R. O. Hansen

Unification of Euler and Werner deconvolution in three dimensions via the generalized Hilbert transform

The extended Euler deconvolution algorithm is shown to be a generalization and unification of 2-D Euler deconvolution and Werner deconvolution. After recasting the extended Euler algorithm in a way that suggests a natural generalization to three dimensions, we show that the 3-D extension can be realized using generalized Hilbert transforms. The resulting algorithm is both a generalization of extended Euler deconvolution to three dimensions and a 3-D extension of Werner deconvolution. At a practical level, the new algorithm helps stabilize the Euler algorithm by providing at each point three equations rather than one. We illustrate the algorithm by explicit calculation for the potential of a vertical magnetic dipole.

Multiple-source Euler deconvolution

Rapid three-dimensional (3-D) source location methods can be extremely useful in framing a subsurface structural model from gravity or magnetic data. However, existing implementations of Euler deconvolution are limited to a single source in each window. This can be a significant limitation in areas of complex structure. We have generalized the method to the multiple-source case, and implemented the 3-D algorithm. Results from synthetic data and from the Gold Acres mining district in Nevada suggest that the new algorithm can be a useful interpretive tool.

Study of the crustal thickness and the subducting lithosphere in Greece from gravity data

The area of Greece has experienced a complex tectonic history dominated by the subduction of the African plate beneath Eurasia. In this study the variations of the crustal thickness in the area of Greece were obtained by means of the multiple-source Werner deconvolution (MSWD) method applied to gravity data. Thicknesses of 40-49 km are estimated beneath the Hellenides mountain belt to the west. Eastward thinning of the crust, to thicknesses ranging from 25 km in the north to 30 km in the south is seen in the Aegean region. These results are in good agreement with recent seismological results, demonstrating that the MSWD method successfully treated the problem. Using the crustal model we derived, we computed the gravity effect of the crust and extracted it from the Bouguer anomaly. We also extracted the gravity effect of the subducting lithosphere from the Bouguer anomaly, producing a residual map where most of the original gravity variation has been successfully removed. The remaining anomalies appear related to near-surface features and an area of low-velocity mantle in the central Aegean Sea.

An analytical expression for the gravity field of a polyhedral body with linearly varying density

Over the past 24 years, the gravity fields of homogeneous polyhedra have been studied extensively (Paul, 1974; Barnett, 1976; Okabe, 1979; Golizdra, 1981; Strakhov et al., 1986a, 1986b; Gotze and Lahmeyer, 1988; Pohanka, 1988; Ivan, 1990; Pohanka, 1990; Holstein and Ketteridge, 1996). This intensive scrutiny is well deserved because polyhedra provide a framework in which complex body shapes can be modeled parsimoniously and because the analytic expressions for the gravity fields of uniform polyhedra involve only elementary transcendental functions available as part of the run‐time library for most computational programming languages.

The rise and fall of early oil field technology: The torsion balance gradiometer

Today elementary, physics students take for granted such quantities as “big G,” the universal gravitational constant. In fact in the late 1700s the value of this quantity was unknown, and the quest to determine it led to some of the earliest geophysical instrumentation. Just after the Revolutionary War in the United States, Cavendish developed the first system to measure the universal gravitational constant, the familiar “big G.” Unfortunately, for geologists (at this time still mostly “gentlemen scientists”), this apparatus produced data which were difficult to interpret geologically, and it was far too large and cumbersome for field use. The geologic limitation was that the system only measured the horizontal derivative of a horizontal component of the gravity field, a quantity which by itself is difficult to interpret. Thus no applications of this elegant yet laboratory‐bound instrument emerged.

On the use of complex attributes and the inferred source parameter estimates in the exploration of archaeological sites

Complex attribute analysis is used to extract parameters of the buried structures that give rise to anomalous total magnetic field. The attributes by themselves aid the interpretation because they can delineate the edges of the concealed targets. Tests on synthetic and real data demonstrate this property. Parameters such as the local depth, dip, dip azimuth, strike and susceptibility contrast are obtained from the complex attributes on the basis of an analytical expression for the effect of a simple subsurface model. The simplest one, the sloping contact model, is used in this study. The local parameters of the concealed targets aid the interpretation further. They can also be used alone to interpret the anomalous field. Interference of the signals caused by nearby buried structures and the noise level limit the extent to which the source parameter estimates can be used. Copyright

Constrained inversion of gravity fields for complex 3-D structures

As part of a research program to develop gravity interpretation tools that can be merged with seismic techniques, a full 3-D complex structural inversion scheme for (possibly multibody) polyhedral models has been developed. The forward modeling algorithm was adopted from previous work. Because the inverse problem is generally very ill posed, several methods of regularizing the inversion were investigated and a combination of the most useful was adopted. The combination includes (i) a structured matrix formulation for the system equations, (ii) an analytical expression for the Jacobian calculation, (iii) first‐derivative damping, (iv) a choice of damping parameter based on a variation of the trust region method, (v) a weighted scheme for parameter correction, and (vi) complete freezing of degrees of freedom found not to influence the gravity field significantly. This combination yields a robust inversion which was successfully demonstrated on data over the Galveston Island salt dome, offshore Texas. Variat...

3D multiple-source Werner deconvolution for magnetic data

Werner deconvolution has been widely used for at least 30 years for rapid interpretation of magnetic data. Since 1993, a multiple-source generalization of the method has been known, and at least two implementations of the algorithm are in use. Recently, Werner deconvolution has been extended to three dimensions through the use of generalized Hilbert transforms. In this paper, a multiple-source extension of the 3D Werner algorithm is developed which also generalizes the 2D multiple-source algorithm. The implementation of this algorithm is tested on both 3D multiple-source synthetic data, for which good agreement with the model is obtained, and on complex data from the Albuquerque basin, which yields results corresponding well with other interpretations and with known geology.

Magnetic modeling for highly permeable bodies with remanent magnetization

A numerical method is given for the solution of the 3D magnetostatic problem for a rb i t ra r i ly shaped bodies wi th h igh s u s c e p t i b i l i t y , including remanent magnet iza t ion . The in tegra l equa t ion i s so lved us ing the method of f in i te e lements and a modified version of Pohankas (1988) a lgor i thm for the magne tos ta t i c case . Some theore t i ca l examples a re cons ide red and demagnetization factors are shown. These methods p romise to be essen t ia l in in te rp re t ing the magne t ic f i e ld over a reas conta in ing h igh ly magnet ic ob jec t s , such as l a n d f i l l s w i t h b u r i a l s i n s t e e l d r u m s .

Two-dimensional inverse filtering for the rectification of the magnetic gradiometry signal

The magnetic difference, the quantity measured by magnetic gradiometers, is considered to be the convolution between a function that controls the anomaly pattern and another that controls the strength signal. These are called shape and amplitude functions, respectively. They are distinct and analytically determined; thus, after the assessment of a suitable model, its shape function can be inverted to serve as a filter. If this filter is next convolved with the measured field, a series of amplitude functions is recovered, provided that the subsurface structures can be simulated by a combination of a number of models similar to the one whose anomaly was inverted. The recovered series is essentially the subsurface distribution of the amplitude function. Alternatively, the scheme can be viewed as a transformation of the original field of magnetic differences. The signals, transformed in this context, comprise rectified monopolar versions of the originals, positioned directly above the centre of the targets. Furthermore, their amplitude is a measure of the magnetization of the targets. Thus, the new anomalies can be viewed as a kind of magnetization or susceptibility mapping. Inversion filters are computed in the Wiener mode by inverting the shape function of the anomalies caused by vertical sided finite prisms. This particular model is appropriate for a wide variety of targets that are commonly met in archaeological prospecting. Converting the magnetic signal from the dual lobe pattern to single monopolar anomalies cause them to resemble the usual outcome of resistivity exploration. In this context, this work aims to form a scheme that makes the magnetic gradiometry outcome directly comparable to the outcome of resistivity mapping. Furthermore, it can be applied without contradicting or excluding any other processing operation. The method is implemented in a FORTRAN program that is reasonably user- friendly. The efficiency of the scheme is tested both on synthetic and real data sets.