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- Volume 34, Issue 6, 1986
Geophysical Prospecting - Volume 34, Issue 6, 1986
Volume 34, Issue 6, 1986
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SAMPLING AND MINIMUM PHASE FROM BOTH A CONTINUOUS AND DISCRETE POINT OF VIEW*
Authors A.R. MITCHELL and W.D. STOKESAbstractExamples show that the sampling operation–i.e., the change from the continuous time domain to the discrete time domain–does not necessarily preserve the minimum‐phase property. Further examples can be constructed to show that the resampling operation on the discrete time domain does not necessarily preserve the minimum‐phase property. Finally it can be shown that the minimum‐phase property can be either created or destroyed by sampling or resampling.
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ON REVERSE‐TIME MIGRATION*
Authors J. GAZDAG and E. CARRIZOABSTRACTMigration is a process whereby events in ‘image space’ are mapped into their correct positions in ‘object space’. The wave equations associated with this mapping may be defined and solved numerically either in image space or in object space. In the former the CMP section, which represents the initial conditions, is extrapolated toward increasing depths, and the migrated data are recovered at zero time. In the latter, the wave‐field extrapolation takes place in the coordinate frame of the depth section, and the CMP data serve as boundary conditions at the surface. Computations begin at the last sample of the record section and continue ‘reverse time’ until time zero.
This paper describes a reverse‐time migration (RTM) method and compares its performance with that of an image‐space method based on the idea of phase shift plus interpolation (PSPI). Synthetic zero‐offset sections serve as examples for migration experiments with the RTM and PSPI methods. It is shown that the RTM approach to migration is rather expensive, but its robustness and accuracy are difficult to surpass.
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AN ALGORITHM FOR FAST TIME‐DOMAIN COMPUTATION OF ONE‐DIMENSIONAL SYNTHETIC VERTICAL SEISMIC PROFILES*
By H. THYBOAbstractThe normal incidence seismic wavefields in a horizontally layered model of the earth have a well‐known matrix description if the earth model has been transformed into a layer stack built up by thin layers, all of equal traveltime. To compute a Synthetic Vertical Seismic Profile (SVSP) the wavefields are iterated through the whole layer stack by means of the matrix description, either expressed in the time domain or in the frequency domain. In conventional methods a single layer is added to a resulting stack of layers at each recursion step. However in the present algorithm, a whole layer stack is added to another layer stack at each recursion step. Improvement in computing time obtained by doing so is approximately linearly dependent on the typical number of layers between receivers. This makes the method very suitable for detailed modeling wherever high resolution of the model is desired.
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SHEAR‐WAVE REFLECTION PROFILING FOR NEAR‐SURFACE LIGNITE EXPLORATION*
Authors B. MILKEREIT, H. STÜMPEL and W. RABBELABSTRACTWe present the results of a shear‐wave reflection experiment and in situ measurements in opencast lignite exploration. Near‐surface coal seams have lower shear‐wave velocities (∼ 200 m/s) and lower densities than sand and clay layers. Due to strong reflection coefficients, a shear‐wave reflection survey provides a powerful tool in lignite prospecting. Due to shorter seismic wavelengths shear waves will yield a higher resolution of shallow subsurface structure than compressional waves. Low shear‐wave velocities and strong lateral velocity variations, however, require a dense data acquisition in the field. The variation of stacking velocities can exceed ± 15% within a profile length of 300 m. The different steps in processing and interpretation of results are described with actual records. The final CMP‐stack shows steep‐angle fault zones with maximum dislocations of 20 m within a coal seam.
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POTENTIAL FIELD OF A STATIONARY ELECTRIC CURRENT USING FREDHOLM'S INTEGRAL EQUATIONS OF THE SECOND KIND*
More LessABSTRACTAn integral equation method is described for solving the potential problem of a stationary electric current in a medium that is linear, isotropic and piecewise homogeneous in terms of electrical conductivity. The integral equations are Fredholm's equations of the ‘second kind’ developed for the potential of the electric field. In this method the discontinuity‐surfaces of electrical conductivity are divided into ‘sub‐areas’ that are so small that the value of their potential can be regarded as constant.
The equations are applied to 3‐D galvanic modeling. In the numerical examples the convergence is examined. The results are also compared with solutions derived with other integral equations. Examples are given of anomalies of apparent resistivity and mise‐a‐la‐masse methods, assuming finite conductivity contrast. We show that the numerical solutions converge more rapidly than compared to solutions published earlier for the electric field. This results from the fact that the potential (as a function of the location coordinate) behaves more regularly than the electric field. The equations are applicable to all cases where conductivity contrast is finite.
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A PORTABLE LOCAL LOOP VLF TRANSMITTER FOR GEOLOGICAL FRACTURE MAPPING*
Authors J.G. HAYLES and A.K. SINHAABSTRACTA portable low‐power Very Low Frequency (VLF) transmitter using a large square loop antenna has been designed, assembled and tested by the Geological Survey of Canada (GSC) for geological studies of fracture patterns in igneous rock masses. Standard laboratory equipment, consisting of a signal generator, a 1100‐W power amplifier and several high‐power tuning capacitors, was used for the purpose.
Field tests at the Chalk River facilities of Atomic Energy of Canada Limited have demonstrated a remarkable similarity between survey results obtained using the VLF signals from the local loop transmitter and from distant US Navy VLF transmitters. The local loop was used to simulate the fields from navy stations NAA in Cutler, Maine and NSS in Annapolis, Maryland. Conductor axes, mapped by using these navy stations, and by using the loop antenna yielded almost identical results. A survey was also done in the same area with the local loop placed in such a manner that the direction of the VLF field was at 45° to the field directions from NAA and NSS. In this case, the same conductor axes were located with only minor shifts in position, indicating that conductors whose axes lie within 45° of the direction of the primary horizontal magnetic field are mapped. Thus, it is probably sufficient to have two sources with orthogonal VLF fields to map all VLF conductors in an area. Since in most areas at least one navy VLF station can be used, the local loop transmitter can be used to generate a signal at right angles to the direction from the navy transmitter to allow a more complete VLF survey coverage.
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AN OPTIMIZED DIGITAL FILTER FOR THE FOURIER TRANSFORM*
More LessAbstractShort filters for calculating Hankel‐transformations, with special attention to the d.c.‐sounding problem, have been published in recent years. These filters, with a typical length of less than 25 coefficients, have made it possible to implement, e.g., VES‐interpretation programs on microcomputers and 3‐D electric and electromagnetic modeling programs on minicomputers. Initially the performance of the short filters was rather poor, but with the introduction of short optimized filters there has been a considerable improvement in the accuracy.
An optimization procedure is applied to design a 20‐point filter for the Fourier sine‐transformation. This filter may be useful in electromagnetic prospecting theory, e.g., in the calculation of the electric and magnetic field from a line source.
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ON THE TIME‐VARYING WIENER FILTER*
By V.P. DIMRIABSTRACTFor a new approach to designing the time‐varying Wiener filter, the input is first divided into sections and then the time‐varying filter is determined from the entire input and the desired output. The technique differs from the existing one in which the time‐invariant filter is determined from each section. Hence, the main difference, between the proposed and the existing technique lies in the arrangement of input data. The proposed technique requires fewer computational operations and performs better than the time‐invariant Wiener filter, as illustrated by numerical examples.
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SOME IMPROVEMENTS IN THE INTERPRETATION OF VERTICAL ELECTRIC SOUNDING CURVES*
By J. CHYBAABSTRACTThe process of VES interpretation is discussed, including the following points.
(a) Preliminary interpretation by means of master curves. It is shown that the positions of the auxiliary points K and Q depend on the resistivity of the substratum. The interpretation is improved if the auxiliary points are determined separately for each master curve.
(b) The individual parts of the measured curves are shifted in overlapping MN electrode positions so that the total sum of squares of the shifts is minimal.
(c) The ambiguity may be reduced by means of supplementary information or assumptions on the resistivities. Fixing the resistivities is not always possible because discrepancies may arise between the ground measurement and the well‐logging data. The simultaneous interpretation of several VES curves is recommended assuming constant resistivities. This assumption may be subsequently verified by means of the F‐test.
(d) A nonlinear algorithm is proposed for the determination of confidence intervals. As the multi‐dimensional confidence intervals are often very complicated, it is recommended to construct only one‐dimensional confidence intervals for the estimable parametric functions.
(e) A ‘double‐least‐squares’ optimization technique is presented. The optimization is performed on the estimable parametric functions, and the individual parameters are determined so that the solution remains near the initial guess. This technique is faster than the singular value decomposition.
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Volumes & issues
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Volume 72 (2023 - 2024)
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Volume 69 (2021)
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Volume 65 (2017)
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Volume 64 (2015 - 2016)
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Volume 63 (2015)
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Volume 62 (2014)
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Volume 61 (2013)
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Volume 60 (2012)
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Volume 59 (2011)
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Volume 58 (2010)
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Volume 57 (2009)
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Volume 56 (2008)
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Volume 55 (2007)
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Volume 54 (2006)
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Volume 53 (2005)
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Volume 52 (2004)
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Volume 51 (2003)
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Volume 50 (2002)
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Volume 49 (2001)
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Volume 48 (2000)
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Volume 47 (1999)
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Volume 46 (1998)
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Volume 45 (1997)
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Volume 44 (1996)
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Volume 43 (1995)
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Volume 42 (1994)
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Volume 41 (1993)
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Volume 40 (1992)
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Volume 39 (1991)
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Volume 38 (1990)
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Volume 37 (1989)
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Volume 36 (1988)
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Volume 35 (1987)
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Volume 34 (1986)
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Volume 33 (1985)
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Volume 32 (1984)
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Volume 31 (1983)
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Volume 30 (1982)
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Volume 29 (1981)
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Volume 28 (1980)
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Volume 27 (1979)
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Volume 26 (1978)
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Volume 25 (1977)
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Volume 24 (1976)
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Volume 23 (1975)
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Volume 22 (1974)
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Volume 21 (1973)
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Volume 20 (1972)
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Volume 19 (1971)
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Volume 18 (1970)
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Volume 17 (1969)
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Volume 16 (1968)
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Volume 15 (1967)
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Volume 14 (1966)
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Volume 13 (1965)
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Volume 12 (1964)
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Volume 11 (1963)
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Volume 10 (1962)
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Volume 9 (1961)
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Volume 8 (1960)
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Volume 7 (1959)
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Volume 6 (1958)
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Volume 5 (1957)
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Volume 4 (1956)
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Volume 3 (1955)
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Volume 2 (1954)
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Volume 1 (1953)