Wave-equation global datuming based on the double square root operator

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. U35-U43 ◽  
Author(s):  
Wenge Liu ◽  
Bo Zhao ◽  
Hua-wei Zhou ◽  
Zhenhua He ◽  
Hui Liu ◽  
...  

Current schemes for removing near-surface effects in seismic data processing use either static corrections or wave-equation datuming (WED). In the presence of rough topography and strong lateral velocity variations in the near surface, the WED scheme is the only option available. However, the traditional procedure of WED downward continues the sources and receivers in different domains. A new wave-equation global-datuming method is based on the double-square-root operator, implementing the wavefield continuation in a single domain following the survey sinking concept. This method has fewer approximations and therefore is more robust and convenient than the traditional WED methods. This method is compared with the traditional methods using a synthetic data example.

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. B153-B165 ◽  
Author(s):  
Kai Yang ◽  
Hong-Ming Zheng ◽  
Li Wang ◽  
Yu-Zhu Liu ◽  
Fan Jiang ◽  
...  

An integrated wave-equation datuming scheme improves the imaging quality of seismic data from overthrust areas. It can be regarded as integrated because upward-layer replacement is included. In this scheme, data are downward continued to a nonplanar datum (such as the base of the weathering layer), followed by upward continuation from the nonplanar datum to a final planar datum using a one-way extrapolator. When compared with a Kirchhoff integral, this method can deal better with the strong lateral velocity variation within the near surface. After a test on synthetic data, the scheme is applied successfully to real 2D overthrust data acquired in the Qi-Lian foothills, western China. Compared with results using static corrections, integrated wave-equation datuming results lead to better reconstruction of the diffractions and reflections, more reliable migration-velocity analyses, and stronger stack and final depth images.


Geophysics ◽  
1998 ◽  
Vol 63 (1) ◽  
pp. 25-38 ◽  
Author(s):  
Xianhuai Zhu ◽  
Burke G. Angstman ◽  
David P. Sixta

Through the use of iterative turning‐ray tomography followed by wave‐equation datuming (or tomo‐datuming) and prestack depth migration, we generate accurate prestack images of seismic data in overthrust areas containing both highly variable near‐surface velocities and rough topography. In tomo‐datuming, we downward continue shot records from the topography to a horizontal datum using velocities estimated from tomography. Turning‐ray tomography often provides a more accurate near‐surface velocity model than that from refraction statics. The main advantage of tomo‐datuming over tomo‐statics (tomography plus static corrections) or refraction statics is that instead of applying a vertical time‐shift to the data, tomo‐datuming propagates the recorded wavefield to the new datum. We find that tomo‐datuming better reconstructs diffractions and reflections, subsequently providing better images after migration. In the datuming process, we use a recursive finite‐difference (FD) scheme to extrapolate wavefield without applying the imaging condition, such that lateral velocity variations can be handled properly and approximations in traveltime calculations associated with the raypath distortions near the surface for migration are avoided. We follow the downward continuation step with a conventional Kirchhoff prestack depth migration. This results in better images than those migrated from the topography using the conventional Kirchhoff method with traveltime calculation in the complicated near surface. Since FD datuming is only applied to the shallow part of the section, its cost is much less than the whole volume FD migration. This is attractive because (1) prestack depth migration usually is used iteratively to build a velocity model, so both efficiency and accuracy are important factors to be considered; and (2) tomo‐datuming can improve the signal‐to‐noise (S/N) ratio of prestack gathers, leading to more accurate migration velocity analysis and better images after depth migration. Case studies with synthetic and field data examples show that tomo‐datuming is especially helpful when strong lateral velocity variations are present below the topography.


Geophysics ◽  
1996 ◽  
Vol 61 (5) ◽  
pp. 1394-1399 ◽  
Author(s):  
Thomas Rühl

Finite‐difference (FD) migration is one of the most often used standard migration methods in practice. The merit of FD migration is its ability to handle arbitrary laterally and vertically varying macro velocity fields. The well‐known disadvantage is that wave propagation is only performed accurately in a more or less narrow cone around the vertical. This shortcoming originates from the fact that the exact one‐way wave equation can be implemented only approximately in finite‐difference schemes because of economical reasons. The Taylor or continued fraction expansion of the square root operator in the one‐way wave equation must be truncated resulting in an approximate version of the one‐way wave equation valid only for a restricted angle range.


Geophysics ◽  
2010 ◽  
Vol 75 (2) ◽  
pp. S23-S34 ◽  
Author(s):  
Jin-Hai Zhang ◽  
Wei-Min Wang ◽  
Shu-Qin Wang ◽  
Zhen-Xing Yao

A wide-angle propagator is essential when imaging complex media with strong lateral velocity contrasts in one-way wave-equation migration. We have developed a dual-domain one-way propagator using truncated Chebyshev polynomials and a globally optimized scheme. Our method increases the accuracy of the expanded square-root operator by adding two high-order terms to the traditional split-step Fourier propagator. First, we approximate the square-root operator using Taylor expansion around the reference background velocity. Then, we apply the first-kind Chebyshev polynomials to economize the results of the Taylor expansion. Finally, we optimize the constant coefficients using the globally optimized scheme, which are fixed and feasible for arbitrary velocity models. Theoretical analysis and nu-merical experiments have demonstrated that the method has veryhigh accuracy and exceeds the unoptimized Fourier finite-difference propagator for the entire range of practical velocity contrasts. The accurate propagation angle of the method is always about 60° under the relative error of 1% for complex media with weak, moderate, and even strong lateral velocity contrasts. The method allows us to handle wide-angle propagations and strong lateral velocity contrast simultaneously by using Fourier transform alone. Only four 2D Fourier transforms are required for each step of 3D wavefield extrapolation, and the computing cost is similar to that of the Fourier finite-difference method. Compared with the finite-difference method, our method has no two-way splitting error (i.e., azimuthal-anisotropy error) for 3D cases and almost no numerical dispersion for coarse grids. In addition, it has strong potential to be accelerated when an enhanced fast Fourier transform algorithm emerges.


Geophysics ◽  
1980 ◽  
Vol 45 (12) ◽  
pp. 1753-1779 ◽  
Author(s):  
Özdoan Yilmaz ◽  
Jon F. Claerbout

Conventional seismic data processing can be improved by modifying wide‐offset data so that dipping events stack coherently. A procedure to achieve this improvement is proposed here, which is basically a “partial” migration of common offset sections prior to stack. It has an advantage over full migration before stack in that, in the case of the latter, the final product is a migrated section. However, the prestack partial migration provides the interpreter with a high‐quality common midpoint (CMP) stacked section which can be subsequently migrated. The theory of prestack partial migration is based on the double square‐root equation, which describes seismic imaging with many shots and receivers. The double square‐root operator in midpoint‐offset space can be separated approximately into two terms, one involving only migration effects and the other involving only moveout correction. This separation provides an analysis of conventional processing. Estimation of errors in the separation yields the equation for prestack partial migration. Extension of the theory for separable approximation to incorporate lateral velocity variation yields a significant term proportional to the product of the first powers of offset, dip, and lateral velocity gradient. This term was used to obtain a rough estimate of lateral velocity variation from a field data set.


Geophysics ◽  
1988 ◽  
Vol 53 (10) ◽  
pp. 1311-1322 ◽  
Author(s):  
V. Shtivelman ◽  
A. Canning

Seismic sections are usually datum corrected by static shifting. For small differences in elevation and slow velocity variations between the input datum and the output datum, static shifting is a sufficiently accurate datum correction procedure. However, for significant differences in elevations and a more complicated velocity model, the accuracy of the static solution may prove to be insufficient; and a more exact method should be used. In this paper, we study the limitations of the static method of datum correction and develop simple and effective extrapolation schemes based on the wave equation, schemes which lead to more accurate datum correction. The distortions of seismic events caused by static correction are illustrated by a number of simple examples. To reduce the distortions, we propose a number of extrapolation schemes based on the asymptories of the Kirchhoff integral solution of the 2-D scalar wave equation. Application of the extrapolation algorithms to synthetic data shows that they provide accurate datum corrections even for a nonplanar input datum and vertical and lateral velocity variations. The algorithms have been successfully applied to real data.


Geophysics ◽  
1983 ◽  
Vol 48 (6) ◽  
pp. 677-687 ◽  
Author(s):  
Dan D. Kosloff ◽  
Edip Baysal

Conventional finite‐difference migration has relied on one‐way wave equations which allow energy to propagate only downward. Although generally reliable, such equations may not give accurate migration when the structures have strong lateral velocity variations or steep dips. The present study examined an alternative approach based on the full acoustic wave equation. The migration algorithm which developed from this equation was tested against synthetic data and against physical model data. The results indicated that such a scheme gives accurate migration for complicated structures.


Geophysics ◽  
1994 ◽  
Vol 59 (6) ◽  
pp. 963-972 ◽  
Author(s):  
Bastian Blonk ◽  
Gérard C. Herman

A method is presented for eliminating near‐surface scattered noise from seismic data. Starting from an appropriately chosen background model, a surface‐consistent scattering model is determined using linearized elastodynamic inverse scattering theory. This scattering model does not necessarily equal the actual scatterer distribution, but it enables one to calculate, approximately, the near‐surface scattered part of the data. The method honors at least some of the complexity of the near‐surface scattering process and can be applied in cases where traditional methods, like wavenumber‐frequency filtering techniques and methods for static corrections, are ineffective. From a number of tests on synthetic data, we conclude that the method is rather robust; its main sensitivity is because of errors in the determination of the background Rayleigh‐wave velocity.


Geophysics ◽  
2007 ◽  
Vol 72 (4) ◽  
pp. U55-U66 ◽  
Author(s):  
Robbert van Vossen ◽  
Jeannot Trampert

Near-surface wavefield perturbations can be very complex and completely mask the target reflections. Despite this complexity, conventional methods rely on parameterizations characterized by simple time and amplitude anomalies to compensate for these perturbations. Determining and compensating for time shifts is generally referred to as (residual) static corrections, whereas surface-consistent deconvolution techniques deal with amplitude anomalies. We present an approach that uses the full waveform to parameterize near-surface perturbations. Therefore, we refer to this method as waveform statics. Important differences from conventional static corrections are that this approach allows time shifts to vary with frequency and takes amplitude variations directly into account. Furthermore, the procedure is fully automated and does not rely on near-surface velocity information. The waveform static corrections are obtained usingblind channel identification and applied to the recordings using multichannel deconvolution. As a result, the method implicitly incorporates array forming. The developed method is validated on synthetic data and applied to part of a field data set acquired in an area with significant near-surface heterogeneity. The source and receiver responses obtained are strongly correlated to the near-surface conditions and show changes, both in phase and frequency content, along the spread. The application of the waveform statics demonstrates that they not only correct for near-surface wavefield perturbations, but also strongly reduce coherent noise. This results in substantial improvements, both in trace-to-trace coherency and in depth resolution. In addition, the procedure delineates reflection events that are difficult to detect prior to our proposed correction. Based on these results, we conclude that complex near-surface perturbations can be successfully dealt with using the multichannel, full-waveform, static-correction procedure.


Geophysics ◽  
1979 ◽  
Vol 44 (7) ◽  
pp. 1175-1192 ◽  
Author(s):  
Kenneth L. Larner ◽  
Bruce R. Gibson ◽  
Ron Chambers ◽  
Ralph A. Wiggins

Seismic surveys on land are frequently conducted along nonlinear survey lines. Familiar examples include crooked lines controlled by existing road networks or by surface typography, lines that are otherwise linear but along which shotpoints occasionally must be offset laterally, and intentionally designed three‐dimensional (3-D) or broad‐line surveys. Departures from linear profiles introduce an element of complexity—crossdip—into the problem of estimating residual near‐surface reflection static time corrections (statics). Crossdip is the component of dip normal to the local profile direction. We have incorporated the effect of crossdip into the system of simultaneous equations that model residual static anomalies. The observed traveltimes of all reflections selected for analysis are represented as linear combinations of source and receiver static anomalies, structural shapes, residual normal moveouts, and crossdip terms. The static time components are taken to be surface‐consistent and independent of reflecting horizon, whereas the other solution parameters are subsurface‐consistent and pertain to specific horizons. Unfortunately, the inclusion of crossdip in the equations increases the degree of nonuniqueness of residual statics solutions. Its inclusion, however, is a necessity wherever horizons having differing crossdips are analyzed simultaneously. Such simultaneous analysis often is the best means for upgrading the reliability of the crosscorrelation estimates (i.e., the traveltime observations) upon which all statics are based. Synthetic‐data examples demonstrate the degree to which crossdip estimates and statics estimates can be separated from one another. Although estimates of crossdips are a useful by‐product, the accuracy of the static corrections is considered of prime importance. When critical crossdip terms are ignored in a statics solution, the quality of the common‐depthpoint (CDP) stacks suffer, as shown in comparison processings of field sections. Moreover, crossdip estimates from 3-D or broad‐line surveys are questionable if crossdip and static corrections are not considered in a unified solution.


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