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The gravity gradient tensor (GGT) has been increasingly used in practical applications, but the advantages and the disadvantages of the analysis of GGT components versus the analysis of the vertical component of the gravity field are still debated. We analyse the performance of joint inversion of GGT components versus separate inversion of the gravity field alone, or of one tensor component. We perform our analysis by inspection of the Picard Plot, a Singular Value Decomposition tool, and analyse both synthetic data and gradiometer measurements carried out at the Vredefort structure, South Africa.We show that the main factors controlling the reliability of the inversion are algebraic ambiguity (the difference between the number of unknowns and the number of available data points) and signal-to-noise ratio. Provided that algebraic ambiguity is kept low and the noise level is small enough so that a sufficient number of SVD components can be included in the regularized solution, we find that: (i) the choice of tensor components involved in the inversion is not crucial to the overall reliability of the reconstructions; (ii) GGT inversion can yield the same resolution as inversion with a denser distribution of gravity data points, but with the advantage of using fewer measurement stations.

We report the results of a multidisciplinary investigation performed across the normal Quaternary faults that ruptured the surface during the August 24 (Mw 6.0) and October 30 (Mw 6.5), 2016 strong earthquakes in the Mt. Vettore-Mt. Bove areas, central Italy. Our aim is to test the effectiveness of the contribution of a multi-scale gravimetric analysis in characterizing seismogenic faults’ geometry at hypocentral depths on well-known outcropping faulty systems with known earthquake distribution. We adopted a multi-scale geophysical/geological approach consisting in the comparison of gravity lineaments inferred by Multiscale Derivative Analysis with the Quaternary structural setting mapped in the study area, the primary coseismic surface ruptures of the 2016-2017 sequence and the earthquakes’ epicentral distribution. Moreover, we performed a combined interpretation of 2D hypocentral sections of the 2016-2017 seismic sequences with images resulting from the Depth from Extreme Points method, to infer the faults’ geometry at depth. Based on our results, the investigated NW-SE Mt. Vettore-Mt. Bove fault system is dipping 60°-70° westward. We also detected the splays of this primary fault and its blind antithetic NW-SE structure, dipping northeastward. In the Norcia basin we highlight two main faults bordering the basin with a dip of about 45°. The one edging the eastern side dips westward, whereas the fault edging the western side dips eastward. Thanks to our analysis we could identify and characterize the geometry of the Norcia and Vettore master faults, as well as other blind/buried and/or silent faults that are related to the 2016 seismogenic structure. Our results show the effectiveness of this approach in potentially high-hazard areas that are structurally poorly known.

Natural remanent magnetization and self-demagnetization on high-susceptibility bodies are two important factors affecting magnetic data inversion. We propose a framework for the inversion and interpretation of magnetic anomalies, in which significant remanent magnetization and self-demagnetization are present simultaneously. The framework is based on the assumptions that the external applied field and internal self-demagnetization field are uniform and the deflection of self-demagnetization in the total magnetization direction is negligible. First, the magnetization vector distributions are obtained from magnetic data by estimating the magnetization direction, then inverting for the magnetization intensity distribution, using the inferred magnetization direction as a constraint. Based on a priori information about the Koenigsberger ratio derived from petrophysical measurements, the direction and intensity of the remanent magnetization are obtained. The self-demagnetization factor is then computed using the finite volume method. Finally, the true-susceptibility distribution is achieved by correcting for the self-demagnetization effect. The method is first applied to synthetic magnetic data produced by a prism-shaped source model that has significant remanent magnetization and high susceptibility. In a case study of the Daye iron-ore deposit, Hubei province, China, the true susceptibility and remanent magnetization are reconstructed. The remanence direction information reveals that local geological activities such as synclines and faults lead to changes in the remanence directions at different local deposits.

Spectral analysis, based on Fourier Transform, provides a high-resolution analysis in frequency domain but it has not resolution in the space domain. In addition, the case of multiple sources is hard to be studied with standard techniques, such as that based on the well-known Spector and Grant's method. We propose to fix these issues by resorting to a scalogram analysis, obtained through the continuous wavelet transform of the potential fields using the Morlet analyzing wavelet. In the scalogram it is indeed possible distinguishing and locating the source contributes for both their space and scale contents. In this framework, we propose a method to analyze locally the scalogram, in bounded sub-regions, so to estimate the depth of each source by Bounded-Region Wavelet Spectra, which provide simultaneously a good space and scale resolution. We tested successfully this approach with magnetic anomalies of synthetic multiple-source models and we finally applied it to the gravimetric data of Sicily (Southern Italy), for estimating a gravity-based Moho depth in Sicily

The estimation of the structural index and of the depth to the source is the main task of many popular methods used to analyze potential field data, such as Euler deconvolution. However, these estimates are unstable even in the presence of a weak amount of noise, and Euler deconvolution of noisy data leads to an underestimation of structural index and depth. We have studied how the structural index and depth estimates are affected by applying low-pass filtering to the data. Physically based low-pass filters, such as upward continuation and integration, have been shown to be the best choice over a range of altitudes (upward continuation) or orders (integration filters), mainly because their outputs have a well-defined physical meaning. In contrast, mathematical low-pass filters require that the filter parameters be tuned carefully by means of several trial tests to produce optimally smoothed fields. The C-norm criterion is a reliable strategy to produce a stabilized vertical derivative, and we discourage Butterworth filters because they tend to a vertical integral filter, for a high cutoff wavenumber, thus complicating the interpretation of the estimated structural index. We found that the estimated structural index and depth to source increase proportionally with the amount of smoothing, unless in the case of overfiltering. In that case, the severe distortion of the original field may cause a decrease of the estimated structural index and depth to source.

In past decades, magnetic surveying had become popular as one of the most effective techniques supporting archaeological prospecting. This is possible because the existence of susceptibility contrasts between the cover soil and several buried finds often causes detectable anomalies. More recently, great advances were made in signal enhancement and boundary analysis of potential field anomalies, thanks to methods allowing a suitable differentiation of the field without making the process unstable. New three-dimensional (3D) imaging techniques provided an estimate of the magnetization distribution within the subsoil by means of high-resolution images of the source distribution. Most of these methods are fast and reliable in the presence of shallow and compact sources, just as in the case of the sources usually occurring in archaeological prospecting. Nevertheless, great effort was spent by the scientific community to overcome serious problems causing low signal-to-noise ratio in the measurements. This chapter provides a step-by-step description of technical solutions adopted to improve the quality of data and to perform a better interpretation of the magnetic anomalies usually associated to the presence of archaeological finds. To this end, a summary of case histories is illustrated giving a general framework of the latest progress in archaeo-magnetism.

Uturuncu volcano (southwestern Bolivia) is localized within one of the largest updoming volcanic zones, the Altiplano Puna Volcanic Complex (APVC). In several geodetic studies the observed uplift phenomenon is analyzed and modeled by considering a deep source, related to the Altiplano Puna Magma Body (APMB). In this framework, we perform a multiscale analysis on the 2003–2010 ENVISAT satellite data to investigate the existence of a multi-source scenario for this region. The proposed analysis is based on Cross-correlation and Multiridge method, pointing out the spatial and temporal multiscale properties of the deformation field. In particular, we analyze the vertical component of ground deformation during two time interval: within the 2005–2008 time interval an inflating source at 18.7 km depth beneath the central zone of the APVC is retrieved; this result is in good agreement with those proposed by several authors for the APMB. Between August 2006 and February 2007, we identify a further inflating source at 4.5 km depth, beneath Uturuncu volcano; the existence of this latter, located just below the 2009–2010 seismic swarm, is supported by petrological, geochemical, and geophysical evidence, indicating as a possible interpretative scenario the action of shallow, temporarily trapped fluids.

We present a constrained inversion procedure based on a priori information derived exclusively from the analysis on the potential field data themselves (self-constrained inversion). The procedure effectively applies to underdetermined problems and involves scenarios where the source distribution can be assumed to be of some simple character, but perturbations from this simplified body might occur. To set up effective constraints we first estimate by an analysis of the gravity or magnetic field some or all of the following source parameters: the source depth-to-the-top, the structural index, the horizontal position of the source body edges and their dip. The second step is incorporating the information related to these constraints in the objective function as depth and spatial weighting functions. We show, through some 2D and 3D synthetic and real data examples, that the sole potential field-based constraints are most of times enough to obtain rather good improvements in the density and magnetization models.

We used a fast method to interpret self-potential data: the depth from extreme points (DEXP) method. This is an imaging method transforming self-potential data, or their derivatives, into a quantity proportional to the source distribution. It is based on upward continuing of the field to a number of altitudes and then multiplying the continued data with a scaling law of those altitudes. The scaling law is in the form of a power law of the altitudes, with an exponent equal to half of the structural index, a source parameter related to the type of source. The method is autoconsistent because the structural index is basically determined by analyzing the scaling function, which is defined as the derivative of the logarithm of the self-potential (or of its [Formula: see text]th derivative) with respect to the logarithm of the altitudes. So, the DEXP method does not need a priori information on the self-potential sources and yields effective information about their depth and shape/typology. Important features of the DEXP method are its high-resolution power and stability, resulting from the combined effect of a stable operator (upward continuation) and a high-order differentiation operator. We tested how to estimate the depth to the source in two ways: (1) at the positions of the extreme points in the DEXP transformed map and (2) at the intersection of the lines of the absolute values of the potential or of its derivative (geometrical method). The method was demonstrated using synthetic data of isolated sources and using a multisource model. The method is particularly suited to handle noisy data, because it is stable even using high-order derivatives of the self-potential. We discussed some real data sets: Malachite Mine, Colorado (USA), the Sariyer area (Turkey), and the Bender area (India). The estimated depths and structural indices agree well with the known information.

We analyse two classes of methods widely diffused in the geophysical community, especially for studying potential fields and their related source distri- butions. The first is that of the homogeneous fractals random models and the second is that of the homogeneous source distributions called “one-point” distributions. As a matter of fact both are depending on scaling laws, which are used worldwide in many scientific and economic disciplines. However, we point out that their appli- cation to potential fields is limited by the simplicity itself of the inherent assumptions on such source distributions. Multifractals are the models, which have been used in a much more general way to account for complex random source distributions of density or susceptibility. As regards the other class, a similar generalization is proposed here, as a multi-homogeneous model, having a variable homogeneity degree versus the position. While monofractals or homogeneous functions are scaling functions, that is they do not have a specific scale of interest, multi-fractal and multi-homogeneous models are necessarily described within a multiscale dataset and specific techniques are needed to manage the information contained on the whole multiscale dataset.

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.

Interpretation of ground-penetrating radar (GPR) data usually involves data processing similar to that used for seismic data analysis, including also migration techniques. Alternatively, in the past few years, microwave tomographic approaches exploiting more accurate models of the electromagnetic scattering have gained interest, owing to their capability of providing accurate results and stable images. Within this framework, this paper deals with the application of a microwave tomography approach, based on the Born Approximation and working in the frequency domain. The case study is a survey performed during the realization of the third lane of the most important highway in southern Italy (the Salerno-Reggio Calabria, near Pontecagnano, Italy). It is shown that such an inversion approach produces well-focused images, from which buried structures can be more easily identified by comparison to traditional radar images. Moreover, the visualization of the reconstruction results is further enhanced through a three-dimensional volumetric rendering of the surveyed region, simply achieved by staggering the reconstructed GPR two-dimensional profiles. By means of this rendering it is possible to follow the spatial continuity of the buried structures in the subsurface thus obtaining a very effective geometrical characterization. The results are very useful in our case where, due to important civil engineering works, a fast diagnosis of the archaeological situation was needed. The quality of our GPR data modelling was confirmed by a test excavation, where a corner of a building and the eastern part of another house, with its courtyard, were found at the depth and horizontal position suggested by our interpretation. Copyright © 2009 John Wiley & Sons, Ltd.

Volcanomagnetic anomalies have been mostly observed during strong eruptions. Our aim is to improve the geomagnetic data analysis to evidence the anomalies occurring in a larger time span, especially in the phases preceding the eruptive events. We developed a time variant statistical approach and applied it to the 2000–2002 Etna geomagnetic temporal series. It is based on an algorithm that statistically predicts the geomagnetic field at the station on the volcanic edifice by that recorded at the remote one. In such a way a number of significant changes in the time series (called statistical innovations), marking the local magnetic field change, were detected. The distribution of such statistical innovations accurately describes the Etna volcanic evolution: we note a progressive increase of the innovation occurrence as the eruptive cycles were approaching and only few and weak innovations at times between the various eruptive cycles. The significance of this analysis is further confirmed by the close agreement among the mean square prediction error, strain release and the volcanic activity behavior. On the contrary, the geomagnetic field at a single station or its difference at two stations do not have any clear correlation with other measured physical quantities. The complex pattern of the prediction error was also investigated by a multifractal analysis. We found that the Holder regularity increases with the intensification of the volcanic activity, implying that innovations tend to be less sporadic and correlated during the major volcanic phases.

We show that potential fields enjoy valuable properties when they are scaled by specific power laws of the altitude. We describe the theory for the gravity field, the magnetic field, and their derivatives of any order and propose a method, called here Depth from Extreme Points (DEXP), to interpret any potential field. The DEXP method allows estimates of source depths, density, and structural index from the extreme points of a 3D field scaled according to specific power laws of the altitude. Depths to sources are obtained from the position of the extreme points of the scaled field, and the excess mass (or dipole moment) is obtained from the scaled field values. Although the scaling laws are theoretically derived for sources such as poles, dipoles, lines of poles, and lines of dipoles, we give also criteria to estimate the correct scaling law directly from the data. The scaling exponent of such laws is shown to be related to the structural index involved in Euler Deconvolution theory. The method is fast and stable because it takes advantage of the regular behavior of potential field data versus the altitude [Formula: see text]. As a result of stability, the DEXP method may be applied to anomalies with rather low SNRs. Also stable are DEXP applications to vertical and horizontal derivatives of a Newtonian potential of various orders in which we use theoretically determined scaling functions for each order of a derivative. This helps to reduce mutual interference effects and to obtain meaningful representations of the distribution of sources versus depth, with no prefiltering. The DEXP method does not require that magnetic anomalies to be reduced to the pole, and meaningful results are obtained by processing its analytical signal. Application to different cases of either synthetic or real data shows its applicability to any type of potential field investigation, including geological, petroleum, mining, archeological, and environmental studies.

Summary We propose a new approach to the study the Normalized Full Gradient, based on assessing the entropy of the normalized downward continued field at each level of continuation. We show that the estimated entropy may be efficiently used to obtain information about potential fields source depth. As entropy decreases when the information is concentrated, we should detect the source depth at the continuation level where the entropy has a local minimum. We test this idea on synthetic and real magnetic data and found rather accurate results.

We describe a multihomogeneity theory for source-parameter estimation of potential fields. Similar to what happens for random source models, where the monofractal scaling-law has been generalized into a multifractal law, we propose to generalize the homogeneity law into a multihomogeneity law. This allows a theoretically correct approach to study real-world potential fields, which are inhomogeneous and so do not show scale invariance, except in the asymptotic regions (very near to or very far from their sources). Since the scaling properties of inhomogeneous fields change with the scale of observation, we show that they may be better studied at a set of scales than at a single scale and that a multihomogeneous model is needed to explain its complex scaling behaviour. In order to performthis task,we first introduce fractional-degree homogeneous fields, to show that: (i) homogeneous potential fields may have fractional or integer degree; (ii) the source-distributions for a fractional-degree are not confined in a bounded region, similarly to some integer-degree models, such as the infinite line mass and (iii) differently from the integer-degree case, the fractional-degree source distributions are no longer uniformdensity functions. Using this enlarged set of homogeneous fields, real-world anomaly fields are studied at different scales, by a simple search, at any local window W, for the best homogeneous field of either integer or fractional-degree, this yielding a multiscale set of local homogeneity-degrees and depth estimations which we call multihomogeneous model. It is so defined a new technique of source parameter estimation (Multi-HOmogeneity Depth Estimation, MHODE), permitting retrieval of the source parameters of complex sources. We test the method with inhomogeneous fields of finite sources, such as faults or cylinders, and showits effectiveness also in a real-case example. These applications showthe usefulness of the new concepts, multihomogeneity and fractional homogeneity-degree, to obtain valid estimates of the source parameters in a consistent theoretical framework, so overcoming the limitations imposed by global-homogeneity to widespread methods, such as Euler deconvolution.

Standard Euler deconvolution is applied to potential-field functions that are homogeneous and harmonic. Homogeneity is necessary to satisfy the Euler deconvolution equation itself, whereas harmonicity is required to compute the vertical derivative from data collected on a horizontal plane, according to potential-field theory. The analytic signal modulus of a potential field is a homogeneous function but is not a harmonic function. Hence, the vertical derivative of the analytic signal is incorrect when computed by the usual techniques for harmonic functions and so also is the consequent Euler deconvolution. We show that the resulting errors primarily affect the structural index and that the estimated values are always notably lower than the correct ones. The consequences of this error in the structural index are equally important whether the structural index is given as input (as in standard Euler deconvolution) or represents an unknown to be solved for. The analysis of a case history confirms serious errors in the estimation of structural index if the vertical derivative of the analytic signal is computed as for harmonic functions. We suggest computing the first vertical derivative of the analytic signal modulus, taking into account its nonharmonicity, by using a simple finite-difference algorithm. When the vertical derivative of the analytic signal is computed by finite differences, the depth to source and the structural index consistent with known source parameters are, in fact, obtained.

SUMMARY The discrete wavelet transform (dwt), using the good property of localization of wavelet bases has been used as a powerful tool in filtering and denoising problems. The continuous wavelet transform (cwt) exploits the upward continuation properties of the field horizontal derivative and allows the location of potential field singularities in a simple geometrical manner. Within the cwt space-scale framework, the lines formed by joining, at different scales, the modulus maxima of the wavelet coefficients (multiscale edge detection method) intersect each other at the position of the point source or along the edges of the causative body. As long as the multiscale edge detection method is applied to experimental data the procedure may, however, fail, since the observed anomalies are the superposition of effects of sources having different density contrast, geometrical size and depths. We show that wavelet transform modulus maxima lines attributed to deep sources do not converge toward the true depths, but yield completely erroneous solutions. On the other hand, use of nth-order derivatives of the potential field allows the enhancement of the shallowest source effects, preventing us from obtaining information on the deeper ones. In this paper we therefore try to overcome this problem by a joint application of cwt and dwt. A localized dwt filter coupled to compactness criterion allows the separation of the effects due to the deeper sources from those of the shallower ones. Hence, the multiscale edge detection method, applied separately to the original and the filtered signals enabled the estimation of the depth of shallower and deeper sources, respectively. This analysis, performed on the gravity anomalies of Sardinia (Italy), has given estimations of the depths to both the Campidano graben and the Moho discontinuity, in good agreement with previous interpretations of gravity and seismic data.

Summary Gravity data inversion is a fundamental tool for geological exploration. A large amount of algorithms have been developed in the past, with different approaches. The choice of the inversion algorithm depends, mainly, on the geological contest, the kind of solution desired, its resolution at the end of the process, the availability of a priori information and on how they can be included in the inversion algorithm. A priori information has, often, a key-role in the inversion process. However, in case of reconstruction of salt and subsalt structures, seismic information may be poor and unfeasible. So we need to constrain the model with other kind of information. In this work, we present a part of recent Eni R&D activity focused on gravity data inversion. The shown results are mainly based on the Data Space Inversion algorithm (Pilkington 2009), originally presented for the magnetic case, and here extended to the gravity problem. We also putted a strong effort on the computational side of the problem, taking advantage from our experience in HPC (High Performance Computing), in order to speed up the inversion process and so enable its use at industrial level. In this paper we present and discuss some results regarding the application of the methodology to the SEAM (SEG Advance Modeling, 2007) demonstrating that the algorithm allows a consistent depth and density model.

Many methods have been used to upward continue potential field data. Most techniques employ the Fast Fourier transform, which is an accurate, quick way to compute level-to-level upward continuation or spatially varying scale filters for level-to-draped surfaces. We here propose a new continuation approach based on the minimum-length solution of the inverse potential field problem, which we call Volume Continuation (VOCO). For real data the VOCO is obtained as the regularized solution to the Tikhonov problem. We tested our method on several synthetic examples involving all types of upward continuation and downward continuation (level-to-level, level-to-draped, draped-to-level, draped-to-draped). We also employed the technique to upward continue to a constant height (2500 m a.s.l.), the high-resolution draped aeromagnetic data of the Ischia Island in Southern Italy. We found that, on the average, they are consistent with the aeromagnetic regional data measured at the same altitude. The main feature of our method is that it does not only provide continued data over a specified surface, but it yields a volume of upward continuation. For example, the continued data refers to a volume and thus, any surface may be easily picked up within the volume to get upward continuation to different surfaces. This approach, based on inversion of the measured data, tends to be especially advantageous over the classical techniques when dealing with draped-to-level upward continuation. It is also useful to obtain a more stable downward continuation and to continue noisy data. The inversion procedure involved in the method implies moderate computational costs, which are well compensated by getting a 3D set of upward continued data to achieve high quality results.

In this paper we present an improvement to the local wavenumber method able to overcome its instability caused by the use of high-order derivatives. We make use of the stable properties of the Depth from EXtreme Points (DEXP) method, in which the depth to the source is determined at the extreme points of the field scaled with a power law of altitude. As DEXP transformation is applied to the local wavenumber function, we show that the scaling law is independent on the structural index. This allows the technique to be fully automatic, so that it may be implemented as a very fast imaging method, mapping every source, of different homogeneity degree, at the correct depth.

In potential field inversion problems we usually solve underdetermined systems and realistic solutions may be obtained by introducing a depth-weighting function in the objective function. The choice of the exponent of such power-law is crucial. It was suggested to determine it from the field-decay due to a single source-block; alternatively it has been defined as the structural index of the investigated source distribution. In both cases, when k -order derivatives of the potential field are considered, the depth-weighting exponent has to be increased by k with respect that of the potential field itself, in order to obtain consistent source model distributions. We show instead that invariant and realistic source-distribution models are obtained using the same depth-weighting exponent for the magnetic field and for its k -order derivatives. A similar behavior also occurs in the gravity case. In practice we found that the depth weighting-exponent is invariant for a given source-model and equal to that of the corresponding magnetic field, in the magnetic case, and of the 1st derivative of the gravity field, in the gravity case. In the case of the regularized inverse problem, with depth-weighting and general constraints, the mathematical demonstration of such invariance is difficult, because of its non-linearity, and of its variable form, due to the different constraints used. However, tests performed on a variety of synthetic cases seem to confirm the invariance of the depth-weighting exponent. A final consideration regards the role of the regularization parameter; we show that the regularization can severely affect the depth to the source because the estimated depth tends to increase proportionally with the size of the regularization parameter. Hence, some care is needed in handling the combined effect of the regularization parameter and depth weighting.

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 Authors 2014. Published by Oxford University Press on behalf of The Royal Astronomical Society.

An integrated review of existing geological and geophysical data – partly acquired during oil and gas exploration – combined with new data provided by deep geothermal studies of selected key areas, was used for the 3D modeling and mapping of the top of the geothermal reservoir developed at a regional scale in Sicily (Central Mediterranean). The resulting 1:500,000 scale map covers the area of the whole Sicily (about 25,700 km2) and is devoted to represent the main input for both the thermal modeling and the evaluation of geothermal potential at a regional scale. As the map indicates the distribution at depth of a likely target for geothermal drilling, it can be also used as a rough indicator of expected drilling cost for geothermal projects. Such a map can be seen as a useful planning tool for any geothermal project, and related exploration to be carried out in the Sicily region in the future.

In this paper we present a new method for source parameter estimation, based on the local wavenumber function. We make use of the stable properties of the Depth from EXtreme Points (DEXP) method, in which the depth to the source is determined at the extreme points of the field scaled with a power-law of the altitude. Thus the method results particularly suited to deal with local wavenumber of high-order, as it is able to overcome its known instability caused by the use of high-order derivatives. The DEXP transformation enjoys a relevant feature when applied to the local wavenumber function: the scaling-law is in fact independent of the structural index. So, differently from the DEXP transformation applied directly to potential fields, the Local Wavenumber DEXP transformation is fully automatic and may be implemented as a very fast imaging method, mapping every kind of source at the correct depth. Also the simultaneous presence of sources with different homogeneity degree can be easily and correctly treated. The method was applied to synthetic and real examples from Bulgaria and Italy and the results agree well with known information about the causative sources.

We processed and interpreted a newly acquired geophysical dataset (WS10 project) integrated with vintage seismic profiles to study the evolution of the Sardinian passive margin of the Sardo-Provencal basin. The most prominent features of the area of study are a consequence of the Messinian Salinity Crisis (MSC) which produced a thick salt layer in the deep basin. Complex halokinetic tectonics deformed the overlying Messinian Upper Unit (UU) and the Pliocene–Quaternary (PQ) sediments, sometimes also bending the sea bottom. A thinner autochthonous salt in the UU suggests at least one further low stand during the late MSC. The Messinian Erosional Surface (MES) separates the erosional truncation of the pre-Messinian sedimentary sequence from the overlaying Plio-Pleistocene sequence with a clear angular discordance, which almost completely masks possible earlier erosional events. The MES is present across the middle and upper slopes of the Sardinian margin and can be easily recognized down to the lower slope, where it interfingers with the Messinian evaporites which gradually thicken basinward. We propose a correlation between the main erosional surface of the lower slope and the salt layer recognized in the UU as a consequence of a falling stage during the last part of the MSC. The seismic profiles now available along the West Sardinian margin provide novel and crucial information to correlate the Messinian depositional units in the deep basin with those present in the lower continental slope and with the erosional surface of the middle and upper slopes. Our data and interpretation highlight the main units of the Messinian trilogy and further internal sequences that are probable related to cyclic fluctuations during their deposition.

In this paper we describe the application of GPR and resistivity (dipole-dipole) methods to an archeological investigation. The aim of this study was to obtain information about buried pre-Roman structures below the Roman residence “Casa dell’Atrio a Mosaico,” in the ancient town of Ercolano (Naples). A number of resistivity (dipole-dipole) and GPR profiles were carried out on the mosaic floor of the house. To perform resistivity profiles in a nondestructive way on the mosaic floor, special flat-base copper electrodes were used. The electrode spacing used was [Formula: see text] and [Formula: see text]. GPR data were collected using [Formula: see text] and [Formula: see text] antennas. The radar profiles were carried out primarily along the electrical profiles in order to obtain a direct comparison between the two methods. Both resistivity and GPR methods evidenced a number of structures buried at shallow depths. Both methods located a structure identified as a conductive body with electric parameters strongly contrasting those of the surrounding material. The archaeological interpretation of such structures is in terms of remnants of walls of pre-Roman houses and of a cistern used for collecting rain, connected to a pipe.

An analysis of the field scaling power spectrum yields useful information about the source distribution, but it is uncertain whether deterministic, random, fractal or mixed approaches have to be used for the interpretation. To this end, the scaling properties of potential field spectra are analysed for a number of different source models of geological interest. Besides the models of Naidu (purely random sources) and Spector and Grant (gross block statistical ensembles) we consider other types of density and magnetization distributions with spectral exponents in the fractal range, such as a single homogeneous body with a random white source distribution. Spectral slopes in the fractal range are obtained. We also study the effects of important natural sources, such as salt domes and sedimentary basins, representing them with simple Gaussians or combinations of Gaussian signals. The same spectral slopes as for gravity signals generated by 3-D fractal source distributions are found for them. Hence the power law decay of the field is not a characteristic only of fractal source models. If a 3-D fractal source distribution is assumed a priori, a way of verifying the goodness of the model is to examine the whitened field at source level. The probability that the whitened field derives from a random white population is estimated for synthetic and real anomalies by applying the usual statistical tests.

Numerous methods have been used for upward continuation, but most of them require data on a regular grid. Gridding can introduce errors that affect the continued data in an unpredictable way. To avoid this problem, we design a continuation operator used for the direct continuation of scattered data on a 3-D basis. In this approach a harmonic function, satisfying the constraints imposed by the measured data, is developed. The continuation is written in the form of a linear combination of the measured data, but it depends on the arbitrary choice of the topographic zero level. However, the coefficients of the linear combination depend only on the position of the data points. This allows the zero level to be estimated on the basis of the continuation of synthetic anomalies calculated between the starting and ending surface. An important feature of the method is its local character, which allows the reduction of computation time. Also, the stability of the method for noisy data is reasonably good. The method is applied to both synthetic and real cases. Synthetic examples show how gridding‐related errors may affect the continuation when an irregular distribution of data points and a variable topography are considered.

A simple and fast determination of the limiting depth to the sources may represent a significant help to the data interpretation. To this end we explore the possibility of determining those source parameters shared by all the classes of models fitting the data. One approach is to determine the maximum depth-to-source compatible with the measured data, by using for example the well-known Bott–Smith rules. These rules involve only the knowledge of the field and its horizontal gradient maxima, and are independent from the density contrast. Thanks to the direct relationship between structural index and depth to sources we work out a simple and fast strategy to obtain the maximum depth by using the semi-automated methods, such as Euler deconvolution or depth-from-extreme-points method (DEXP). The proposed method consists in estimating the maximum depth as the one obtained for the highest allowable value of the structural index ( N max ). N max may be easily determined, since it depends only on the dimensionality of the problem (2D/3D) and on the nature of the analyzed field (e.g., gravity field or magnetic field). We tested our approach on synthetic models against the results obtained by the classical Bott–Smith formulas and the results are in fact very similar, confirming the validity of this method. However, while Bott–Smith formulas are restricted to the gravity field only, our method is applicable also to the magnetic field and to any derivative of the gravity and magnetic field. Our method yields a useful criterion to assess the source model based on the (∂ f /∂ x ) max / f max ratio. The usefulness of the method in real cases is demonstrated for a salt wall in the Mississippi basin, where the estimation of the maximum depth agrees with the seismic information.

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.

In potential field inversion problems we usually solve underdetermined systems and this leads to a very shallow solution, typically known as minimum length solution. This may be avoided introducing a depth weighting function in the objective function (Li and Oldenburg, 1996). In this paper we derive invariance rules for either the minimum norm minimization and for the regularized inversion with depth weighting and positivity constraints. For a given source class, corresponding to a specific structural index N, the invariance rule assures that the same solution is obtained inverting the magnetic (or gravity) field or any of its nth order vertical derivatives. Although we demonstrate mathematically this invariance rule for the minimum norm minimization only, it is shown to occur also for the regularized inversion with depth weighting and positivity constraints. In this case, a source-class invariant form of depth weighting is derived, referring to that of the magnetic field, in the magnetic case, and to the 1st derivative of the gravity field, in the gravity case. We also illustrate how the combined effect of regularization parameter and depth weighting influences the estimated source model depth in the regularized inversion with depth weighting and positivity constraints.

We compare the effect of the smoothing of magnetic data on the estimation of the structural index by Euler deconvolution. We analyze physically-based as well as mathematical filters. In general, we found that low-pass filtering the data leads to an increase of the estimated structural index.

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.

Summary We describe a multiscale method for source parameter estimations (depth to the source, homogeneity degree) based on a new theory, which we propose for studying explicitly inhomogeneous field. The theory consists in enlarging the set of homogeneous fields with those having a fractional homogeneity-degree and in relaxing the standard homogeneous equation to a local homogeneity equation. We show the validity of the method as applied to the inhomogeneous-degree gravity fields of a finite vertical cylinder.

[1] In potential field problems, the continuous wavelet transform (CWT) has allowed the estimation of the source properties, such as the depth to the source and the structural index (N). The natural choice for the analyzing wavelets has been the set belonging to the Poisson kernel. However, a much larger set of analyzing wavelets has been used for analyzing signals other than potential fields. Here we extend the CWT of potential fields to other wavelet families. Since the field is intrinsically dilated with Poissonian wavelets from the source depth to the measurement level, distortions are unavoidably introduced when CWT uses a different wavelet from the measurement level to other scales. To fix the problem, we define a new form for the continuous wavelet transform convolution product, called “composite continuous wavelet transform” (CCWT). CCWT removes the field dilations with Poisson wavelets, intrinsically contained at the measurement level and replaces them with dilations performed with any other kind of wavelet. The method is applied to synthetic and real cases, involving sources as poles, dipoles, intrusions in complex magnetized basement topography and buried steel drums, from measurements taken at the Stanford University test site. CCWT takes advantage from the special features of the several considered wavelets, e.g., the Gaussian wavelet is useful for its low pass filtering characteristic and Morlet wavelet for its localization property. Hence, depending on the case, an important parameter for the choice of the analyzing wavelet is its central frequency.

An efficient way to deal with the inherent ambiguity in potential fields interpretation is to determine some model parameters common to classes of models fitting the data. One of these approaches is to determine the maximum depth to sources, for example by the well known Bott-Smith rules. These rules involve only the maximum of the field and of its horizontal gradient and are independent from the density contrast. Thanks to the direct relationship between structural index and depth to sources we work out a simple and fast strategy to obtain the same results by using the DEXP method. It consists in varying the structural index, related to the scaling exponent of the DEXP transformation, and selecting as the maximum depth the one obtained for the highest allowable value of the structural index. We tested our approach on synthetic models against the results obtained by the classical Bott-Smith formulas and the results are in fact very similar, confirming the validity of this method. The same approach could be valid also for other semi-automated methods, such as Euler deconvolution, demonstrating that this class of methods have potentialities still not fully exploited.

A 14-year-long microgravity data set (October 1994 to September 2007) collected along a 24-km east-west trending profile of 19 stations was analyzed to detect underground mass redistributions related to the volcanic activity involving the southern flank of Mt Etna (Italy). A multiresolution wavelet analysis was applied to separate the volcano-related anomalies from the unwanted components. The residual image having both spatial extension and temporal duration evidenced two complete gravity increase/decrease cycles mainly affecting the central and eastern stations of the profile. The first gravity increase (early 1995 to the end of 1996) and decrease (end of 1996 to late in 1998) cycle reached a maximum amplitude of approximately 90 micro Gal. The second gravity increase (mid-1999 to mid-2000) and decrease (mid-2000 to early-2004) cycle attained an amplitude of about 80 micro Gal. After about 5 years of a persistent negative gravity anomaly, a new semicycle started at the end of 2006 and continued during the last survey carried out in September 2007. The density changes, modeled over time since 1994 using a Quadratic Programming algorithm, are mainly located at a depth of 2–4 km bsl in a region recognized to be a preferential pathway of magma rising and an intermediate zone of magma storage/withdrawal. The computed positive mass variations of about 105 x 10^9 kg were interpreted as magma accumulation, while negative mass changes of about -120 x 10^9 kg were associated with either magma drainage or opening of new voids by tectonic stresses within a source volume, where tensional earthquakes occurred.

We use the Depth from Extreme Points method (DEXP) for self-potential (SP) data interpretation. We describe how to estimate in a fast way the depth to the source, the structural index and the shape factor of the causative sources. The method is demonstrated for synthetic examples and shows a great agreement with the true model parameters. Also, it is applied to real data from a Malachite Mine, Jefferson County, Colorado, USA. The estimated depth and structural index are in a good agreement with the known geology.

This work is focused on the integration of infrared thermography and ground penetrating radar for the inspection of architectonic structures. First, laboratory tests were carried out with both techniques by considering an ad hoc specimen made of concrete and with the insertion of anomalies of a different nature and at different depths. Such tests provided helpful information for ongoing inspections in situ, which were later performed in two important Italian archaeological sites, namely Pompeii (Naples) and Nora (Cagliari). In the first site, the exploration was devoted to the analysis of the wall paintings of Villa Imperiale with the aim of evaluating the state of conservation of frescoes as well of the underneath masonry structure. As main findings, the applied techniques allowed outlining some areas, which were damaged by ingression in-depth of moisture and/or by disaggregation of the constituent materials, and also for recognition of previous restoration. In the archaeological area of Nora, instead, the attention was driven towards the evaluation of the state of degradation of the theatre remnants. Our prospections show that the front side of the theatre, being more strongly affected by degradation, needs a massive restoration work. As a general result, we demonstrated that a joint interpretation of infrared thermography and ground penetrating radar data supplies detailed 3D information from near-surface to deep layers, which may assist in restoration planning.

A new automated technique, based on the continuous wavelet transform (CWT), has been developed in order to analyze potential field data. Estimations of horizontal location, depth and structural index are obtained as results utilizing an automated approach. The technique selects the ridges along a user- specified profile and performs an automatic source parameters research. The estimated source position and structural index are characterized by solution attributes that are used to assess the quality of the estimation. A solution selection on the basis of these parameters is subsequently performed. Applications to either synthetic or real-case data show the usefulness of the automated approach.

Since the Sixties, the turning point marked by the Plate Tectonics global theory has provided new ideas for the interpretation of the complex geodynamic evolution of the Mediterranean area. The renewed interest that followed gave a strong impulse to the geological and geophysical investigations of the Mediterranean area and, more specifically, of the Tyrrhenian Basin. Therefore, large scale geophysical surveys and oceanographic cruises were carried out until the end of the Eighties to fill the gap of geophysical information existing in the area until then. Afterwards, short scale surveys were prevalent to improve the detail of the information in areas of geodynamic interest. The gathered data sets allowed new models to be formulated, improving the knowledge of the crustal and lithospheric structure of the Tyrrhenian Basin (and surrounding areas) and the reconstruction of its complex geodynamic evolution. In this frame, the contribution of gravity and magnetic investigations has been unquestionable and deserves a wide-ranging review both on large and small scale. The main features of the potential fields resulting from these surveys will be described and the interpretative models suggested by several authors will be summarized.

In this paper we will explore the performances of the joint inversion, through the Euler reduced equations, of more ridges converging to the same source (Euler deconvolution for a multiridge set). The application to the total field due to synthetic sources demonstrate the improvements in the source parameters estimation with respect to the single ridge inversion even in the case of the simultaneous estimation of two parameters. The method yields a good clustering of the solutions and could not require any acceptance criterion or a-posteriori processing.

A new method based on 3D multiridge analysis of potential fields assumes a 3D subset in the harmonic region and studies the behavior of potential field ridges, which are built by joining extreme points of the analyzed field computed at different altitudes. Three types of ridges are formed by searching for the zeros of the first horizontal and first vertical derivatives of the potential field (types I and II, respectively) and the zeros of the potential field itself (type III). This method uses a redundant set of ridges, called a multiridge set, to determine source type and location. For homogeneous potential fields generated by simple sources, all of the ridges are straight lines converging to the source position. This method analyzes the multiridges by using a geometric criterion to find the source position at the intersection of the multiridge set and by solving the three reduced Euler equations associated with ridge types I, II, and III. The reduced Euler type I and II equations are used to obtain the structural index and the vertical and horizontal source positions; equation type III estimates the horizontal and vertical source positions. Tests on synthetic as well as the Bishop model field yield good results even with noise-corrupted data. Results obtained using magnetic data collected over the wreck of a military ship in the Tyrrhenian Sea successfully determine its vertical and horizontal positions and the structural index.

Summary We present a method to determine the source properties of potential fields by exploring the field at multiple scales. At each scale the depth to the source and the structural index are determined by analyzing the scaling function τ, a dimensionless quantity which defines the scaling properties of the field, and the differential scaling function β. For homogeneous fields, the estimated results are scale invariant. The source properties are determined either by a geometrical method applied to the a single ridge or by solving for the zeros of β.

In this paper we present a study of the structural setting of the volcanic island of Ischia by the analysis and interpretation of high-resolution aeromagnetic and self-potential data recently acquired over the island. The magnetic data allowed us to locate the main anomaly sources and lineaments of the island and its offshore surroundings, while the self-potential (SP) data provided information on both the structural pattern of the resurgent caldera and the high-temperature fluid circulation. An inversion of the acquired magnetic and SP data, and a joint modelling of the magnetic data and of a previous gravity data set along a SW–NE profile allowed us to build a model of the island. The model is characterized by the presence of an igneous–very likely trachytic–structure, whose top is located at 1200–1750 m b.s.l. Such a body, possibly formed by several neighbouring intrusions, has a density contrast with the pyroclastic cover of about 0.4 gr/cm 3 and its central-western part, below Mt. Epomeo, seems to be demagnetized. The demagnetization should be connected to the high geothermal gradient measured in this portion of the island and may be due to hydro-chemical alteration processes and/or to the possible presence of partially melted spots within the intrusion. Our outcome is consistent with the results of previous geophysical, geo-volcanological and geothermal studies.

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