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  • 1
    Electronic Resource
    Electronic Resource
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical prospecting 38 (1990), S. 0 
    ISSN: 1365-2478
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences , Physics
    Notes: Inversion of multicomponent seismic data can be subdivided in three main processes: (1) Surface-related preprocessing (decomposition of the multicomponent data into ‘primary’ P-and S-wave responses). (2) Prestack migration of the primary P- and S-wave responses, yielding the (angle-dependent) P-P, P-S, S-P and S-S reflectivity of the subsurface. (3) Target-related post-processing (transformation of the reflectivity into the rock and pore parameters in the target). This paper deals with the theoretical aspects of surface-related preprocessing.In a multicomponent seismic data set the P- and S-wave responses of the subsurface are distorted by two main causes: (1) The seismic vibrators always radiate a mixture of P- and S-waves into the subsurface. Similarly, the geophones always measure a mixture of P- and S-waves. (2) The free surface reflects any upgoing wave fully back into the subsurface. This gives rise to strong multiple reflections, including conversions.Therefore, surface-related preprocessing consists of two steps: (1)Decomposition of the multicomponent data (pseudo P- and S-wave responses) into true P- and S-wave responses. In practice this procedure involves (a) decomposition per common shot record of the particle velocity vector into scalar upgoing P- and S-waves, followed by (b) decomposition per common receiver record of the traction vector into scalar downgoing P- and S-waves. (2) Elimination of the surface-related multiple reflections and conversions. In this procedure the free surface is replaced by a reflection-free surface. The effect is that we obtain ‘primary’ P-and S-wave responses, that contain internal multiples only.An interesting aspect of the procedure is that no knowledge of the subsurface is required. In fact, the subsurface may have any degree of complexity. Both the decomposition step and the multiple elimination step are fully determined by the medium parameters at the free surface only. After surface-related preprocessing, the scalar P- and S-wave responses can be further processed independently by existing scalar algorithms.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical prospecting 33 (1985), S. 0 
    ISSN: 1365-2478
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences , Physics
    Notes: Part I of this series starts with a brief review of the fundamental principles underlying wave field extrapolation. Next, the total wave field is split into downgoing and upgoing waves, described by a set of coupled one-way wave equations. In cases of limited propagation angles and weak inhomogeneities these one-way wave equations can be decoupled, describing primary waves only. For large propagation angles (up to and including 90°) an alternative choice of sub-division into downgoing and upgoing waves is presented. It is shown that this approach is well suited for modeling as well as migration and inversion schemes for seismic data which include critical angle events.
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  • 3
    Electronic Resource
    Electronic Resource
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical prospecting 24 (1976), S. 0 
    ISSN: 1365-2478
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences , Physics
    Notes: The least squares estimation procedures used in different disciplines can be classified in four categories:〈list xml:id="l1" style="custom"〉a. Wiener filtering,b. b. Autoregressive estimation,c. c. Kalman filtering,d. d. Recursive least squares estimation.The recursive least squares estimator is the time average form of the Kalman filter. Likewise, the autoregressive estimator is the time average form of the Wiener filter. Both the Kalman and the Wiener filters use ensemble averages and can basically be constructed without having a particular measurement realisation available.It follows that seismic deconvolution should be based either on autoregression theory or on recursive least squares estimation theory rather than on the normally used Wiener or Kalman theory. A consequence of this change is the need to apply significance tests on the filter coefficients.The recursive least squares estimation theory is particularly suitable for solving the time variant deconvolution problem.
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  • 4
    Electronic Resource
    Electronic Resource
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical prospecting 40 (1992), S. 0 
    ISSN: 1365-2478
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences , Physics
    Notes: Elastic redatuming can be carried out before or after decomposition of the multicomponent data into independent PP, PS, SP, and SS responses. We argue that from a practical point of view, elastic redatuming is preferably applied after decomposition. We review forward and inverse extrapolation of decomposed P- and S-wavefields. We use the forward extrapolation operators to derive a model of discrete multicomponent seismic data. This forward model is fully described in terms of matrix manipulations.By applying these matrix manipulations in reverse order we arrive at an elastic processing scheme for multicomponent data in which elastic redatuming plays an essential role. Finally, we illustrate elastic redatuming with a controlled 2D example, consisting of simulated multicomponent seismic data.
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  • 5
    Electronic Resource
    Electronic Resource
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical prospecting 32 (1984), S. 0 
    ISSN: 1365-2478
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences , Physics
    Notes: In migration procedures, the velocity profile of the subsurface is the most important input information. Since, in general, this information is only approximately known, errors in the migration output due to errors in the velocity input occur in all practical applications. In migration, velocity errors and depth errors can be interchanged. This interchange property is perfect in the paraxial approximation. From this result it follows that migration with incorrect velocities may still yield correctly migrated data if the imaging principle is modified. This attractive property can be used in the stripping version of migration (recursive migration).
    Type of Medium: Electronic Resource
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  • 6
    Electronic Resource
    Electronic Resource
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical prospecting 32 (1984), S. 0 
    ISSN: 1365-2478
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences , Physics
    Notes: Random amplitude and phase errors in seismic input data introduce a coherent distribution of migration half-circles or “smiles”, the occurrence of which may cause a significant decrease of signal-to-noise ratio. In addition, the effect of quantization errors is discussed for different wordlengths, used both during acquisition and during data processing. Results of sign-bit recordings are shown.
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical prospecting 34 (1986), S. 0 
    ISSN: 1365-2478
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences , Physics
    Notes: Wave field extrapolation including critical angle events in modeling, migration and inversion can be handled with algorithms based on both the one-way wave equations and the two-way wave equation. It is shown that for 1-D inhomogeneous media, critical angle events as well as multiple reflections may elegantly be included in pre-stack modeling, pre-stack migration and velocity inversion. For 2-D and 3-D inhomogeneous media a powerful pre-stack migration scheme can be developed which includes critical angle events as well as multiple reflections. Finally, suggestions for practical applications are given.
    Type of Medium: Electronic Resource
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  • 8
    Electronic Resource
    Electronic Resource
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical prospecting 34 (1986), S. 0 
    ISSN: 1365-2478
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences , Physics
    Notes: In one-way wave field extrapolation downgoing and upgoing waves are treated independently, which is allowed if propagation at small angles against the vertical in (weakly) inhomogeneous media is considered. In practical implementation the slow convergence of the square-root operator causes numerical deficiencies. On the other hand, in two-way wave field extrapolation no assumptions need to be made on the separability of downgoing and upgoing waves. Furthermore, in practical implementation the use of the square-root operator is avoided. To put the two-way techniques into perspective, it is shown that two-way wave field extrapolation could be described in terms of one-way processes, namely: (1) decomposition of the total wave field into downgoing and upgoing waves; (2) one-way wave field extrapolation; (3) composition of the total wave field from its downgoing and upgoing constituents. This alternative description of two-way wave field extrapolation is valid for media which are homogeneous along the z-coordinate as well as for small dip angles in arbitrarily inhomogeneous media. In addition, it is shown that this description is also valid for large dip angles in 1-D (vertically) inhomogeneous media, including critical-angle events, when the WKBJ one-way wave functions discussed in part I of this paper are considered.For large dip angles in arbitrarily inhomogeneous media the two-way wave equation is solved by means of Taylor series expansion. For practical implementation a truncated operator is designed, assuming gentle horizontal variations of the medium properties. This operator is stable and converges already in the first order approximation, also for critical-angle events.
    Type of Medium: Electronic Resource
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  • 9
    Electronic Resource
    Electronic Resource
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical prospecting 27 (1979), S. 0 
    ISSN: 1365-2478
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences , Physics
    Notes: A method is presented to derive approximate versions of the wave equation which allow finite-difference migration for very steep dips (〉 50°). It is shown that for conventional finite-difference schemes, in addition to the dip limitation, the maximum acceptable frequency should be specified. A finite-difference migration technique is proposed in the frequency domain. It is derived that finite-difference wave field extrapolation in the frequency domain consists of a space-variant convolution procedure for each frequency component, the space-variance being defined by the lateral variation in the velocity.Finally it is shown that with finite-difference migration, particle velocity data can be easily obtained from pressure data.
    Type of Medium: Electronic Resource
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  • 10
    Electronic Resource
    Electronic Resource
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical prospecting 27 (1979), S. 0 
    ISSN: 1365-2478
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences , Physics
    Notes: The relationship between two finite-difference schemes (15° and 40°) and the Kirchhoff summation approach is discussed by using closed form solutions of Claerbout's approximate versions of the wave equation. Forward extrapolation is presented as a spatial convolution procedure for each frequency component. It is shown that downward extrapolation can be considered as a wavelet deconvolution procedure, the spatial wavelet being given by the wave theory. Using this concept, a three-dimensional model for seismic data is proposed.The advantages of downward extrapolation in the space-frequency domain are discussed.Finally, it is derived that spatial sampling imposes an upper limit on the aperture and a lower limit on the extrapolation step.
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