Maurizio Fedi

On the application of Euler deconvolution to the analytic signal

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.

Spatiotemporal gravity variations to look deep into the southern flank of Etna volcano

[1]xa0A 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 μGal. The second gravity increase (mid-1999 to mid-2000) and decrease (mid-2000 to early-2004) cycle attained an amplitude of about 80 μGal. 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 × 109 kg were interpreted as magma accumulation, while negative mass changes of about −120 × 109 kg were associated with either magma drainage or opening of new voids by tectonic stresses within a source volume, where tensional earthquakes occurred.

Composite continuous wavelet transform of potential fields with different choices of analyzing wavelets

[1]xa0In 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.

Regularity Analysis Applied to Well Log Data

Well logs are largely used for oil exploration and production in order to obtain geological information of rocks. Several parameters of the rocks can be scanned and interpreted in term of lithology and of the quantity and kind of fluids within the pores. Generally the drilled rocks are mostly sedimentary and the modelling is mainly petrophysical. Here we analyze four logs from the KTB Main Borehole, drilled for the German Continental Deep Drilling Program. The hole cuts across crystalline rocks like amphibolites, amphibolite-metagabbros, gneiss, variegated units and granites.

Euler Deconvolution of Vertical Profiles of Potential Field Data

Euler deconvolution has become a standard tool in rapid, semi-automated interpretation of potential fields. Its success derives from its flexibility, providing the source position and a parameter describing the shape of the source (structural index, SI) for all the possible one point sources (point, horizontal or vertical line, dike, contact). The Euler algorithms are particularly prone to regional (even constant) fields and noise (e.g.: FitzGerald et al., 2004). The presence of regional fields was the main reason for the strategy proposed in the first formulation of the method, that considered a moving window approach in which the regional field was approximated as a constant. This constant background (B) and the structural index are coupled in the Euler interpretation formula, and it was noted that the simultaneous estimation of SI, B and z0 was not stable. For this reason Thompson (1982) suggested to input the value of the structural index to obtain the position (x0, z0) of the point-source and B. Anyway this approach suffers of a certain degree of subjectivity because the choice of SI influences directly the depth estimate. Thus different approaches were devised to deal with the problem of the background fields allowing to obtain the SI as an unknown (e.g.: Stavrev, 1997; Hsu, 2002; Nabighian and Hansen, 2001). Meanwhile, other methods recovering the same parameters (position and SI) were proposed (e.g.: Smith et al., 1998; Salem and Ravat, 2003). Among them, the Continuous Wavelet Transform method (e.g.: Hornby et al., 1999; Sailhac and Gibert, 2003) that uses the information coming from the field upward continued to a set of different altitudes. This method has reduced sensitivity to noise with respect to the Euler deconvolution, because in the upward continued field the highwavenumbers are naturally attenuated. This method, however, tries to recover the information about the source using simultaneously data at as much altitudes as possible, both with the geometrical method (to recover z0) and the linear regression approach (to recover z0 and SI). In this paper we present a formulation of the Euler deconvolution method that uses a vertical profile of data above an anomaly source. It can be applied to almost all the Euler algorithms to tackle the problem of the regional field and, using upward continued fields, it efficiently deals with the problem of noise. Method

Contour map of the top of the regional geothermal reservoir of Sicily (Italy)

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 25,700u2005km2) 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.

Nonstationary analysis of geomagnetic time sequences from Mount Etna and North Palm Springs earthquake

[1]xa0Volcanomagnetic and/or seismomagnetic effects are geomagnetic variations generated before eruptions and/or seismic events. Our aim is to analyze geomagnetic time series to detect the volcanomagnetic and/or seismomagnetic effects among a number of other variations. Two advanced signal-processing techniques are proposed to analyze the geomagnetic time series. The first technique, called Continuous Wavelet Transform Singularity Analysis (CWTSA), is based on the Continuous Wavelet Transform; the second, called Time-Variant Statistical Analysis of Nonstationary Signals (TVANS), is based on a time-varying adaptive algorithm (Recursive Least Squares). Both techniques are very effective in detecting the geomagnetic variations at the time instants likely linked to volcanic and/or seismic activity. The application of these methodologies to geomagnetic time sequences, respectively, recorded on Mount Etna during the volcanic activity of 1981 and in North Palm Springs during the seismic events of 8 July 1986 yields a good correspondence between events detected by both techniques and volcanic end seismic events. The statistical significance of geomagnetic time series was also assessed to verify the obtained results from CWTSA and TVANS. It was defined at significance level of 95% in the wavelet power spectrum for the difference of the geomagnetic time series aiming at distinguishing the most “significant” events when they are upon this one.

DEXP: a fast method to determine the depth to the sources of potential fields

Summary A new method is described to determine the depth to sources of potential fields. The theory is here developed for sources such as poles or dipoles, but it may be extended to lines of poles, line masses, lines of dipoles, dykes, ribbons and so on. The field is scaled at several altitudes following a specific law analytically determined. The depth to the source is finally obtained by determining the scaled field extreme points. The method is fast and stable. It may be applied to any vertical derivative of a Newtonian potential, by using theoretically determined scaling functions for each order of derivative.

Scaling Laws in Geophysics: Application to Potential Fields of Methods Based on the Laws of Self-similarity and Homogeneity

We analyse two classes of methods widely diffused in the geophysical community, especially for studying potential fields and their related source distributions. 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 application 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.

Research Note: Compact Depth from Extreme Points: a tool for fast potential field imaging

We propose a fast method for imaging potential field sources. The new method is na variant of the “Depth from Extreme Points,” which yields an image of a quantity nproportional to the source distribution (magnetization or density). Such transformed nfield is here transformed into source-density units by determining a constant with nadequate physical dimension by a linear regression of the observed field versus the nfield computed from the “Depth from Extreme Points” image. Such source images are noften smooth and too extended, reflecting the loss of spatial resolution for increasing naltitudes. Consequently, they also present too low values of the source density. nWe here show that this initial image can be improved and made more compact to nachieve a more realistic model, which reproduces a field consistent with the observed none. The new algorithm, which is called “Compact Depth from Extreme Points” niteratively produces different source distributions models, with an increasing degree nof compactness and, correspondingly, increasing source-density values. This is done nthrough weighting the model with a compacting function. The compacting function nmay be conveniently expressed as a matrix that is modified at any iteration, based non the model obtained in the previous step. At any iteration step the process may nbe stopped when the density reaches values higher than prefixed bounds based on nknown or assumed geological information. As no matrix inversion is needed, the nmethod is fast and allows analysing massive datasets. Due to the high stability of nthe “Depth from Extreme Points” transformation, the algorithm may be also applied nto any derivatives of the measured field, thus yielding an improved resolution. The nmethod is investigated by application to 2D and 3D synthetic gravity source distributions, nand the imaged sources are a good reconstruction of the geometry and density ndistributions of the causative bodies. Finally, the method is applied to microgravity ndata to model underground crypts in St. Venceslas Church, Tovacov, Czech Republic.