Standard Euler deconvolution is applied to potential-field\ud functions that are homogeneous and harmonic. Homogeneity\ud is necessary to satisfy the Euler deconvolution equation itself,\ud whereas harmonicity is required to compute the vertical\ud derivative from data collected on a horizontal plane, according\ud to potential-field theory. The analytic signal modulus of a\ud potential field is a homogeneous function but is not a harmonic\ud function. Hence, the vertical derivative of the analytic signal\ud is incorrect when computed by the usual techniques for\ud harmonic functions and so also is the consequent Euler deconvolution.\ud We show that the resulting errors primarily affect\ud the structural index and that the estimated values are\ud always notably lower than the correct ones. The consequences\ud of this error in the structural index are equally important\ud whether the structural index is given as input (as in standard\ud Euler deconvolution) or represents an unknown to be solved\ud for. The analysis of a case history confirms serious errors in\ud the estimation of structural index if the vertical derivative of\ud the analytic signal is computed as for harmonic functions.We\ud suggest computing the first vertical derivative of the analytic\ud signal modulus, taking into account its non-harmonicity, by\ud using a simple finite-difference algorithm.When the vertical\ud derivative of the analytic signal is computed by finite differences,\ud the depth to source and the structural index consistent\ud with known source parameters are, in fact, obtained
A B S T R A C TThe founder of the Russian school of direct interpretation of potential fields (with minimal prior geological-geophysical information) was V.M. Berezkin, who introduced the operator of total normalized gradient for the 2D interpretation of profile gravity data sets. This operator was successfully applied in searches of hydrocarbon reservoirs. The further development of this approach (the so-called quasi-singular points method) has allowed solution also to various structural problems, using mathematical criteria for the transition from extremes of total normalized gradient fields to coordinates of anomalous sources. The main numerical evaluation strategy is based on stabilized downward continuation of field derivatives and specific use of the filtration properties of Fourier series approximation. The characteristic properties of the quasi-singular points method are: 1) presentation of a more general total normalized gradient function through additional parameters (derivative order m, form of smoothing function Q, number of Fourier coefficients N * with maximal N), optimum values being chosen during a peak-spectrum analysis of the interpreted function; 2) calculation of the set of total normalized gradient fields for various values of N * /N, representing coordinate systems {x,N * /N} as an 'axes tree' of extrema, where each 2D total normalized gradient field is representationally compressed in a 1D line, permitting a) immediate overview of the positions of the axes in all variants of the calculated fields and b) reduction of the retained information, as required in subsequent interpretation; 3) development of two criteria for transition from extrema of total normalized gradient fields to the coordinates of anomaly sources. The quasisingular points method is intended for tracing limiting gently-sloping boundaries, if their micro-relief features are sources of the interpreted anomaly but sub-vertical contacts may also be traced. The method has been tested in delineating various geological structures. One of the most challenging, successfully achieved, was tracing of the Moho discontinuity and study of the upper mantle, using only Bouguer anomaly data along interpretation profiles. This is attested in an example of two regional profiles intersecting the European part of Russia. The central part of one of them coincides with the results from a deep seismic profile.
We analyze gravitational effects of distant topographic and bathymetric relief beyond the (earth-centered) angular distance of [Formula: see text] (i.e., beyond the outer limit of the Hayford-Bowie zone O, or approximately [Formula: see text]) using a spherical earth model. Our results support current procedures that neglect distant relief effects for most local gravity surveys but show their potential importance for continental- and global-scale surveys. The distant relief can produce horizontal gradients as large as [Formula: see text] [Formula: see text]. In mountainous areas, the gravitational effect of the distant relief is significant, even for local surveys, although the vertical gradient of the distant relief effect never exceeds [Formula: see text] [Formula: see text] on the earth’s surface.
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