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- Volume 55, Issue 1, 2007
Geophysical Prospecting - Volume 55, Issue 1, 2007
Volume 55, Issue 1, 2007
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Migration methods for imaging different‐order multiples
Authors Zhiyong Jiang, Jianming Sheng, Jianhua Yu, Gerard T. Schuster and Brian E. HornbyABSTRACTMultiples contain valuable information about the subsurface, and if properly migrated can provide a wider illumination of the subsurface compared to imaging with VSP primary reflections. In this paper we review three different methods for migrating multiples. The first method is model‐based, and it is more sensitive to velocity errors than primary migration; the second method uses a semi‐natural Green's function for migrating multiples, where part of the traveltimes are computed from the velocity model, and part of the traveltimes (i.e., natural traveltimes) are picked from the data to construct the imaging condition for multiples; the third method uses cross‐correlation of traces. The last two methods are preferred in the sense that they are significantly less sensitive to velocity errors and statics because they use “natural data” to construct part of the migration imaging conditions. Compared with the interferometric (i.e., crosscorrelation) imaging method the semi‐natural Green's function method is more computationally efficient and is sometimes less prone to migration artifacts.
Numerical tests with 2‐D and 3‐D VSP data show that a wider subsurface coverage, higher‐fold and more balanced illumination of the subsurface can be achieved with multiple migration compared with migration of primary reflections only. However, there can be strong interference from multiples with different orders or primaries when multiples of high order are migrated. One possible solution is to filter primaries and different orders of multiples before migration, and another possible solution is least squares migration of all events. A limitation of multiple migration is encountered for subsalt imaging. Here, the multiples must pass through the salt body more than twice, which amplifies the distortion of the image.
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Paraxial ray methods for anisotropic inhomogeneous media
Authors Tijmen Jan Moser and Vlastislav ČervenýABSTRACTA new formalism of surface‐to‐surface paraxial matrices allows a very general and flexible formulation of the paraxial ray theory, equally valid in anisotropic and isotropic inhomogeneous layered media. The formalism is based on conventional dynamic ray tracing in Cartesian coordinates along a reference ray. At any user‐selected pair of points of the reference ray, a pair of surfaces may be defined. These surfaces may be arbitrarily curved and oriented, and may represent structural interfaces, data recording surfaces, or merely formal surfaces. A newly obtained factorization of the interface propagator matrix allows to transform the conventional 6 × 6 propagator matrix in Cartesian coordinates into a 6 × 6 surface‐to‐surface paraxial matrix. This matrix defines the transformation of paraxial ray quantities from one surface to another. The redundant non‐eikonal and ray‐tangent solutions of the dynamic ray‐tracing system in Cartesian coordinates can be easily eliminated from the 6 × 6 surface‐to‐surface paraxial matrix, and it can be reduced to 4 × 4 form. Both the 6 × 6 and 4 × 4 surface‐to‐surface paraxial matrices satisfy useful properties, particularly the symplecticity. In their 4 × 4 reduced form, they can be used to solve important boundary‐value problems of a four‐parametric system of paraxial rays, connecting the two surfaces, similarly as the well‐known surface‐to‐surface matrices in isotropic media in ray‐centred coordinates. Applications of such boundary‐value problems include the two‐point eikonal, relative geometrical spreading, Fresnel zones, the design of migration operators, and more.
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Assessment and compensation of inconsistent coupling conditions in point‐receiver land seismic data
Authors Claudio Bagaini and César Barajas‐OlaldeABSTRACTWe introduce a method to detect and compensate for inconsistent coupling conditions that arise during onshore seismic data acquisitions. The reflected seismic signals, the surface waves, or the ambient‐noise records can be used for the evaluation of the different coupling conditions of closely spaced geophones. We derive frequency‐dependent correction operators using a parametric approach based upon a simple model of the interaction between geophone and soil. The redundancy of the measurements available permits verification of the assumptions made on the input signals in order to derive the method and to assess the validity of the model used. The method requires point‐receiver data in which the signals recorded by the individual geophones are digitized.
We have verified the accuracy of the method by applying it to multicomponent ambient‐noise records acquired during a field experiment in which the coupling conditions were controlled and modified during different phases of the experiment. We also applied the method to field data, which were acquired without the coupling conditions being controlled, and found that only a few geophones showed an anomalous behaviour. It was also found that the length of the noise records routinely acquired during commercial surveys is too short to provide enough statistics for the application of our method.
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Validity of the long‐wave approximation in periodically layered media
More LessABSTRACTIn seismic modelling, a stack of thin layers is often replaced by an effective equivalent anisotropic homogeneous slab. For waves with finite wavelength, this is an approximation, and the error thus introduced can be quantified by considering the relative error in the phase velocity between the layer stack and the effective medium. For periodic layering, the relative phase‐velocity error can be expressed in closed form as a function of wavelength, reflection coefficients and layer thicknesses. By comparing the relative phase‐velocity error with laboratory measurements and numerical simulations, we find that the difference in seismic response between a periodic layer stack and an equivalent effective medium depends not only on wavelength, but it also depends significantly on reflection coefficients and the ratio between layer thicknesses. For a 1% relative error in the phase velocity, and if all layers have the same thickness measured in vertical traveltime, we find that the wavelength must be larger than approximately three times the layer period for a reflection coefficient of 0.1, but this increases to 13 times the layer period for a reflection coefficient of 0.9, which is highly unrealistic in a geological setting.
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Negative bending in seismic reflection associated with time‐advanced and time‐retarded fields
Authors Luc T. Ikelle and Anthony F. GangiABSTRACTWe present an analysis of scattering diagrams (i.e., Feynman‐like diagrams for wave scattering) of the correlation‐type representation theorem for ordinary inhomogeneous media with both positive stiffnesses and positive Poisson's ratios. This analysis reveals scattering events whose scattering diagrams include “negative” bending (i.e., bending in the opposite direction of that of scattering diagrams in ordinary inhomogeneous media). Unlike common scattering events, these events are inconsistent with the current interpretation of some of the basic physical laws, such as Snell's law, just like the so‐called “negative refraction” in optics. Yet we find them very useful, for instance, in suppressing some undesired events from scattering data.
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Stoneley wave generation by an incident P‐wave propagating in the surrounding formation across a horizontal fluid‐filled fracture
More LessABSTRACTThis study investigates the generation of the low‐frequency borehole Stoneley wave (tube wave) by a plane P‐wave propagating through the surrounding elastic formation, which is intersected by a fluid‐filled fracture. A model is constructed taking into account the dynamic fluid coupling between the borehole interior and the fluid‐filled fracture of infinite extent with parallel walls. The basic mechanism of such coupling is due to the contraction of the fracture walls by the incident P‐wave, leading to seismic radiation into the fracture. The dynamic fluid flux from the fracture into the borehole interior, and vice versa, is the source of the low‐frequency Stoneley wave. An expression for the monopole pressure source, exciting the tube wave, is obtained.
The tube‐wave equation in the long‐wave approximation is derived in the presence of a fluid‐filled fracture of infinite extent. Amplitudes and waveforms of Stoneley waves are analysed in the seismic wavelength range for P‐wave pulses of various shapes. It is shown that the amplitude and waveform of the Stoneley wave depends significantly on the two dimensionless parameters of the problem: (1) the ratio of the borehole radius to the dominant wavelength of the incident pulse; (2) the ratio of the fracture width to the borehole radius. It is found that the amplitude of the generated Stoneley wave can be of the order of the P‐wave amplitude in the borehole fluid. Stoneley waveforms are found to be completely different from those of the incident pulse.
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Hankel transform filters for dipole antenna radiation in a conductive medium
By F.N. KongABSTRACTWe discuss the Hankel transforms related to a particular application, i.e. the dipole antenna radiation in conductive media, such as the antenna radiation in sea‐bed electromagnetic applications. In this application, the electromagnetic wavefields decay very rapidly with distance. A good filter means that it can be used to evaluate weak fields.
Exponential sampling transforms a Hankel transform into a convolution equation, which must be solved to obtain the filter coefficients. Here, we use a direct matrix inversion method to solve the convolution equation in the sample domain, instead of the Fourier transform method and the Wiener–Hopf method, previously used to solve the convolution equation. This direct method is conceptually simple and is suitable for our optimization process: by using the Sommerfeld identity, we search for the optimum sampling interval, which corresponds to the minimum wavefield, evaluated for a given length filter.
The performances of the new filters obtained are compared with some well‐known filters. We find that our filters perform better for our application; that is, for the same length filters, our filters are able to calculate weaker fields.
For users working in similar applications, three sets of filters with lengths 61, 121, 241 are available from the author.
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Calculation of magnetic anomalies caused by 2D bodies of arbitrary shape with consideration of demagnetization
More LessABSTRACTForward calculations of magnetic anomalies caused by two‐dimensional bodies of any shape and magnetic properties may be performed either without considering demagnetization as in the equivalent source technique or taking demagnetization into account as in the volume integral equation (VIE) approach, in which, for this purpose, magnetized bodies are divided into a set of rectangular prismatic cells. Ignoring demagnetization may result in distortion of the shape and the amplitude of an anomaly, whereas rectangular cells may not be an optimal representation of the source. Moreover, an inaccurate form approximation in the VIE technique may lead to inconsistent results in the near‐body region. In this paper, a method is proposed, based on the VIE approach but differing by applying triangular elementary cells. The method largely overcomes the above‐mentioned limitations of the VIE technique. It allows us to delineate large and complex structures exactly and only requires the source to be divided into a few elementary cells to take demagnetization into account satisfactorily. These improvements have been attained through analytical calculation of the Green's function in the complex plane, using the theory of the Cauchy‐type integral. Comparing numerical solutions with analytical solutions for homogeneous elliptic cylinders without remanence, the method is found to be consistent with the theory in the range of relative magnetic permeability of 2–20, not only far from but also at subcell distances from the body. The method is appropriate for modelling highly and inhomogeneously magnetized 2D bodies of any shape. It may be of value in interpreting underground measurements or topographic effects, as well as in modelling regional geomagnetic profiles, and it is also a convenient tool for testing questionable geological hypotheses. In the framework of the method, the gravitational anomaly for the same causative bodies can be easily calculated. However, at higher and geologically uncommon values of relative magnetic permeability, the algorithm may become unstable but may be stabilized with SVD regularization. The fact that discrepancies were found with the method employed is a basis for further research.
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1D and 2D Cole‐Cole‐inversion of time‐domain induced‐polarization data
Authors Mark Hönig and Bülent TezkanABSTRACTA new method for the 2D inversion of induced polarization (IP) data in the time domain has been developed. The entire IP transients were observed and inverted into 2D Cole‐Cole earth models, including resistivity, chargeability, relaxation time and the frequency constant. Firstly, a modified 1D time‐domain electromagnetic algorithm was used to calculate the response of a layered polarizable ground. The transient signals were then inverted using the Marquardt method to derive the Cole‐Cole parameters of each layer. However, model calculations showed that the EM effects could be neglected for the time range (>1 ms) and for the transmitter–receiver distances (<50 m) used in this study. Therefore, the induction effects were not considered for the solution of the 2D inverse problem and a DC solution was applied. An approximative forward algorithm was introduced in order to calculate the IP transients directly in the time domain and in order to speed up the inverse procedure. The approximation is highly accurate, and this is demonstrated by comparing the approximations with their exact solutions up to 3D.
The inverse algorithm presented consists of two steps. The transient voltages of an array data set were inverted separately into a two‐dimensional resistivity model for each time channel. The time‐dependent resistivity of each cell was then interpreted as the response of a homogeneous half‐space. In the 2D inversion algorithm, a 3D DC algorithm was used as a forward operator. The method only requires a standard 2D DC inversion and a homogenous half‐space Cole‐Cole inversion.
The developed algorithm has been successfully applied to synthetic data sets and to a field data set obtained from a waste site situated close to Düren in Germany.
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Volumes & issues
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Volume 72 (2023 - 2024)
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Volume 71 (2022 - 2023)
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Volume 70 (2021 - 2022)
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Volume 69 (2021)
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Volume 68 (2020)
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Volume 67 (2019)
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Volume 66 (2018)
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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)