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- Volume 35, Issue 6, 1987
Geophysical Prospecting - Volume 35, Issue 6, 1987
Volume 35, Issue 6, 1987
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COMPUTATIONAL ASPECTS OF THE CHOICE OF OPERATOR AND SAMPLING INTERVAL FOR NUMERICAL DIFFERENTIATION IN LARGE‐SCALE SIMULATION OF WAVE PHENOMENA*
By O. HOLBERGABSTRACTConventional finite‐difference operators for numerical differentiation become progressively inaccurate at higher frequencies and therefore require very fine computational grids. This problem is avoided when the derivatives are computed by multiplication in the Fourier domain. However, because matrix transpositions are involved, efficient application of this method is restricted to computational environments where the complete data volume required by each computational step can be kept in random access memory.
To circumvent these problems a generalized numerical dispersion analysis for wave equation computations is developed. Operators for spatial differentiation can then be designed by minimizing the corresponding peak relative error in group velocity within a spatial frequency band. For specified levels of maximum relative error in group velocity ranging from 0.03% to 3%, differentiators have been designed that have the largest possible bandwidth for a given operator length.
The relation between operator length and the required number of grid points per shortest wavelength, for a required accuracy, provides a useful starting point for the design of cost‐effective numerical schemes. To illustrate this, different alternatives for numerical simulation of the time evolution of acoustic waves in three‐dimensional inhomogeneous media are investigated. It is demonstrated that algorithms can be implemented that require fewer arithmetic and I/O operations by orders of magnitude compared to conventional second‐order finite‐difference schemes to yield results with a specified minimum accuracy.
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INVERSE Q‐FILTERING. A SPECTRAL BALANCING TECHNIQUE*
By L.J. GELIUSABSTRACTInverse Q‐filtering (IQF) is a technique designed to correct for transmission losses due to inelastic attenuation. It is based on the constant‐Q model and is derived from a Taylor series solution of a standard convolutional‐trace model (primaries only). To avoid a non‐causal solution, the attenuation is assumed to be minimum‐phase. Band limitation is introduced to make IQF a stable process in the presence of noise. The main features of IQF are demonstrated using both synthetic and field data.
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EXPERIMENTAL COMPARISON BETWEEN SPECTRAL RATIO AND RISE TIME TECHNIQUES FOR ATTENUATION MEASUREMENT*
Authors P. TARIF and T. BOURBIEABSTRACTTwo techniques for the measurement of attenuation–spectral ratio and rise time techniques–were tested and compared in the laboratory. The spectral ratio technique proved to be reliable and easy to implement for intermediate values (5 < Q < 50) of attenuation. For low (Q > 50) and high attenuations, the spectral ratio technique is inaccurate. Calculating the rise time on simulated signals, we found a relation between rise time τ and the ratio travel‐time to quality factor T/Q which could be approximated in intervals by the linear relation τ=τ+C*T/Q. The constants τ and C depend on the absolute value of T/Q and on the initial source signal. The rise time technique, performed on the first quarter period of the signal, enables high attenuations (Q < 5) to be measured. The determination of the relation between τ and T/Q is possible if one knows the initial source. We theoretically approximate this relation through a simulation using a realistic propagation model. With laboratory measurements made on Fontainebleau sandstone, we show that the rise time technique using the theoretical relation τ=τ(T/Q) gives comparable values of Q to those obtained from the spectral ratio technique. In borehole seismics, where it is often difficult to remove undesired signals, the rise time technique applied with the right (τ, T/Q) relation is the best method to use.
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THE PERFECTLY REFLECTING WEDGE USED AS A CONTROL MODEL IN SEISMIC DIFFRACTION MODELLING*
By G.D. HUTTONABSTRACTThe numerical modelling of seismic diffraction, e.g., at faults and other discontinuities, generally requires the use of fast approximate methods. The geophysicist responsible for the development of such numerical methods has a real need of exact solutions to certain ideal geometries to check the accuracy of his calculations.
One such exact solution, which is available, is the acoustic wave solution to the perfectly reflecting wedge. The solution is three‐dimensional and the source is an explosive point source. This model is ideal for seismic diffraction; the solution has the advantage of being exact, truly three‐dimensional and of being in the convenient form of the temporal and spatial impulse response. More complicated sources which are extended in either space or time can, therefore, be modelled exactly by numerical integration.
This paper presents some examples of the use of the perfectly reflecting wedge as a control model for an asymptotic high frequency diffraction modelling method. This control model has revealed that certain survey and wedge configurations can yield significant disagreement with, e.g., the Kirchhoff approximation. Such configurations could occur during VSP modelling when the survey lies in the near field or in the shadow zone of a high contrast fault. This control model has also been instructive in demonstrating why the high frequency, asymptotic, approximation is generally very good and has indicated a possible improvement to the Kirchhoff approximation for edge diffraction.
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VERTICAL HETEROGENEITY AND MOVEOUT IN THE ONE‐DIMENSIONAL MEDIUM*
By W.J. VETTERABSTRACTSince the important contributions of Dürbaum and Dix, 30 years ago, velocity profile estimation procedures on horizontally layered and vertically heterogeneous media from seismic probing data have been based largely on hyperbolic moveout models and RMS and stacking velocity concepts. Re‐examination of the fundamentals reveals that quantitative velocity heterogeneity and canonical valocity profiles have been implicit factors for moveout modelling and for profile inversion in the use of the Dix procedure. Heterogeneity h is the ratio (and vRMS the geometric or harmonic mean) of the path‐average and time‐average velocities for a raypath or, in a more restricted sense, for the normal ray belonging to a velocity profile. The canonical profile for a given velocity profile or profile segment is a moveout‐equivalent monotonically increasing ramp‐like profile.
The ramp or constant gradient in depth is the simplest velocity profile approximator which can explicitly accommodate velocity heterogeneity. A ramp model structure is detailed which facilitates moveout simulation and model parameter estimation, and the parametric effects are explored. The horizontal offset range is quantified for which this model can give good moveout approximations.
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CORRECTIONS FOR CONDUCTIVITY ESTIMATES IN INDUCTION PROSPECTING OF SULPHIDE DYKES IN A LAYERED ENVIRONMENT*
Authors J.G. NEGI, O.P. GUPTA and M.S. JOSHIABSTRACTIn a weathered environment estimates of depth and conductance of metallic sulphide dykes from conventional anomaly index diagrams for a vertical half‐plane in air have to be corrected, besides the usual corrections, for:
1. moderate conductivity of the host rocks, and
2. finiteness of strike length S and depth extent D.
Model experiments have been carried out to evaluate the response variation of a vertical planar conductor with varying depth extent and strike length for both insulating and conductive surroundings. The results indicate:
1. A conductor with finite depth extent (D/L < 2.5) or strike length (S/L < 5.0) in an insulating medium yields a lower estimate of conductance (mineralization) and a greater depth.
2. A moderately‐conductive host rock enhances the anomaly and rotates the phase so that the conductor appears to be more resistive (less mineralized) and shallower.
The results have practical significance since in weathered surroundings a highly‐mineralized body of finite size could be missed, or misjudged, because of low estimates of conductivity and depth.
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Volume 72 (2023 - 2024)
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Volume 69 (2021)
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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 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)