Discrete plane‐wave decomposition by least‐mean‐square‐error method

Geophysics ◽  
1994 ◽  
Vol 59 (6) ◽  
pp. 973-982 ◽  
Author(s):  
Orhan Yilmaz ◽  
M. Turhan Taner
Keyword(s):  
Plane Wave ◽  
Point Source ◽  
Mean Square Error ◽  
Radon Transform ◽  
Mean Square ◽  
Least Mean Square ◽  
Spherical Waves ◽  
Plane Wave Decomposition ◽  
Wave Decomposition ◽  
Error Method

The recording of a point source wavefield can be decomposed into a set of plane‐wave components, each corresponding to different angles of propagation. Such plane‐wave seismograms have a far simpler structure than the spherical waves of the point source records, which makes them desirable in many steps of seismic data processing such as predictive deconvolution, migration, inversion, etc. The implementation of the plane‐wave decomposition requires the computation of the Radon transform in the discrete data domain. A straightforward application of the integral solutions to geophysical problems fails to compensate for the sampled and limited aperture nature of the actual data. In this paper, we give a new method in which the x-t domain is shown to relate to the p-τ domain by a linear system of equations in the time‐space domain. An iterative least‐mean‐square‐error method is introduced to solve the set of equations. This method is combined with a unique method of alias suppression which uses the reasonable range of dips possible at a given (x, t) location and acts as interpolation of the x-t data. This combination improves the initial estimates and speeds up convergence. Our transform is independent of the number of plane‐waves and selected ray parameter range. We present synthetic and real data examples to demonstrate the accuracy and robustness of the method. The examples are compared against results using the generalized Radon transform approach used by Beylkin (1987) and against conventional slant stack.

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.

An algorithm for the plane‐wave decomposition of point‐source seismograms

Geophysics ◽  
1990 ◽  
Vol 55 (10) ◽  
pp. 1380-1385 ◽  
Author(s):  
M. Dietrich
Keyword(s):  
Plane Wave ◽  
Point Source ◽  
Bessel Functions ◽  
Real Data ◽  
Hankel Transform ◽  
Considerable Reduction ◽  
Correct Formulation ◽  
Plane Wave Decomposition ◽  
Wave Decomposition ◽  
Discrete Integration

The correct formulation of the plane‐wave decomposition of point‐source seismograms involves a sequence of Fourier and Hankel transforms which can be evaluated in several ways. The procedure which is proposed here exploits the fact that the plane‐wave response is bandlimited along the horizontal slowness axis. This property permits to expand the Hankel transform into a Fourier‐Bessel series. In practice, this algorithm requires an interpolation in distance of the recorded dataset, but allows a considerable reduction of Bessel functions calculations. Numerical applications performed with synthetic and real data show that the Fourier‐Bessel summation technique yields results which are equivalent to a discrete integration of the Hankel transform.

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.

Mixed‐delay wavelet deconvolution of the point‐source seismogram

Geophysics ◽  
1991 ◽  
Vol 56 (9) ◽  
pp. 1405-1411 ◽  
Author(s):  
M. Tygel ◽  
H. Huck ◽  
P. Hubral
Keyword(s):  
Free Surface ◽  
Plane Wave ◽  
Point Source ◽  
Half Space ◽  
Stratified Medium ◽  
Model Parameters ◽  
Plane Wave Decomposition ◽  
Mixed Delay ◽  
Time Space ◽  
Wave Decomposition

The problem of extracting a mixed‐delay source wavelet from a point‐source seismogram for an acoustic, horizontally stratified medium (bounded by a free surface above and a half‐space below or between two half‐spaces) can be completely solved without any further assumptions about the source pulse or the model parameters. The solution relies on information contained in the so‐called evanescent part of the point‐source seismogram, which can be extracted via a plane‐wave decomposition, i.e., by a transformation of the point‐source seismogram from the time‐space domain into the frequency‐rayparameter domain.

Wavenumber‐based filtering of marine point‐source data

Geophysics ◽  
1993 ◽  
Vol 58 (9) ◽  
pp. 1335-1348 ◽  
Author(s):  
Lasse Amundsen
Keyword(s):  
Plane Wave ◽  
Point Source ◽  
Line Source ◽  
Hankel Transform ◽  
Source Pressure ◽  
Function Type ◽  
Plane Wave Decomposition ◽  
Source Data ◽  
Processing Techniques ◽  
Wave Decomposition

In seismic processing, plane‐wave decomposition has played a fundamental role, serving as a basis for developing sophisticated processing techniques valid for depth‐dependent models. By comparing analytical expressions for the decomposed wavefields, we review several processing algorithms of interest for the geophysicist. The algorithms may be applied to marine point‐source data acquired over a horizontally layered viscoelastic and anisotropic medium. The plane‐wave decomposition is based on the Fourier transform integral for line‐source data and the Hankel transform integral for point‐source data. The involved wavenumber integrals of the cosine or Bessel‐function type are difficult to evaluate accurately and efficiently. However, a number of the processing techniques can easily be run as a filtering operation in the spatial domain without transforming to the wavenumber domain. The mathematical expressions for the spatial filters are derived using plane wave analysis. With numerical examples, we demonstrate the separation of upgoing and downgoing waves from the pressure, the removal of the source ghost from the pressure, and the transformation of point‐source pressure data to the corresponding line‐source data. The filters for these three processes work satisfactorily. Limited spatial aperture is discussed both for point‐source and line‐source data. The resolution kernels relating finite‐aperture decomposed data to infinite‐aperture decomposed data are given. The kernels are approximately equal in the asymptotic limit when the minimum offset is zero.

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.

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.

Transient plane‐wave transforms for the point‐source seismogram

Geophysics ◽  
1985 ◽  
Vol 50 (12) ◽  
pp. 2889-2891 ◽  
Author(s):  
M. Tygel ◽  
P. Hubral
Keyword(s):  
Plane Wave ◽  
Point Source ◽  
Short Note ◽  
Seismic Exploration ◽  
Stratified Medium ◽  
Inverse Transformation ◽  
Plane Wave Decomposition ◽  
Transformation Formulas ◽  
The Time Domain ◽  
Wave Decomposition

The point‐source seismogram for a horizontally stratified medium needs—contrary to a generally accepted belief—only to be decomposed into a finite continuous spectrum of plane wave seismograms in order to guarantee its exact recovery from the spectrum of plane wave seismograms. In this short note, we present some new and updated forward and inverse transformation formulas with which exact decomposition and composition of a point‐source seismogram is achieved in the time domain. The new transformation formulas result from combining certain recently observed fundamental properties of point‐source seismograms with well‐known formulas of the existing theory of plane wave decomposition (Müller, 1971; Chapman, 1980; Phinney et al., 1981. The theory of plane‐wave decomposition of point‐source seismograms is now well established in seismic exploration. It is associated with such concepts as the Fourier‐Bessel transform, the Radon transform, backprojection, and slant‐stacking.

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