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- Volume 33, Issue 5, 1985
Geophysical Prospecting - Volume 33, Issue 5, 1985
Volume 33, Issue 5, 1985
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SEISMIC MODELING IN THE IMPLICIT MODE*
By I. R. MUFTIAbstractFinite‐difference seismic models are often saddled with huge memory requirements for data manipulation, a prohibitive amount of CPU time and even approximate results. At least part of these costs may be due to the fact that most of the work reported on this subject is devoted to the development of explicit models which suffer from severe limitations of stability and necessitate extremely fine time sampling of the wavefield.
A new method of seismic modeling which works in the implicit mode and is unconditionally stable is put forward. It is based on the self‐adjoint version of the acoustic wave equation. The evaluation of the wavefield is done by using a highly efficient splitting algorithm which does not require transposing the field data at the various time steps. Moreover, it can accommodate anisotropic media as well as three‐dimensional structures. Computational efficiency is achieved by introducing an unconventional procedure which yields the sum of the values of the wavefield corresponding to a new time step and a previous time step. The new value can be obtained from this sum by a simple subtraction.
A number of numerical examples, some obtained by using time steps about five times larger than the largest permissible time step in the corresponding explicit model, are presented. The effect of topographic changes on the growth of the wavefield due to a source located near the surface of the ground is investigated.
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LEAST‐SQUARES FILTERS WITHOUT TRANSIENT ERRORS: AN EXAMINATION OF THE ERRORS IN LEAST‐SQUARES FILTER DESIGN*
Authors A. W. H. BUNCH and R. E. WHITEAbstractA new approach has been developed for the design of cross‐equalization filters by the least‐squares method. The filters estimated by this new exact method are subject to only two types of error: bias and random error. Cross‐equalization filters estimated by a more conventional least‐squares method are further subject to “transient error”. This type of error becomes important when designing filters from a data gate of a length comparable with the length of the filter, i.e., less than four times the length of the filter.
The effect of altering various design parameters has been investigated for the new method. It has been found that the proportion of bias in the filter decreases as the effective filter length increases, whereas the random error in the filter decreases with increase in either the signal‐to‐noise ratio of the data or the ratio of the data duration to the filter length. The level of whitening applied to the auto‐correlation matrix before inversion was not found to be a critical design parameter. Also, two techniques have been tested for reducing any anomalous d.c. component in the calculated filter.
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INVERSION METHODS FOR τ‐p MAPS OF NEAR OFFSET DATA—LINEAR INVERSION*
More LessAbstractConventional velocity analysis, based on the ideas of rms velocity and hyperbolic reflection events in the x‐t domain, is restricted in validity to near vertical incidence. Thus analysis of near‐offset datasets usually requires the muting of wide‐angle reflections from shallow interfaces before the rms velocities are determined. The ray‐theoretical integral for the delay time τ, which depends on the slowness p and the velocity function, is valid for all angles. The wide‐angle reflections can be used to improve the accuracy of the derived velocity function in the near surface region, if the recorded x‐t data are mapped into the τ‐p domain. By representing the velocity function between reflectors as a series of gradient zones, i.e. regions with a uniform increase in velocity with depth, the recovery of the velocities may be posed as a matrix linear inverse problem for the slopes of the gradient zones. In order to convert the problem to a linear one, the velocity discontinuities at the reflecting interfaces must be fixed in advance. Their positions are based on the behaviour of the τ‐p map of the data. Finding a stable velocity model may require several iterations with the reflecting interfaces at different positions. An understanding of the workings of the inversion algorithm allied with an analysis of the causes of instability aids the search for a stable model.
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VERTICAL SEISMIC PROFILING IN COAL*
Authors S. A. GREENHALGH and M. SUPRAJITNOAbstractThe intellection of seismic wave propagation in coal measures demands direct observation of the wavefield progression. Two vertical seismic profiles with high spatial and temporal sampling, were recently recorded in the Sydney Basin coalfields as part of an experimental coal seismic program.
Static corrections and interval velocities were obtained by an automated system to determine first kicks and pulse rise times. Upgoing and downgoing waves were separated in the f—k‐plane using a novel technique of contour slice filtering. The isolated upgoing waves clearly display reflections from the major coal seams within the stratigraphic sequence. The downgoing wave spectra were subjected to attenuation analysis. The deduced specific quality factor Q for Permian coal measure rocks lies in the range 20–70. Similar estimates were obtained in the time domain from measurements of pulse broadening.
Synthetic VSP seismograms, computed using an exact recursive formulation, are an indispensable aid to interpretation. They illustrate the filtering effects of coal seams and sequences, and the effects of the contribution of internal and free‐surface multiple reflections in the recorded wavetrains.
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SOME REMARKS ON THE BEHAVIOR OF THE MAGNETOTELLURIC PHASE*
By G. FISCHERAbstractThe magnetotelluric phase in the medium of incidence over a good conductor varies from its value at the conductor surface, usually close to 45°, to the characteristic standing‐wave phaseshift of 90° over a distance of a few conductor skin‐depths, much shorter than the vacuum wavelength at the frequencies considered. Over the top of a graben or at the surface of a two‐layer structure the basement can be looked upon as the matrix. If the graben or the overburden is a better conductor it will carry an extra current density in phase with the electric field, thus giving rise to an additional magnetic field also in phase with the electric field. This drives the argument of the surface impedance toward smaller values. For a more resistive graben or overburden the current is depressed, corresponding to a reduction of the surface magnetic field by an amount in phase with the electric field. This effect drives the phase of the impedance to values above the characteristic 45° of a uniform conductor.
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EM AND IP RESPONSE OF A STEEL WELL CASING FOR A FOUR‐ELECTRODE SURFACE ARRAY. PART I: THEORY*
Authors J. R. WAIT and J. T. WILLIAMSAbstractThe basic formulation for the mutual coupling between grounded circuits in the presence of a vertical conductor is used to examine the full electromagnetic and induced polarization response for a homogeneous half‐space model. Both interfacial polarization at the steel/electrolyte boundary and induced eddy currents in the casing are accounted for in the analysis. It is shown that for mild steel casings the eddy currents, hitherto neglected, play an important role because of their strong frequency dependence. Calculated curves of the appar‐ ent complex resistivity as a function of frequency show that the overall response can be influenced by the state of corrosion at the surface of the casing.
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EM AND IP RESPONSE OF A STEEL WELL CASING FOR A FOUR‐ELECTRODE SURFACE ARRAY. PART II: NUMERICAL RESULTS*
Authors J. T. WILLIAMS and J. R. WAITAbstractSome numerical results for the apparent resistivity of a homogeneous half space containing a vertical steel well casing as measured by a dipole‐dipole array are presented. The basic model includes the electromagnetic and induced polarization effects by allowing for frequency dependent eddy currents and an electrochemical boundary of the well casing. It is shown that the well casing produces an apparent complex resistivity response that could be mistaken for an induced polarization anomaly existing above a hydrocarbon deposit. This response of the well casing is strongly dependent on frequency.
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A NOTE ON THE INTERPRETATION OF MAGNETIC ANOMALIES OF INFINITE THIN SHEETS BY USING RELATION FIGURES*
Authors RAMA CH. RAO, V. VIJAYAKUMAR, G. VIRUPAKSHI and M. B. S. V. RAOAbstractThe concept of relation figures is utilized in the case of a thin infinite sheet for identification and determination of the causative source geometry, and its various parameters. The plot of ▵T versus ▵V is a conic, the plots between the respective symmetric components and the respective asymmetric components are straight lines, and the resulting diagram between the first horizontal versus the first vertical derivative of either ▵T or ▵V is a cardioid.
The authors are grateful to Shri V. Babu Rao and Dr R. K. Verma for their encouragement. They would like to thank the Director of the N. G. R. I. for permission to publish this paper. The neat drawing of Sri P. Sundara Rao is appreciated.
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Volumes & issues
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Volume 72 (2023 - 2024)
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Volume 70 (2021 - 2022)
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Volume 69 (2021)
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Volume 67 (2019)
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Volume 66 (2018)
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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)