On plane‐wave decomposition: Alias removal

Geophysics ◽  
1989 ◽  
Vol 54 (10) ◽  
pp. 1339-1343 ◽  
Author(s):  
S. C. Singh ◽  
G. F. West ◽  
C. H. Chapman
Keyword(s):  
Plane Wave ◽  
Seismic Data ◽  
P Wave ◽  
S Wave ◽  
Plane Wave Decomposition ◽  
Time Migration ◽  
Time Distance ◽  
P Domain ◽  
Wave Decomposition ◽  
Rapid Modeling

The delay‐time (τ‐p) parameterization, which is also known as the plane‐wave decomposition (PWD) of seismic data, has several advantages over the more traditional time‐distance (t‐x) representation (Schultz and Claerbout, 1978). Plane‐wave seismograms in the (τ, p) domain can be used for obtaining subsurface elastic properties (P‐wave and S‐wave velocities and density as functions of depth) from inversion of the observed oblique‐incidence seismic data (e.g., Yagle and Levy, 1985; Carazzone, 1986; Carrion, 1986; Singh et al., 1989). Treitel et al. (1982) performed time migration of plane‐wave seismograms. Diebold and Stoffa (1981) used plane‐wave seismograms to derive a velocity‐depth function. Decomposing seismic data also allows more rapid modeling, since it is faster to compute synthetic seismograms in the (τ, p) than in the (t, x) domain. Unfortunately, the transformation of seismic data from the (t, x) to the (τ, p) domain may produce artifacts, such as those caused by discrete sampling, of the data in space.

On: “Comparison of plane‐wave decomposition and slant stacking of point‐source seismic data” by Rakesh Mithal and Emilio E. Vera (GEOPHYSICS, 52, 1631–1638, December 1987)

Geophysics ◽  
1990 ◽  
Vol 55 (3) ◽  
pp. 378-379 ◽  
Author(s):  
Douglas W. McCowan
Keyword(s):  
Data Analysis ◽  
Plane Wave ◽  
Point Source ◽  
Seismic Data ◽  
Stratified Media ◽  
Cylindrically Symmetric ◽  
Plane Wave Decomposition ◽  
Source Radiation ◽  
Wave Decomposition ◽  
Slant Stacking

Mithal and Vera give the impression that the correct cylindrically symmetric slant stack (e.g., Chapman, 1981; Harding, 1985; Brysk and McCowan, 1986a) needed to represent point‐source radiation in vertically stratified media is both expensive and unnecessary in ordinary data analysis.

Reply by the authors to Douglas W. McCowan

Geophysics ◽  
1990 ◽  
Vol 55 (3) ◽  
pp. 379-379 ◽  
Author(s):  
Rakesh Mithal ◽  
Emilio E. Vera
Keyword(s):  
Plane Wave ◽  
Point Source ◽  
Seismic Data ◽  
Wave Reflection ◽  
Suitable Alternative ◽  
Plane Wave Decomposition ◽  
Main Emphasis ◽  
Source Data ◽  
Wave Decomposition ◽  
Slant Stacking

In his discussion, McGowan directs his attention exclusively to which method should be used to produce a plane-wave decomposition of point-source seismic data. Although the choice of method is an important point, it was not the main emphasis of our paper which, as its title indicates, was the comparison between plane-wave decomposition (cylindrical slant stacking) and simple slant stacking. We demonstrated the differences between these two processes and clearly indicated the necessity of using cylindrical slant stacking in order to get the correct plane-wave reflection response of point-source data. McGowan criticizes our method because it makes use of the standard asymptotic approximation of the Bessel function [Formula: see text] and considers only outward traveling waves. In our paper we acknowledged that these simplifications do not produce accurate results for ray parameters near zero and explicitly mentioned the method of Brysk and McGowan (1986) as a suitable alternative to deal with this problem.

A plane‐wave decomposition for elastic wave fields applied to the separation of P‐waves and S‐waves in vector seismic data

Geophysics ◽  
1986 ◽  
Vol 51 (2) ◽  
pp. 419-423 ◽  
Author(s):  
A. J. Devaney ◽  
M. L. Oristaglio
Keyword(s):  
Plane Wave ◽  
Elastic Wave ◽  
Seismic Data ◽  
Seismic Profile ◽  
P Waves ◽  
Wave Fields ◽  
S Waves ◽  
Plane Wave Decomposition ◽  
Wave Decomposition ◽  
Seismic Measurements

We describe a method to decompose a two‐dimensional (2-D) elastic wave field recorded along a line into its longitudinal and transverse parts, that is, into compressional (P) waves and shear (S) waves. Separation of the data into P-waves and S-waves is useful when analyzing vector seismic measurements along surface lines or in boreholes. The method described is based on a plane‐wave expansion for elastic wave fields and is illustrated with a synthetic example of an offset vertical seismic profile (VSP) in a layered elastic medium.

Plane‐wave decomposition of seismograms

Geophysics ◽  
1982 ◽  
Vol 47 (10) ◽  
pp. 1375-1401 ◽  
Author(s):  
Sven Treitel ◽  
P. R. Gutowski ◽  
D. E. Wagner
Keyword(s):  
Plane Wave ◽  
Three Dimensional ◽  
Radial Symmetry ◽  
Angle Of Incidence ◽  
Plane Wave Decomposition ◽  
Time Migration ◽  
Predictive Deconvolution ◽  
Seismic Recording ◽  
Wave Decomposition ◽  
Slant Stacking

A point‐source seismic recording can be decomposed into a set of plane‐wave seismograms for arbitrary angles of incidence. Such plane‐wave seismograms possess an inherently simple structure that make them amenable to existing inversion methods such as predictive deconvolution. Implementation of plane‐wave decomposition (PWD) takes place in the frequency‐wavenumber domain under the assumption of radial symmetry. This version of PWD is equivalent to slant stacking if allowance is made for the customary use of linear recording arrays on the surface of a three‐dimensional medium. An imaging principle embodying both kinematic as well as dynamic characteristics allows us to perform time migration of the plane‐wave seismograms. The imaging procedure is implementable as a two‐dimensional filter whose independent variables are traveltime and angle of incidence.

Use of the slant stack for geologic or geophysical map lineament analysis

Geophysics ◽  
1997 ◽  
Vol 62 (6) ◽  
pp. 1774-1778 ◽  
Author(s):  
Robert S. Pawlowski
Keyword(s):  
Plane Wave ◽  
Radon Transform ◽  
Seismic Data ◽  
Seismic Events ◽  
Seismic Data Processing ◽  
Coherent Noise ◽  
Plane Wave Decomposition ◽  
Lineament Analysis ◽  
The Individual ◽  
Wave Decomposition

The slant‐stack technique (also known as Radon transform, τ-p transform, and plane‐wave decomposition) used in seismic data processing for discriminating between and separating seismic events of differing dips (or moveout) is applied here to the problem of geologic or geophysical map lineament analysis. The latter problem is analogous to the seismic coherent noise problem in the sense that lineaments associated with one geologic event or episode are often underprinted by the lineaments of preceding geologic disturbances and overprinted by the lineaments of subsequent disturbances. Consequently, it can be difficult to distinguish between the individual lineament sets.

Elastic least-squares reverse time migration with hybrid l1/l2 misfit function

Geophysics ◽  
2017 ◽  
Vol 82 (3) ◽  
pp. S271-S291 ◽  
Author(s):  
Bingluo Gu ◽  
Zhenchun Li ◽  
Peng Yang ◽  
Wencai Xu ◽  
Jianguang Han
Keyword(s):  
Least Squares ◽  
Seismic Data ◽  
Synthetic Data ◽  
Adjoint Equations ◽  
P Wave ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
Time Migration ◽  
Wave Image

We have developed the theory and synthetic tests of elastic least-squares reverse time migration (ELSRTM). In this method, a least-squares reverse time migration algorithm is used to image multicomponent seismic data based on the first-order elastic velocity-stress wave equation, in which the linearized elastic modeling equations are used for forward modeling and its adjoint equations are derived based on the adjoint-state method for back propagating the data residuals. Also, we have developed another ELSRTM scheme based on the wavefield separation technique, in which the P-wave image is obtained using P-wave forward and adjoint wavefields and the S-wave image is obtained using P-wave forward and S-wave adjoint wavefields. In this way, the crosstalk artifacts can be minimized to a significant extent. In general, seismic data inevitably contain noise. We apply the hybrid [Formula: see text] misfit function to the ELSRTM algorithm to improve the robustness of our ELSRTM to noise. Numerical tests on synthetic data reveal that our ELSRTM, when compared with elastic reverse time migration, can produce images with higher spatial resolution, more-balanced amplitudes, and fewer artifacts. Moreover, the hybrid [Formula: see text] misfit function makes the ELSRTM more robust than the [Formula: see text] misfit function in the presence of noise.

Comparison of plane‐wave decomposition and slant stacking of point‐source seismic data

Geophysics ◽  
1987 ◽  
Vol 52 (12) ◽  
pp. 1631-1638 ◽  
Author(s):  
Rakesh Mithal ◽  
Emilio E. Vera
Keyword(s):  
Phase Shift ◽  
Plane Wave ◽  
Point Source ◽  
Bessel Function ◽  
Seismic Data ◽  
Weighting Function ◽  
Real Data ◽  
Plane Wave Decomposition ◽  
Wave Decomposition ◽  
Slant Stacking

The plane‐wave decomposition and slant stacking of point‐source seismic data are not identical processes; they are, however, related. We have found that the algorithm for slant stacking can be used for plane‐wave decomposition if we apply a weighting function (depending on frequency and offset, and including a π/4 phase shift) before slant stacking, and a p-dependent correction after the slant stacking. This procedure requires only a small extra effort to incorporate the geometrical spreading and phase shift not accounted for by the slant stacking. In this process we use the asymptotic approximation for the zeroth‐order Bessel function. This approximation reduces the number of computations significantly, but it is valid only for ωpx greater than 2 or 3. Using this approximation, we have been able to obtain the correct plane‐wave decomposition of expanding spread profile data for ray parameters as low as 0.03 s/km; for smaller p, the exact Bessel function should be used. We have performed model studies to compare plane‐wave decomposition and slant stacking. Using a possible velocity model for the North Atlantic Transect (NAT) expanding spread profile (ESP 5), we computed synthetic seismograms at a 50 m spacing using the reflectivity method, and then computed the plane‐wave decomposition and slant stacks of these seismograms. On comparing these with the exact τ-p seismograms for this model, we found that the waveforms, the frequency content, and the amplitudes were exactly reproduced in the plane‐wave decomposition, but were significantly different in the slant stacks. We also computed the plane‐wave decomposition and slant stacks of real data (NAT ESP 5). The results in this case show that the amplitudes of deep crustal arrivals in plane‐wave decomposition are higher than in slant stacks, and therefore these arrivals can be identified much better in the plane‐wave decomposition.

Efficient angle-domain common-image gathers using Cauchy-condition-based polarization vectors

Geophysics ◽  
2016 ◽  
Vol 81 (2) ◽  
pp. S69-S77 ◽  
Author(s):  
Xiongwen Wang ◽  
Jianliang Qian ◽  
Huazhong Wang
Keyword(s):  
Plane Wave ◽  
Model Building ◽  
Computational Cost ◽  
Decomposition Algorithm ◽  
Reverse Time ◽  
Imaging Condition ◽  
Plane Wave Decomposition ◽  
Time Migration ◽  
Imaging Result ◽  
Wave Decomposition

Because angle-domain common-image gathers (ADCIGs) from reverse time migration (RTM) are capable of obtaining the correct illumination of a subsurface geologic structure, they provide more reliable information for velocity model building, amplitude-variation versus angle analysis, and attribute interpretation. The approaches for generating ADCIGs mainly consist of two types: (1) indirect approaches that convert extended image gathers into ADCIGs and (2) direct approaches that first obtain propagating angles of wavefronts and then map the imaging result to the angle domain. In practice, however, generation of ADCIGs usually incurs high computational cost, poor resolution, and other drawbacks. To generate efficient ADCIGs using RTM methods, we have introduced a novel approach to obtain polarization vectors — directions of particle motion — from the Cauchy wavefield (CWF) and an efficient localized plane-wave decomposition algorithm to implement the angle-domain imaging condition. The CWF is a wavefield constructed from the Cauchy condition of the wave equation at any given time, and it only contains negative frequencies of the original wavefield so that the polarization vector is obtained from the local CWF in the wavenumber domain. With polarization vectors at our disposal, we have further developed an efficient localized plane-wave decomposition algorithm to implement the angle-domain imaging condition. Numerical examples have indicated that the new approach is able to handle complex wave phenomenon and has advantages in illuminating subsurface structure.

Comparison of methods for extracting ADCIGs from RTM

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. S89-S103 ◽  
Author(s):  
Hu Jin ◽  
George A. McMechan ◽  
Huimin Guan
Keyword(s):  
Plane Wave ◽  
Low Noise ◽  
Direction Vector ◽  
Reverse Time ◽  
Imaging Condition ◽  
Plane Wave Decomposition ◽  
Time Migration ◽  
Local Plane ◽  
Local Shift ◽  
Wave Decomposition

Methods for extracting angle-domain common-image gathers (ADCIGs) during 2D reverse-time migration fall into three main categories; direction-vector-based methods, local-plane-wave decomposition methods, and local-shift imaging condition methods. The direction-vector-based methods, which use either amplitude gradients or phase gradients, cannot handle overlapping events because of an assumption of one propagation direction per imaging point per imaging time; however, the ADCIGs from the direction-vector-based methods have the highest angle resolution. A new direction-vector-based method using instantaneous phase gradients in space and time gives the same propagation directions and ADCIGs as those obtained by the Poynting vector or polarization vector based methods, where amplitudes are large. Angles calculated by the phase gradients have larger uncertainties at smaller amplitudes, but they do not significantly degrade the ADCIGs because they contribute only small amplitudes. The local-plane-wave decomposition and local-shift imaging condition methods, implemented either by a Fourier transform or by a slant stack transform, can handle overlapping events, and produce very similar angle gathers. ADCIGs from both methods depend on the local window size in which the transforms are done. In small local windows, both methods produce ADCIGs with low noise, but also with low angle resolution; in large windows, they have high angle resolution, but contain smeared artifacts.

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