaline Province, central region of Brazil, suggest that those intrusions have a strong
remanent magnetization, in agreement with a previous independent study in the same area.
... Avvikelsen från antagandet om variation inom beräkningsnivåerna försämrar naturligtvis upplösningen på de större djupen. I det magnetiska fallet finns det publicerat en tumregel om att uppräkningsnivåerna skall halveras för att få det verkliga djupet som de magnetiska orsakerna kan härledas till (Pedersen 1991). För tyngdkraftsdata finns ej motsvarande analys gjord ännu. ...
... Vattenförande sprickzoner kan lokaliseras genom att kombinera magnetiska och elektromagnetiska anomalikartor (Jacobsen 1987, Pedersen 1991. Den magnetiska anomalibilden är visualiserad i en figur av gråskalor ( fig. ...
The geological map of the county of Jönköping covers parts of two different geological provinces: the Transscandinavian Igneous Belt (TIB) and the Eastern Segment of the Sveconorwegian Province. The Protogine Zone, a prominent N–S trending shear zone, is a divider between the two areas and represents the easternmost part of the Sveconorwegian Province. (In Swedish with summary in English)
... 3-D imaging is an alternative tool for producing the equivalent distribution of density in the subsurface or estimating the depth, dip or edges of sources. Cribb [16] and Pedersen [17] proposed the generalized inversion approach in the wavenumber domain for potential field data, the result of which is an equivalent distribution of density that is proportional to the upward continuation of the vertical derivative of gravity field. Patella [18], Mauriello and Patella [19], and Guo et al. [20] presented the 3-D probability tomography or correlation imaging approaches for producing the equivalent density distribution describing the probability of occurrence of a pole mass. ...
... The above wavenumber-domain imaging equation of Kobrunov and Varfolomeev [24] (Equation (6)) is similar to those presented by Pedersen [17] and Fedi and Pilkington [30]. ...
We present a wavenumber-domain iterative approach for rapid 3-D imaging of gravity anomalies and gradients data, which is based on 3-D mesh model with a flat observational surface. The approach deconvolves the spectra of gravity anomalies or gradients by a 2-D deconvolution filter describing the spectrum of imaging operator, and then transforms the resultant spectra into the space domain to derive the density distribution. This 2-D deconvolution filtering is operated layer by layer from top to bottom in the subsurface and finally all the results are merged to generate the 3-D density distribution. We improve previous 2-D deconvolution filters by involving a depth-scaling factor and utilize a priori constraint and the iteration algorithm for imaging, enable the presented approach to produce a density model with a considerable resolution and accuracy. The wavenumber-domain algorithm makes the imaging faster than the conventional space-domain inversions. Tests on the synthetic data and the real data from a metallic deposit area in Northwest China verified the feasibility and high efficiency of the presented approach.
... Due to the inherent nonuniqueness of potential field methods, any given set of measured magnetic data can be reproduced by (infinitely) many magnetization distributions that may have very diverse characteristics. Therefore, based solely on the data, no earth model can be selected as representative with any level of confidence (Pedersen, 1991;Blakely, 1996). In addition, the number of data points is usually smaller than the number of model parameters, and, hence, the problem is underdetermined (Menke, 2012). ...
... We believe that this behavior is a result of the regularization term in equation 10, whereby the norm of the solution was minimized. With a fixed magnetization direction, small-scale positive and negative magnetization structures would be produced (Pedersen, 1991). Figure 11b displays the amplitude of the magnetization for the model resulting from a 3D inversion with the magnetization direction fixed along the inducing field direction. ...
The inherent non-uniqueness in modeling magnetic data can be partly reduced by adding prior information, either as mathematical constructs or simply as bounds on magnetization obtained from laboratory measurements. If a good prior model can be used as a reference model, then the quality of estimated models through an inverse approach
can be greatly improved. But even though data on magnetic properties of rocks might exist, their distribution may often be quite irregular on both local and regional scales so that it is difficult to define representative classes of rock types suitable for constraining geophysical models of magnetization. Here, we introduce a new way of constructing a reference model that varies only laterally and is confined to the part of the terrain that lies above the lowest topography in the area. To obtain this model, several estimated 2D magnetization distributions were constructed by data inversion as a function of iteration number. Then, a suitable 2D model of the topography was chosen as a starting point for constructing a 3D reference model by modifying it with a vertical decay such that its average source depth was the same for all horizontal positions. The average source depth of the reference model was chosen to satisfy the average source depth obtained from analyzing the radial power spectrum of the area studied. Finally the measured magnetic data was inverted in 3D using the given reference model. For a selected reference model, shallow structures showed a better overall correlation with large remanent magnetizations measured on rock samples from the area. Throughout the entire model, the direction of magnetization was allowed to vary freely. We show that the Euclidean norm of the estimated model was reduced compared with the case where the magnetization direction was fixed.
... Many multiscale methods have been developed for potential field interpretation. This class of methods is mainly based on upward continuation of potential fields (Pedersen, 1991;Sailhac and Gibert, 2003;Fedi and Florio, 2006;Fedi, 2007;Florio et al., 2009;Fedi et al., 2009Fedi and Abbas, 2013;Abbas et al., 2014;Baniamerian et al., 2016). Recently Fedi and Pilkington (2012) have demonstrated that all imaging methods, e.g., Sandwich model (Pedersen, 1991), correlation (Patella, 1997), migration (Zhdanov, 2002), wavelet transform (Moreau et al., 1997) and depth from extreme points (DEXP; Fedi, 2007) are, in practice, multiscale methods and involve upward continuation or a simultaneous application of upward continuation and differentiation to potential fields. ...
... This class of methods is mainly based on upward continuation of potential fields (Pedersen, 1991;Sailhac and Gibert, 2003;Fedi and Florio, 2006;Fedi, 2007;Florio et al., 2009;Fedi et al., 2009Fedi and Abbas, 2013;Abbas et al., 2014;Baniamerian et al., 2016). Recently Fedi and Pilkington (2012) have demonstrated that all imaging methods, e.g., Sandwich model (Pedersen, 1991), correlation (Patella, 1997), migration (Zhdanov, 2002), wavelet transform (Moreau et al., 1997) and depth from extreme points (DEXP; Fedi, 2007) are, in practice, multiscale methods and involve upward continuation or a simultaneous application of upward continuation and differentiation to potential fields. Therefore, when using imaging methods it is very important to perform the whole procedure as accurately as possible and, in particular, reduce noise/edge effects as well. ...
Imaging of potential fields yields a fast 3D representation of the source distribution of potential fields. Imaging methods are all based on multiscale methods allowing the source parameters of potential fields to be estimated from a simultaneous analysis of the field at various scales or, in other words, at many altitudes. Accuracy in performing upward continuation and differentiation of the field has therefore a key role for this class of methods. We here describe an accurate method for performing upward continuation and vertical differentiation in the space-domain. We perform a direct discretization of the integral equations for upward continuation and Hilbert transform; from these equations we then define matrix operators performing the transformation, which are symmetric (upward continuation) or anti-symmetric (differentiation), respectively. Thanks to these properties, just the first row of the matrices needs to be computed, so to decrease dramatically the computation cost. Our approach allows a simple procedure, with the advantage of not involving large data extension or tapering, as due instead in case of Fourier domain computation. It also allows level-to-drape upward continuation and a stable differentiation at high frequencies; finally, upward continuation and differentiation kernels may be merged into a single kernel. The accuracy of our approach is shown to be important for multi-scale algorithms, such as the continuous wavelet transform or the DEXP (depth from extreme point method), because border errors, which tend to propagate largely at the largest scales, are radically reduced. The application of our algorithm to synthetic and real-case gravity and magnetic data sets confirms the accuracy of our space domain strategy over FFT algorithms and standard convolution procedures.
... For instance, it can be proved that there is an upper bound but no lower bound for the vertical location of a structure and only if there is an upper bound for the contrast in the model (Parker, 1974(Parker, , 1975. Infinitely, many shallower structures explain the data equally well, including such pathological solutions as equivalent thin sheets (Pedersen, 1991). A highly variable sensitivity does not cause any problems in itself. ...
... If the near-surface part of the anomalous body had been disregarded, the determination of its lower boundary would have been equivalent to the problem of determining the topography of a uniaxially magnetized half-space with a varying upper burial surface. Once the magnetization contrast is specified, the interface is uniquely determined from perfect (i.e., all wavenumbers available, and noise free) measurements (Pedersen, 1991). However, in this synthetic example, we assumed that we only have knowledge of a certain interval within which the contrasts were allowed to vary. ...
ABSTRACT
We have studied a gabbro intrusion in northern Sweden, using
3D inversion of airborne magnetic data, ground-based gravity
data, and petrophysical measurements on outcrop samples. Gabbro
intrusions are of interest because they are potential hosts of
Cu-Ni and platinum group element mineralization. We developed
a joint inversion algorithm and applied it to both potential-field
data sets to obtain spatial distributions of density and magnetic
susceptibility. The distributions were coupled through a nonrigidly
enforced parameter relationship determined from the petrophysical
samples. We managed the problem of balancing the
influence of the two data sets by a novel adaptive reweighting
scheme that enforced the discrepancy principle for each data set
independently. We demonstrated in tests with synthetic data that
neither individual nor joint inversions gave reliable estimates for
the depth extension of the intrusive body, the near-surface details,
or any complex geometrical features. However, the joint inversion
improved the image of the interface between the intrusion and the
surrounding rocks and revealed that the density and susceptibility
models satisfied the observed petrophysical relationship, which, in
turn, caused the structures in the models to align. The geometry of
the intrusion was an intrinsic result of the inversion, based on the
two distinct petrophysical trends for the gabbro and the surrounding
rocks. The inferred shape was simple and concise, and was
therefore a useful and testable hypothesis about the subsurface
geology that was in agreement with both potential-field data sets
and the petrophysical information.
... Apart from having most of the solution concentrated near the surface, the generalized inverse also leads to negative values which are physically unrealistic. Pedersen (1991) derived similar expressions for the magnetization distribution in a layered model and noted that, for the data set examined, solutions became smoother and larger in amplitude with increasing depth. He also discussed weighting the layers differently but did not outline ways of determining the weighting factors. ...
... The exponent in z results from the equivalence in behavior between a block-like source and a simple dipole. This kind of weighting function can be included easily in the frequency domain expressions of Cribb (1978) and Pedersen (1991) for the inversion of magnetic data in terms of layered models. Equation (1) can be written ...
A 3-D inversion approach is outlined that determines a distribution of susceptibility that produces a given magnetic anomaly The subsurface model consists of a 3-D array of rectangular blocks, each with a constant susceptibility, The inversion incorporates a model norm that allows smoothing and depth-weighting of the solution. Since the number of parameters can be many thousands, even for small problems, the linear system of equations is inverted using a preconditioned conjugate gradient approach. This reduces memory requirements and avoids large matrix multiplications, The method is used to determine the 3-D susceptibility distribution responsible for the Temagami magnetic anomaly in southern Ontario, Canada.
... Recently, alternative approaches have been introduced for fast imaging of subsurface physical properties, such as the sandwich model (Pedersen, 1991), the continuous wavelet transform of potential field data (Hornby et al., 1999), correlation imaging of potential field data (Mauriello & Patella, 2001), migration of potential field data (Zhdanov, 2002) and the depth from extreme points (DEXP) method (Fedi, 2007). Migration of potential fields is a rapid imaging tool to map an anomalous target by transforming the measured data into a 3D density or susceptibility image. ...
Gravity and magnetic data contain information about the density and susceptibility distributions of subsurface rocks and minerals. The physical property distributions are determined by inversion or iterative migration of potential field data. The information in the gravity and magnetic data can complete each other. Therefore, joint interpretation of potential field data can reduce the ambiguity and uncertainty of the information. The iterative migration method is a fast algorithm to recover physical property models from potential field data. This paper introduces a novel joint iterative migration with a focusing stabilizer for potential field data. The joint migration with the focusing stabilizer enforces the structural correlation between migration models because physical properties change rapidly in the same areas. The proposed method is applied to the migration modeling of potential field data from two synthetic examples, and real data from the San Nicolas massive sulfide deposit in Mexico. The joint migration models are more focused and enhance the resolved edges of the geological structures.
... Cribb (1976) obtained the same results as unconstrained inversion in the spatial domain (leastsquares solution of linear matrix) by fi ltering according to the general theoretical framework of potential-field linear inversion. Pedersen (1991) proposed using a Winner fi lter to estimate the magnetic distribution. The fi lter can be simplifi ed to the Cribb form. ...
The practical application of 3D inversion of gravity data requires a lot of computation time and storage space. To solve this problem, we present an integrated optimization algorithm with the following components: (1) targeting high accuracy in the space domain and fast computation in the wavenumber domain, we design a fast 3D forward algorithm with high precision; and (2) taking advantage of the symmetry of the inversion matrix, the main calculation in gravity conjugate gradient inversion is decomposed into two forward calculations, thus optimizing the computational efficiency of 3D gravity inversion. We verify the calculation accuracy and efficiency of the optimization algorithm by testing various grid-number models through numerical simulation experiments.
... Vattenförande sprickzoner sammanfaller ofta lokaliserade till dalar med glaciala och postglaciala avsättningar. de kan lokaliseras genom att kombinera magnetiska och elektromagnetiska anomalikartor (Jacobsen 1987, Pedersen 1991. den magnetiska anomalibilden är visualiserad i en gråskalefigur, figur 24, där kartbilden är optimerad för att redovisa de zoner som är presumtiva, vattenförande krosszoner. ...
The geological map of the county of Kalmar (Kalmar län) can be divided into four main areas, the Svecofennian area, the Västervik area, the Transscandinavian Igneous Belt (TIB) area, and the Phanerozoic area. (In Swedish with English summary)
... Kobrunov & Varfolomeev (1981) presented a wavenumber-domain deconvolution approach for imaging of the gravity anomaly. Pedersen (1991) analyzed the power spectrum of magnetic field data to estimate magnetization distributions in the wavenumber domain. Pilkington (1997) used Fast Fourier Transform (FFT) to raise the calculation efficiency during his presented imaging approach. ...
Three-dimensional magnetic inversion, based on the least-square and regularization algorithm in the space domain, is an important tool for quantitative interpretation of magnetic data. However, the common 3D inversion approaches usually require great numbers of forward and inversion calculations and cause low efficiency for inverting large-scale data. Three-dimensional imaging is an alternate rapid tool for qualitative and quantitative interpretation of magnetic data. In this paper, we present a wavenumber-domain iterative approach for 3D imaging of magnetic anomalies and gradients, which could increase imaging efficiency and is suitable for rapidly imaging large-scale data. The wavenumber-domain formulas for forward modeling and imaging of total magnetic anomaly, three magnetic components, magnetic gradients and magnetic full-tensor gradients are deduced and provided. A depth-scale factor and the constraints of magnetic interface are included into the imaging formulas to enhance depth resolution. An iterative algorithm is adopted for the imaging to reduce the fitting error and improve the imaging accuracy. Tests on synthetic and real data from the Sichuan basin, China, verified the feasibility of the presented approaches.
... Received 6 June 2018; Received in revised form 11 October 2018; Accepted 7 November 2018 multiplication notation using Fourier Transform (Bhattacharyya, 1965;Spector and Grant, 1970;Garcia-Abdeslem and Ness, 1994;Salem et al., 2014). Spector and Grant (1970) method has become an important technique for top depth estimation of anomalous sources in frequency/ wavenumber domain (Shuey et al., 1977;Connard et al., 1983;Okubu et al., 1985;Blakely, 1988;Pedersen, 1991;Garcia-Abdeslem and Ness, 1994;Okubu and Matsunaga, 1994;Tanaka et al., 1999). Spector and Grant (1970) concept is based on an ensemble of prisms of frequencyindependent randomly and uncorrelated distribution of magnetic sources equivalent to white noise distribution. ...
The high-resolution aeromagnetic data of parts of the lower and middle Benue Trough (Nigeria) has been analyzed to estimate the magnetic basement depths using the scaling spectral method. Magnetic basement depths beneath the area of study are found dominantly between 3 km and 5 km with the 3 km depths corresponding to the areas of Ogam/Obangedde, Ugba and Makurdi whereas the 5 km depths are found around the areas of Abakaliki, Gboko, Igumale, Oturkpo, Gbajimba and Arufu. Shallower magnetic basement depths of 1 km are estimated around Ebeel and the area of Ugba and 2 km depths from the areas of Akwana and Otukpa. We have also calculated few higher values of 6 km around the areas of Igbor, near Igumale and the area north of Oturkpo, 7 km depths at Onuweyi and Ikumbur areas and 9 km depths near Makurdi, Yandev area and east of Ogoja area. Our estimated basement depths are in good agreement with the results of the earlier studies in the region. The obtained scaling exponent values are found between 0 and 3 and are not correlated everywhere with the estimated depths and the geology of the region. That may be because of the effects of adjacent geological units and the presence of the volcanic intrusions within the region.
... The use of equivalent sources (Pedersen, 1991) and other techniques discussed by Fedi and Pilkington (2012) could yield a magnetisation distribution that can reproduce the observed field; however, the question of filtering topography from buried sources would still need to be solved. Bansal and Dimri (2014) present an algorithm based on scaling behaviour of rocks to interpret data at different depths. ...
Airborne magnetic surveys over rugged topography generate noticeable magnetic signatures that are likely
equivalent in amplitude with the signal of geological interest. Synthetic models demonstrate that this magnetic terrain
effect has the same wavelength as the topography and cannot be overcome by drape flying. The magnetic terrain effect
amplifies negative and positive magnetic signals over steep valleys and ridges, respectively. These magnetic artefacts
may induce incorrect geological interpretation of magnetic features. In order to remove these spurious signals, we develop
a semiquantitative methodology based on 3D magnetic modelling of the topographic effect. Observed total magnetic field
is then corrected by subtracting the synthetic field related to the topographic effect. The key element in this approach is the
appropriate estimate of the magnetisation associated with the topography, which is especially difficult to determine in areas
characterised by rugged terrain. We estimate the magnetic signal related to the topographic effect by: (1) filtering the
magnetic data based on the wavelength band in which magnetic and topographic data show maximum coherency,
(2) inverting the filtered magnetic data in order to obtain a model of the magnetic susceptibility distribution associated
with the topography, and finally (3) calculating the magnetic signal response of the topography-related susceptibility
model. We successfully tested this approach in the Río Blanco-Los Bronces and El Teniente porphyry copper districts
(Andes of Central Chile), which are characterised by rugged topography and the presence of highly magnetised volcanic
rocks. Validation is achieved by comparing the magnetic response over zones with a good geological and petrophysical
knowledge. In these examples, the topography-corrected magnetic data show the distribution of geological units and
susceptibility better than the non-corrected magnetic data.
... The general model takes into account the shape of the observed spectrum and allows for a general multilayer case and considers the effects of sources at a number of distinct spectral/depth levels. The physical significance of the number of layers and their thicknesses remains questionable given the equivalence inherent in such a modelling procedure (Pedersen, 1991). ...
The deep crustal magnetic structure of Britain has not previously been described in a uniform manner. We provide a new assessment of the deep crustal magnetic bodies responsible for the long wavelength magnetic features. The study area contains deep crustal relics of the destruction of early Palaeozoic oceanic lithosphere along the Thor-Tornquist Suture and primarily the Iapetus Suture separating Baltica and Avalonia from the Laurentian terranes. Spectral decomposition is applied to a merged onshore and offshore magnetic anomaly data set. Thirty idealised basement bodies are compared with a representation of the subsurface obtained by a coarse 3D inversion of the data. The central area separating Laurentia and Avalonia, is largely characterised by an absence of high susceptibilities throughout the whole crustal volume. We find that the idealised basement bodies are largely consistent with relatively high susceptibility zones at depths in excess of 10 km. The zones of higher relative susceptibility are referenced to the tectonic-terrane framework of the area and possible geological explanations for the contrasts are reviewed. In the north, the Laurentian terranes are diverse, comprising crust first created in the Archaean (Hebridean Terrane), Palaeoproterozoic (Rhinns Terrane), Mesoproterozoic? (Midland Valley Terrane), Neoproterozoic (sub-Southern Upland rocks) and Ordovician. Magnetic anomalies further record the assembly of the Gondwanan (Eastern Avalonian) part of the country through Neoproterozoic and Ordovician (Tornquist) arc magmatism and accretion. The convergence zones between Laurentia, Avalonia and Baltica have all left a magnetic imprint, as has Variscan convergence to the south.
... (2), has been also retrieved for other different source geometries (Naidu, 1972;Hahn et al., 1976;Pedersen, 1991;Pawlowski, 1994;Garcia-Abdeslem and Ness, 1994). Then, Maus and Dimri (1995) improved the Spector and Grant approach by developing an inversion method that is able to estimate the depth of magnetic and gravity anomaly sources and to take into account the scaling behavior of the corresponding geophysical parameters, i.e. magnetic susceptibility and density. ...
Spectral analysis of the self-potential (SP) field for geometrically simple anomalous bodies is studied. In particular, three spectral techniques, i.e. Periodogram (PM), Multi Taper (MTM) and Maximum Entropy (MEM) methods, are proposed to derive the depth of the anomalous bodies. An extensive numerical analysis at varying the source parameters outlines that MEM is successful in determining the source depth with a percent error less than 5%. The application of the proposed spectral approach to the interpretation of field datasets has provided depth estimations of the SP anomaly sources in very good agreement with those obtained by other numerical methods.
... This strong decay of the amplitude of the magnetic field with increasing elevation also suggests that the demagnetized region caused by hydrothermal alteration is relatively shallow and thin. The distance at which the magnetic field decays is roughly of the same order as the depth interval in which the magnetic source is confined [Pedersen, 1991;Tivey, 1994;Gee et al., 2001]. This can be seen in the 3-D magnetization model shown in Figure 4, where the depth range of the demagnetized body is~100 m. ...
High-resolution geophysical data have been collected using the Autonomous Underwater Vehicle (AUV) Sentry over the ASHES (Axial Seamount Hydrothermal Emission Study) high-temperature (~348°C) vent field at Axial Seamount, on the Juan de Fuca Ridge. Multiple surveys were performed on a 3-D grid at different altitudes above the seafloor, providing an unprecedented view of magnetic data resolution as a function of altitude above the seafloor. Magnetic data derived near the seafloor show that the ASHES field is characterized by a zone of low magnetization, which can be explained by hydrothermal alteration of the host volcanic rocks. Surface manifestations of hydrothermal activity at the ASHES vent field are likely controlled by a combination of local faults and fractures and different lava morphologies near the seafloor. Three-dimensional inversion of the magnetic data provides evidence of a vertical, pipe-like upflow zone of the hydrothermal fluids with a vertical extent of ~100m.
... The difference in depth of investigation between the magnetic method and the different EM apparatus is thus a main parameter to consider when simultaneously inverting EM and magnetic data. The equivalent stratum problem is well known in magnetism: the magnetic response of a feature at a given depth is equivalent to the response of a thin sheetlike feature at a shallower depth (Pedersen, 1991). The reciprocal effect of this equivalence can be a problem in the interpretation of the filtered EM data if a true susceptibility variation inside the near-surface layer is interpreted as generated by a deeper feature. ...
... In the absence of heat flow data, the two magnetic profiles (Fig. 2) could be used to compute the depth to the Curie isotherm. The depth to the deepest interface was calculated both from the conventional power spectrum (Treitel et al., 1971;Pederson, 1991) and from maximum likelihood methods (Fisher, 1958;Davies, 1982). The latter is more useful because of its limited subjectivity. ...
Ocean bottom magnetometer (OBM) investigations were made across the Ninetyeast Ridge in the Bay of Bengal and geomagnetic field variations were recorded for a period of about one month. Analysis of data indicates that the thickness as well as resistivity of the oceanic crust/upper mantle reduces from 17 - 14 km and 439 - 308 Ohm-m respectively, as one moves from the Bay of Bengal to the Andaman arc region. The highly conducting Andaman arc region may have formed due to subduction related processes at the Andaman trench. Analysis of OBM data collected earlier across the Barren Island in the Andaman Sea indicates the presence of two conductivity zones, at 17 - 27 km and 80 - 100 km depth. The enhanced conductivity at a shallow depth may be due to the conducting material/magma that has been emplaced by upwelling of the mantle material at the tectonic zone. The thickness of this material increases from north to south of the Barren Island and seems to concentrate the induced currents within the island. The enhanced conductivity at greater depths may be due to the H2O and CO2 that may be released by metamorphic reactions of the subducting Indian plate beneath the Burmese plate.
... The estimation of the depth of anomalous sources is usually carried out by Spector and Grant ( Spector and Grant, 1970) method and its variants in frequency domain ( Bhattacharyya, 1966;Naidu, 1968;Spector and Grant 1970;Treitel et al., 1971;Hahn et al., 1976;Connard et al., 1983). These methods assume different assemblage of sources like statistical ensemble of prisms ( Spector and Grant, 1970), white noise of vertical needles with constant magnetization ( Naidu, 1972: Hahn et al., 1976), a sandwich model of uniaxially magnetic sheets ( Pedersen, 1991), equivalent density layer ( Pawlowski, 1994) etc. These methods are in continuous use since their development because of simplicity ( Ofoegbu and Hein, 1991: Cowan and Cowan, 1993: Hildenbrand et al., 1993: Gracia-Abdeslem and Ness, 1994). ...
... 3). The inverse problem in that setting was considered in various aspects by many authors [Kobrunov, 1982[Kobrunov, , 2008Dolgal, 2002;Pedersen, 1991;Tarantola, Valette, 1982]. However, in our case, the weight functions ( , , θ) for the density model of the medium are determined through entropic characteristics of velocity-density interrelations (Eq. ...
The complex geophysical 3D model of the Earth's crust and the upper mantle is created for the Archaean Karelian Craton and the Late Palaeoproterozoic accretionary Svecofennian Orogen of the southeastern Fennoscandian Shield with the use of methods of complex inversion of geophysical data based on stochastic description of interrelations of physical properties of the medium (density, P-wave velocity, and heat generation). To develop the model, we use results of deep seismic studies, gravity and surficial heat flow data on the studied region. Numerical solutions of 3D problems are obtained in the spherical setting with an allowance for the Earth's surface topography. The geophy-sical model is correlated with the regional geological data on the surface and results of seismic CMP studies along 4B, FIRE-1 and FIRE-3-3A profiles. Based on results of complex geophysical simulation and geological interpretation of the 3D model, the following conclusions are drawn. (1) The nearly horizontal density layering of the continental crust is superimposed on the previously formed geological structure; rock differentiation by density is decreasing with depth; the density layering is controlled by the recent and near-recent state of the crust, but can be disturbed by the latest deformations. (2) Temperature variations at the Moho are partially determined by local variations of heat generation in the mantle, which, in turn, are related to local features of its origin and transformation. (3) The concept of the lower continental crust being a reflectivity zone and the concept of the lower continental crust being a layer of high density and velocity are not equivalent: the lower crust is the deepest, high-density element of near-horizontal layering, whereas the seismic image of the reflectivity zone is primarily related to transformation of the crust as a result of magmatic under-and intraplating under conditions of extension and mantle-plume activity. (4) At certain combinations of crustal thickness and temperature at the level of Moho discontinuity, the crust in a platform region can be transformed into eclogites. In this case, the crust–mantle boundary is determined by quantitative proportions of the rocks that underwent eclogitization or escaped this process and by corresponding density and velocity values. (5) High compaction of rocks in the crust under lithostatic loading cannot be explained by «simple» concepts of me-tamorphism and/or rock compaction, which are based on laboratory studies of rock samples and mathematical simulations ; this is an evidence of the existence of additional, quite strong mechanisms providing for reversible changes of the rocks.
... Separation filtering of regional and residual magnetic fields has been the subject of extensive research, and a large number of space-domain and wavenumber-domain methods have been developed. Important contributions include the 'matched' filtering approach of Spector and Grant (1970), wavenumber filtering and equivalent-layer analysis by Pedersen (1979Pedersen ( , 1991 and differential-upward-continuation based separation filtering by Jacobsen (1987) and Cowan and Cowan (1993). ...
... Vattenförande sprickzoner kan lokaliseras genom att kombinera magnetiska och elektromagnetiska anomalikartor (Jacobsen 1987, Pedersen 1991. Den magnetiska anomalibilden är visualiserad i en figur av gråskalor ( fig. ...
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... Majority of the structures are rotated into the prevalent, large-scale sinistral fabric of the overall N-S-trending Pajala shear zone system (Fig. 2). Depth estimates of individual shear zones vary between 2.5 km and 5 km in the south (Pedersen, 1991) and 10-15 km in the central parts (Patison et al., 2006;Niiranen et al., 2007). Berthelsen and Marker (1986) suggested that an older phase of dextral movement preceded the sinistral phase in the Pajala shear zone. ...
... The second and third scales of magnetic anomalies in this type of study are deeper geologic sources and shallow, potentially interesting sources at the archaeological and/or environmental scale. We typically separate these sources into equivalent layers using either differencing of upward continuations following [6] or matched bandpass filtering based on equivalent sources [7]. Employing matched bandpass filtering for anomaly separation has a long history [8] in the application of aeromagnetic data to tectonics, structure, and resource exploration, but not in archaeology. ...
... These properties are derived from applying the DEXP transformation to the ratio between any pair of different-order vertical derivatives of the field or to the partial derivatives of this ratio. Comparing this method to previous imaging methods (Cribb 1976; Pedersen 1991;Patella 1997;Zhdanov 2002;Fedi 2007), we note that our method is the only one able to giving correct depth and structural index estimation in case of a field generated by a multiset of different types of sources, such as for instance spherical, cylindrical or fault sources. For instance, migration assumes intrinsically a single type of scaling-law and so gives either correct or incorrect depth estimates, depending on the type of source involved (Fedi & Pilkington 2012); DEXP of potential fields yields correct source estimates if the right scaling-law is assumed, but the multisource case cannot be accounted in a single step of imaging. ...
We illustrate a new imaging method to estimate the depth to the sources of potential fields and the structural index. The method consists of applying the Depth from EXtreme Point (DEXP) transformation to the ratio (ℜ) between two different-order partial derivatives of the field. While the scaling function of the potential field depends on the structural index, we show that the scaling function of ℜ merely depends on the difference between the two used orders of differentiation. This allows three main features to be established for the DEXP transformation of ℜ: (1) it is independent from the structural index; (2) the estimation of the source depths is fully automatic, simply consisting in the search of position of the extreme points of the DEXP image and (3) the structural index of each source is finally determined from the scaling function or the extreme points using the estimated depth. Besides the well-known characteristics of the DEXP transformation, such as high-resolution and stability, the DEXP transformation of ℜ enjoys one more relevant feature: it can be applied to multisource cases, yielding simultaneously correct estimations of structural index and depth for each source in the same image. However, while the DEXP transformation is a linear transformation of the field, the DEXP transformation of ℜ is non-linear, and a procedure is described to circumvent the non-linear effects.
The method is tested with synthetic examples and the estimated source parameters show a good agreement with the true values. The method was applied also to real magnetic data from the Pima copper mine, Arizona, USA, Hamrawien area, Egypt and Cataldere, Bala district of Turkey. The results are consistent with the known information about the causative sources.
... Our method performs a multiscale analysis of potential fields, similar to previous papers using explicitly the upward continued data (McGrath 1991;Pedersen 1991;Fedi & Rapolla 1999;Paoletti et al. 2007;Cella et al. 2009) or the continuous wavelet transform (Moreau et al. 1999;Sailhac & Gibert 2003;Fedi & Cascone 2011). This fundamentally implies that the gravity or magnetic source could have any shape. ...
We describe a multiscale method to estimate the excess mass of gravity anomaly sources,
based on the theory of source moments. Using a multipole expansion of the potential field and
considering only the data along the vertical direction, a system of linear equations is obtained.
The choice of inverting data along a vertical profile can help us to reduce the interference effects
due to nearby anomalies and will allow a local estimate of the source parameters. A criterion
is established allowing the selection of the optimal highest altitude of the vertical profile data
and truncation order of the series expansion. The inversion provides an estimate of the total
anomalous mass and of the depth to the centre of mass. The method has several advantages
with respect to classical methods, such as the Gauss’ method: (i) we need just a 1-D inversion
to obtain our estimates, being the inverted data sampled along a single vertical profile; (ii) the
resolution may be straightforward enhanced by using vertical derivatives; (iii) the centre of
mass is also estimated, besides the excess mass; (iv) the method is very robust versus noise;
(v) the profile may be chosen in such a way to minimize the effects from interfering anomalies
or from side effects due to the a limited area extension. The multiscale estimation of excess
mass method can be successfully used in various fields of application. Here, we analyse the
gravity anomaly generated by a sulphide body in the Skelleftea ore district, North Sweden,
obtaining source mass and volume estimates in agreement with the known information. We
show also that these estimates are substantially improved with respect to those obtained with
the classical approach.
... The difference in depth of investigation between the magnetic method and the different EM apparatus is thus a main parameter to consider when simultaneously inverting EM and magnetic data. The equivalent stratum problem is well known in magnetism: the magnetic response of a feature at a given depth is equivalent to the response of a thin sheetlike feature at a shallower depth (Pedersen, 1991). The reciprocal effect of this equivalence can be a problem in the interpretation of the filtered EM data if a true susceptibility variation inside the near-surface layer is interpreted as generated by a deeper feature. ...
Magnetic and electromagnetic measurements are influenced by magnetic susceptibility and, thus, are widely used in geophysical surveys for archeology or pedology. To date, the data inversion is performed separately. A filtering process incorporating both types of data is presented here. After testing the algorithm with synthetic data, the algorithm is used in several case studies in archeological prospecting. This approach presents two advantages: establishing the presence of remanent magnetizations (viscous or thermoremanent), and achieving more refined depth analysis of the anomaly.
... In the absence of heat flow data, the two magnetic profiles (Fig. 2) could be used to compute the depth to the Curie isotherm. The depth to the deepest interface was calculated both from the conventional power spectrum (Treitel et al., 1971;Pederson, 1991) and from maximum likelihood methods (Fisher, 1958;Davies, 1982). The latter is more useful because of its limited subjectivity. ...
The Geological Survey of India conducted a magnetic survey and geological studies in 1993 around Barren Island in the Andaman Sea on board the RV Samudra Manthan. Five ocean bottom magnetometers were deployed by the Indian Institute of Geomagnetism as a part of this cruise. The Curie isotherm profiles prepared from the sea-surface magnetic data indicate a high heat flow in the east of Barren Island. These profiles also indicate the presence of a north-south structural barrier which has prevented upflow of volcanic material to the west of Barren Island. Ocean-bottom magnetometer data were recorded simultaneously at five sites for about 15 days and these have been used to determine the electrical conductivity structure beneath Barren Island. Magnetic variations recorded at the seafloor stations indicate a concentration of electric currents near the island instead of the usual effect where currents are deflected away from the island. Transfer functions, showing the relationship between the horizontal components of the seafloor stations and land station, have been computed and the quantitative estimates of the transfer functions across Barren Island indicate a high conducting zone at a depth of about 17-27 km. This zone may have been produced by an upwelling of the mantle material through the magma chamber. The structure of this conductive zone at the north and south of Barren Island seems to concentrate the flow of the subsurface electrical currents within the island and the current flow takes a sharp southward turn beneath the island. This north-south conducting zone may have caused a rise in the depth of the Curie isotherm mapped by a shipborne magnetic survey of this region. Most probably, a partial melting of this conductive zone (magma chamber) may have given rise to the volcanic activity on Barren Island.
... The study of gravity and magnetic fields at different altitudes, so constituting a multiscale data set, is an important part of several interpretation methods (e.g. Paul et al. 1966;Paul & Goodacre 1984;McGrath 1991;Pedersen 1991). More recently, other methods involving a multilevel data set were proposed, among which the continuous wavelet transform method (CWT; e.g. ...
We use a multiscale approach as a semi-automated interpreting tool of potential fields. The depth to the source and the structural index are estimated in two steps: first the depth to the source, as the intersection of the field ridges (lines built joining the extrema of the field at various altitudes) and secondly, the structural index by the scale function. We introduce a new criterion, called 'ridge consistency' in this strategy. The criterion is based on the principle that the structural index estimations on all the ridges converging towards the same source should be consistent. If these estimates are significantly different, field differentiation is used to lessen the interference effects from nearby sources or regional fields, to obtain a consistent set of estimates. In our multiscale framework, vertical differentiation is naturally joint to the low-pass filtering properties of the upward continuation, so is a stable process. Before applying our criterion, we studied carefully the errors on upward continuation caused by the finite size of the survey area. To this end, we analysed the complex magnetic synthetic case, known as Bishop model, and evaluated the best extrapolation algorithm and the optimal width of the area extension, needed to obtain accurate upward continuation. Afterwards, we applied the method to the depth estimation of the whole Bishop basement bathymetry. The result is a good reconstruction of the complex basement and of the shape properties of the source at the estimated
... When the number of parameters is large, the calculation is done more efficiently by using fast Fourier transforms. If J l r is the susceptibility distribution within the lth layer, and r is the position vector , then the Fourier transform of the field resulting from all NZ layers is Pedersen, 1991; Blakely, 1995and z l is the distance from the observation level to the layer dipoles center. Expression t contains the direction cosines of earth's magnetic field; k is the wave vector, whose components in the x,y,z coordinate system are k x ,k y ,0; and F denotes Fourier transformation. ...
I have developed an inversion approach that determines a 3D susceptibility distribution that produces a given magnetic anomaly. The subsurface model consists of a 3D, equally spaced array of dipoles. The inversion incorporates a model norm that enforces sparseness and depth weighting of the solution. Sparseness is imposed by using the Cauchy norm on model parameters. The inverse problem is posed in the data space, leading to a linear system of equations with dimensions based on the number of data, N. This contrasts with the standard least-squares solution, derived through operations within the M-dimensional model space (M being the number of model parameters). Hence, the data-space method combined with a conjugate gradient algorithm leads to computational efficiency by dealing with an N x N system versus an M x M one, where N << M. Tests on synthetic data show that sparse inversion produces a much more focused solution compared with a standard model-space, least-squares inversion. The inversion of aeromagnetic data collected over a Precambrian Shield area again shows that including the sparseness constraint leads to a simpler and better resolved solution. The degree of improvement in model resolution for the sparse case is quantified using the resolution matrix.
... These filters simply turn out to be combinations of upwardcontinuation operators and thus provide a theoretical basis for using upward continuation as a regional/residual separation method (e.g., Gupta and Ramani, 1980). Pedersen (1991) extends this approach by allowing different magnetization amplitudes in each layer which he estimates from the observed field power spectrum. ...
Separating the fields produced by sources at different depths is a
common requirement in the interpretation of potential field data.
Approaches to this problem are generally data- or model-based.
Data-based methods rely on breaks in the slope of the logarithmic power
spectrum of the observed field to design filters that can effect the
separation. When the power spectrum shows no identifiable spectral slope
breaks, other approaches are necessary. We outline a model-based method
that does not depend on power spectral information but requires
estimates of the average depths of the source distributions. An ensemble
of models is computed based on these known values and a set of filter
parameters are determined that produce the closest fit (in a
least-squares sense) to the theoretical fields generated by each source
distribution. This approach is used to separate basement effects from
intrasedimentary sources in magnetic data collected over the Colville
Hills, Northwest Territories, Canada.
... Separation filtering of regional and residual gravity and magnetic fields has been the subject of extensive research and a large number of different space-domain and wavenumber-domain methods have been developed. Important contributions include the 'matched' filtering approach of Spector and Grant (1970), wavenumber filtering and equivalent-layer analysis by Pedersen (1979Pedersen ( , 1991 and differential upward continuation-based separation filtering by Jacobsen (1987) and Cowan and Cowan (1993). ...
The enhancement of magnetic data for qualitative interpretation involves manipulating magnetic relief and magnetic texture. Magnetic relief consists primarily of anomaly amplitude and is relatively objective. Magnetic texture consists of shape, size and continuity of adjacent anomalies and is more subjective. Conventional filtering responds primarily to amplitude variations within the data and high-amplitude anomalies often mask more subtle anomalies of interest. Changes in geology, or specifically rock magnetisation, within a survey area cause changes in both texture and relief. When rock magnetisation is weak, anomalies are subdued and particular filtering and enhancement methods are required. However, the effects of these on areas of greater magnetic relief must also be considered. Three fundamentally different approaches to enhancing subtle anomalies have been implemented and tested on a wide range of sedimentary basin and low magnetic gradient areas. The first approach uses separation or layer filtering to deconvolve the effects of magnetic sources around a mean depth. The second uses texture filtering using grey-level co-occurrence matrix (GLCM) based filters and the third uses the gradient tilt angle. The responses of these filters are illustrated using a regional aeromagnetic dataset from the Proterozoic Arunta block in the Northern Territory of Australia. Results show that each method has its advantages and limitations. Each shows different amplitude and wavelength responses for a given dataset, but all three are relatively broadband compared with a conventional filter such as a first vertical derivative. No single method performed well on all areas of the test dataset. The GLCM filters were almost pure texture with very limited tonal content. The gradient tilt angle provides good texture content, but retained more tonal content than the GLCM filters. Combining filters with different bandwidth such as the separation filter and tilt angle can be very effective and provides the best results.
Magnetic anomalies are interpreted for subsurface geological information. Three-dimensional inversion of magnetic data is a challenging quantitative approach for interpreting the data. The rapid iterative migration technique could be a good and fast alternative for the inversion method. The focusing iterative migration that employs a focusing stabilizer can generate focused migration models which justify geological interfaces, adequately. This paper introduces a new algorithm using a relaxed steepest descent method and a sigmoid stabilizer for fast focusing migration of magnetic fields. The developed method can improve the computational efficiency of focusing iterative migration by reducing the required iterations. The better performance of the new method is demonstrated by two numerical models and a real case study. The magnetic anomaly over San Nicolas massive sulfide deposit in Mexico is used for the case study. Compared with the drilling information, the iterative migration methods produce robust migration models for the San Nicolas deposit.
It is known from potential theory that a continuous and planar layer of dipoles can exactly reproduce the total-field anomaly produced by arbitrary 3D sources. We prove the existence of an equivalent layer having an all-positive magnetic-moment distribution for the case in which the magnetization direction of this layer is the same as that of the true sources, regardless of whether the magnetization of the true sources is purely induced or not. By using this generalized positivity constraint, we present a new iterative method for estimating the total magnetization direction of 3D magnetic sources based on the equivalent-layer technique. Our method does not impose a priori information either about the shape or depth of the sources, does not require regularly spaced data, and presumes that the sources have a uniform magnetization direction. At each iteration, our method performs two steps. The first one solves a constrained linear inverse problem to estimate a positive magnetic-moment distribution over a discrete equivalent layer of dipoles. We consider that the equivalent sources are located on a plane and have an uniform and fixed magnetization direction. In the second step we use the estimated magnetic-moment distribution and solve a nonlinear inverse problem for estimating a new magnetization direction for the dipoles. The algorithm stops when the equivalent layer yields a total-field anomaly that fits the observed data. Tests with synthetic data simulating different geological scenarios show that the final estimated magnetization direction is close to the true one. We apply our method to a field data from the Goi’s Alkaline Province (GAP), over the Montes Claros complex, center of Brazil. The results suggest the presence of intrusions with remarkable remanent magnetization, in agreement with the current literature for this region.
Depth from extreme points method is an important tool to estimate the depth of sources of gravity and magnetic data. In order to interpret gravity gradient tensor data conveniently, formulas for the tensor data form regarding depth from extreme points method were calculated in this paper. Then all of the gradient tensor components were directly used to interpret the causative source. Beyond the gzz component, also the gxx and gyy components can be used to obtain depth information. In addition, the total horizontal derivative of the depth from extreme points of the gradient tensor can be used to describe the edge information of geologic sources. In this paper, we investigated the consistency of the homogeneity degree calculated by using the different components, which leads to the calculated depth being confirmed. Therefore, a more integrated interpretation can be obtained by using the gradient tensor components. Different synthetic models were used with and without noise to test the new approach, showing stability, accuracy and speed. The proposed method proved to be a useful tool for gradient tensor data interpretation. Finally, the proposed method was applied to full tensor gradient data acquired over the Vinton Salt Dome, Louisiana, USA, and the results are in agreement with those obtained by previous researches. This article is protected by copyright. All rights reserved
Magnetic surveys around Barren and Narcondam Island have brought out different anomaly pattern during different period of surveys since 1990. This variation in amplitude of anomaly pattern in the magnetic signal, which is responsible for Curie Isotherm Depth (CID), shows distinct picture of thermal eruption from 1990 to recent past. This in turn will raise or lower the CID, which is reflected in the present study. Data collected from the various cruises of R/V Samudra Manthan (SM-61,78,113,136 and 157) of the Geological Survey of India are analysed here.
Recent progresses in geosciences make more and more multidisciplinary data available for mining exploration. This allows developing methodologies to compute predictivity for gold zones by the statistical analysis of variable input parameters. Using newly developed software, the spatial distribution and the topology of polygons (e.g. intrusions) and polylines (e.g. shear zones) are controlled by parameters defined by users (e.g. density, length, surface, etc.). The distance of points of interest (gold deposits) with respect to a given type of objects (polygons or polylines) is given using a probability distribution function. The statistical analyses of output results from the direct modeling process show that i) values of relative surface mean of polygons, relative length mean of polylines, the number of objects and their clustering are critical to statistical appraisals, ii) the validity of the different tested inversion methods strongly depends on the relative importance and on the dependency between the parameters used, and iii) the robustness of the inferred distribution points of interest laws with respect to the quality of the input data. This approach was applied to the geological and geophysical data of the Yenissei ridge of the total area of 75730 km2 for the predictivity mapping of 29 new gold zones with the total area of 1811 km2. The newly developed method allows reducing up to four times of the area of predictivity gold zones, compared with previous studies. For more accurate construction of gold zones, a 3D density model of the Yenisei ridge was constructed. This model is based on surface gravity and aeromagnetic data (numerical grids of 1x1km), ―Batholite‖ and ―Shpate‖ seismic and magnetotelluric profiles, respectively. The 3D density model shows that: a) the Yenissei ridge has a cover-folded structure, formed during a Neopretorozoic collisional event, b) only γNPta Tatarsky-Ayhta granites and shear zones have spatial relationships with gold mineralization.
Self potential method which is one of the electrical methods of prospecting was employed to investigate the mineralization potential of some communities in Isialangwa south local government area of Abia-state; Nigeria. The survey was done using one fixed-one movable electrode technique on a grid with measuring separation of 20 meters. The two electrode movable " leap frog " method was also employed on a 30 meter spacing interval from Ugba junction towards Owerrinta. It was observed that large portion of the study area recorded negative SP anomalies ranging from-15mV to-90mV which is indicative of mineralization endowment of the area and also suggestive that ore bodies exist therein in the subsurface. There was also zone s of positive potential values ranging from +25 to +70 mV which is indicative of sulphide ore deposits.
We present a reformulation of reduction to the pole (RTP) of magnetic data at low latitudes and the equator using equivalent sources. The proposed method addresses both the theoretical difficulty of low-latitude instability and the practical issue of computational cost. We prove that a positive equivalent source exists when the magnetic data are produced by normal induced magnetization, and we show that the positivity is sufficient to overcome the low-latitude instability in the space domain. We further apply a regularization term directly to the recovered RTP field to improve the solution. The use of equivalent source also naturally enables the processing of data acquired on uneven surface. The result is a practical algorithm that is effective at the equatorial region and can process large-scale data sets with uneven observation heights.
Randomly scattered debris, some of it ferromagnetic, on or near the ground surface regularly degrades magnetic data acquired for archaeological purposes. These sources create short-wavelength high-amplitude dipolar anomalies in total field magnetic intensity maps and they can dominate maps of the vertical magnetic gradient. Thus, we generally wish to separate longer wavelength anomalies created by features of interest, such as foundations or building perimeters, from the shorter wavelength anomalies created by debris on or near the ground surface. Matched bandpass filtering, employed extensively in the aeromagnetic industry, is an effective way to separate magnetic anomalies arising from different depths. It entails fitting the radially averaged power spectrum of the total field magnetic data with a series of power spectra corresponding to simple equivalent layers at the archaeological site. We show that applying matched bandpass filtering to a set of total field magnetic intensity data yields two equivalent layers with excellent separation of near-surface sources from deeper sources of interest. The benefit of this approach over enhancing surface features using magnetic gradiometry, analysis by upward continuation, or the analytic signal is that we isolate the source layers without losing information.
Recent implementations of Euler deconvolution allow to solve simultaneously for the source position and the structural index. This opens the way to a comparison between this technique and the Continuous Wavelet Transform (CWT) method, that allows the estimation of essentially the same parameters. Direct comparison of Euler deconvolution and CWT methods is possible only by applying the first method to a potential field upward continued to many altitudes. While the two methods give very similar results when the gravity or magnetic field of a one-point source is concerned, they behave different for those sources characterized by fractional structural index, as many real geologic structures are (for example a limited throw fault). In this paper, the variation of the Euler estimated parameters at many altitudes above a magnetized prism is described. Such a variation gives additional information on source geometric parameters and position, which may be recovered from plots of estimated depth and structural index vs. altitude: a) the extended or one-point nature of the source results clearly; b) it is possible to understand to which part of the source the depth estimate is related to; c) it is possible to get indications about the source thickness and lateral dimensions. On the other hand, for sources of finite extent, the CWT analysis may be made only for sets of levels and not at any level, differently from the above outlined Euler deconvolution approach. Nevertheless the results from these two methods are substantially consistent at high or low altitudes.
Interpretation of magnetic data can be carried out either in the space or frequency domain. The interpretation in the frequency domain is computationally convenient because convolution becomes multiplication. The frequency domain approach assumes that the magnetic sources distribution has a random and uncorrelated distribution. This approach is modified to include random and fractal distribution of sources on the basis of borehole data. The physical properties of the rocks exhibit scaling behaviour which can be defined as P(k) = Ak−β, where P(k) is the power spectrum as a function of wave number (k), and A and β are the constant and scaling exponent, respectively. A white noise distribution corresponds to β = 0. The high resolution methods of power spectral estimation e.g. maximum entropy method and multi-taper method produce smooth spectra. Therefore, estimation of scaling exponents is more reliable. The values of β are found to be related to the lithology and heterogeneities in the crust. The modelling of magnetic data for scaling distribution of sources leads to an improved method of interpreting the magnetic data known as the scaling spectral method. The method has found applicability in estimating the basement depth, Curie depth and filtering of magnetic data.
Frequency-domain controlled-source electromagnetic surveys can be negatively affected by near-surface geologic features, resulting in static shifts in the observed electric field amplitude data. We propose using an equivalent source method to denoise electric field data from frequency-domain controlled-source electromagnetic (CSEM) surveys. The equivalent source method can either be applied individually to data from each measured frequency or simultaneously to data from all frequencies measured. We demonstrate the effectiveness of the method by denoising an individual synthetic CSEM survey with data contaminated with noise from near-surface conductive bodies, as well as a synthetic time-lapse survey representing resistivity changes in a target due to, for instance, CO2 sequestration. We also apply the equivalent source method to denoise a set of controlled-source audio-magnetotelluric data affected by static shift from a field survey in western China.
Two granite-granodiorite-tonalite-trondhjemite (TTG) complexes intruded mafic-intermediate volcanic and volcaniclastic rocks at 1954-1940 Ma in the central part of the Knaften area, which lies in the northern part of the Palaeoproterozoic Bothnian Basin, in the central part of the Fennoscandian Shield, northern Sweden. The regional metamorphism was upper greenschist to amphibolite facies. The granitoid complexes are subdivided texturally into three types of intrusions. Contact relationships show that the most coarse-grained type is the oldest, followed by a medium-grained type and finally an aplitic-pegmatitic type intrusion. Felsic dykes that cut the granitoids form the youngest intrusive event. Myrmekitic/symplectitic textures or graphic intergrowths show that the later intrusive phases were more volatile-rich. The granitoids are high-level intrusions, and formed the root-zone of a volcano. Close to, but outside the granitoid complexes, compositionally similar felsic porphyry dykes occur. A package of reworked felsic tuffaceous rocks occurs in the northern part of the Knaften area and may constitute extrusive equivalents of the intrusions. These clastic rocks display primary structures such as cross bedding, which indicate deposition in shallow water. Geochemically, the Knaften intrusions are granites, granodiorites, tonalites and trondhjemites. The northern granitoids are more fractionated than the southern ones, and some felsic dykes are geochemically similar to the southern intrusions. The tuffaceous rocks are intermediate in composition between the felsic and more mafic intrusive phases. All Knaften felsic phases have similar trace element patterns and, their trace element chemistry is similar to other rocks from collisional settings in island arc terrains.
Magnetic anomalies may be interpreted in terms of weighted averages of magnetization (WAM) by a simple transformation. The WAM transformation consists of dividing at each measurement point the experimental magnetic field by a normalizing field, computed from a source volume with a homogeneous unit-magnetization. The transformation yields a straightforward link among source and field position vectors. A main WAM outcome is that sources at different depths appear well discriminated. Due to the symmetry of the problem, the higher the considered field altitude, the deeper the sources outlined by the transformation. This is shown for single and multi-source synthetic cases as well as for real data. We analyze the real case of Mt. Vulture volcano (Southern Italy), where the related anomaly strongly interferes with that from deep intrusive sources. The volcanic edifice is well identified. The deep source is estimated at about 9 km depth, in agreement with other results.
We propose a method for inverting the power spectrum of gravity and magnetic data. The method is demonstrated on aeromagnetic and borewell data from the German Continental Deep Drilling Project (KTB). Density and susceptibility distributions in the Earth's crust exhibit scaling behavior with power spectra proportional to f-beta, where f is the wavenumber and beta is the scaling exponent of the source distribution. We model the sources of the potential field by a random function with scaling properties, defined on a half-space with its top at a specified depth beneath the observation plane. Comparing the theoretical power spectrum for this model with the power spectrum of the measured data, we obtain the best values for the depth to source and the scaling exponent as a global minimum of the misfit function. Despite the simplicity of the model, it offers a new understanding of the factors influencing the shape of the potential field power spectrum. In particular, the low wavenumber part of the power spectrum can be dominated by the scaling properties of the source distribution and not by the depth to some kind of basement. The scaling exponent of the field varies with the type of surface geology. The question of whether the scaling exponent can actually be used to identify different types of geology gives an interesting new aspect to power spectrum inversion.
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Geophysics is the science of using physical observations of the Earth to infer its inner structure. Generally, this is done with a variety of numerical modeling techniques and inverse problems. The development of new algorithms usually involves copy and pasting of code, which leads to errors and poor code reuse. Fatiando a Terra is a Python library that aims to automate common tasks and unify the modeling pipeline inside of the Python language. This allows users to replace the traditional shell scripting with more versatile and powerful Python scripting. The library can also be used as an API for developing stand-alone programs. Algorithms implemented in Fatiando a Terra can be combined to build upon existing functionality. This flexibility facilitates prototyping of new algorithms and quickly building interactive teaching exercises. In the future, we plan to continuously implement sample problems to help teach geophysics as well as classic and state-of-the-art algorithms.
The problem of reduction of magnetic and gravity data, when observed on an arbitrary surface in a region of high topographic relief, is studied with equivalent source representation at the points of observation. It is shown that the analytical relationship between the total magnetic field or the gravity effect and equivalent magnetization or density on an arbitrary observational surface is given by a Fredholm integral equation of the second kind. A rapidly convergent iterative scheme is described for the solution of the integral equation, yielding the surface distribution of magnetization or density. With this distribution, the field at any other surface can be easily computed. Then it has been demonstrated with model examples that the gravity or magnetic field observed on a rough terrain can be accurately reduced to a horizontal plane for processing and interpretation. A new method has been suggested for minimization of terrain‐induced anomalies on a magnetic or gravity map. This method is based on the concept that when the anomalous field observed on an arbitrary surface is continued to a surface parallel to the topography, the terrain effect in the continued field is sharply reduced relative to the field created by bodies of finite extent in the crust. Model examples are presented to show the accuracy and reliability of the method.
Separation filtering is incomplete even under the ideal synthetic condition of known power spectra of the regional and residual fields. The author has designed some Wiener filters, which minimize the inevitable separation error, from previous statistical source models of Naidu, and Spector and Grant. This formulation includes the classic separation filters of Strakhov and of Elkins as Wiener filters. A proposed generalization of Wiener filters, denoted as uniformly suboptimum filters, quantitatively supports the statement that a wide span of separation problems may be solved adequately using some convenient, small standard filter family. A uniform random-source model without assumed vertical correlations invokes upward continuation filters. In addition to this role as a Wiener filter, the upward continuation operator is given by elementary functions in both space and wavenumber domains, is numerically stable, and is also physically comprehensible when applied to real, nonrandom anomalies.
When attempting to get an overview of continuation of potential fields in exploration geophysics, one cannot avoid reading the much quoted article by Bhattacharyya and Chan (hereafter referred to as BC). However, it is unfortunate that the article is unclear on some fundamental points, which this comment attempts to clarify.
Analytical expressions for the Fourier transforms of potential fields caused by inclined dikes and vertical circular cylinders are evaluated. Their spectra are used for a statistical analysis along the lines proposed by Spector and Grant (Geophys., 35, 293-302 (1970)). Special attention is paid to the term related to the effect of the lateral extent of the sources. This term is found to decay via an inverse power relationship. Thus, it decays fairly slowly, and it may become quite important to correct the spectrum prior to interpretation. Numerical calculations of the term for the cylinder and the prism indicate that an inverse third power relationship is appropriate for correction of practical power spectra. 11 figures.
An equivalent source algorithm is described for continuing either 1-or 2-D potential fields between arbitrary surfaces. In the 2-D case, the dipole surface is approximated as a set of plane faces with constant moments over each face. In the 1-D case, the plane faces of the dipole surface reduce to straight line segments. Application of the algorithm to model and field examples of aeromagnetic data shows the method to be effective and accurate even when the terrain has strong topographic relief and is composed of highly magnetic volcanic rocks. -Authors
A mathematical basis for the application of power spectrum analysis to aeromagnetic map interpretation is developed. An ensemble of blocks of varying depth, width, thickness, and magnetization is considered as a statistical model. With the use of the fundamental postulate of statistical mechanics, a formula which can be used to analyze the power spectrum of an aeromagnetic map is developed. The influences of horizontal size, depth, thickness, and depth extent of the blocks on the shape of the power spectrum are assessed. Examples which include power spectra of maps from Canada and Central America demonstrate the application of the approach. In the cases studied a double ensemble of blocks appears to best explain the observed power spectrum characteristics.