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#### Frequency-wavenumber migration in practice

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Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2018 SEG International Exposition and Annual Meeting, October 14–19, 2018

Paper Number: SEG-2018-2995921

... for the separation of the two components is scattering angle filtering based upon equation 6 (Wu and Alkhalifah, 2015, 2017; Yao et al., 2018). As can be seen from equation 6, for a given

**frequency**and velocity, the tomographic component has smaller**wavenumbers**than the**migration**component. This is illustrated...
Abstract

ABSTRACT Full-waveform inversion (FWI) includes both migration and tomography modes. The tomographic component of the gradient from reflections usually is much weaker than the migration component. In order to use the tomography mode of FWI, it is necessary to extract the tomographic component from the gradient. We analyze the characteristics of wavenumbers of the migration and tomographic components, and then introduce a new method to extract the tomographic component based upon non-stationary smoothing. We demonstrate the effectiveness of the proposed method for enhancing the tomographic mode of FWI throughout theoretical analysis and numerical examples. Presentation Date: Wednesday, October 17, 2018 Start Time: 1:50:00 PM Location: Poster Station 8 Presentation Type: Poster

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2002 SEG Annual Meeting, October 6–11, 2002

Paper Number: SEG-2002-1152

... Summary 3-D prestack

**migration**of constant-offset data can be formulated in the**frequency**-**wavenumber**domain as a stationary phase approximation to equivalent Kirchhoff expressions. This formulation constitutes a proof of Dubrulle''s (1983) heuristic algorithm. Numerical implementations are 3-D...
Abstract

Summary 3-D prestack migration of constant-offset data can be formulated in the frequency-wavenumber domain as a stationary phase approximation to equivalent Kirchhoff expressions. This formulation constitutes a proof of Dubrulle''s (1983) heuristic algorithm. Numerical implementations are 3-D extensions of his algorithms and lead to very efficient migration under the assumption of lateral invariance in velocity and acquisition geometry. Introduction Dubrulle (1983) presented a method for migrating 2-D common-offset data in the frequency-wavenumber domain. His algorithm for finite offset data is based on a heuristic extension of an algorithm devised for zero offset data. The phase shifts required to migrate the data are computed by the numerical solution of a pair of algebraic equations relating the traveltime and midpoint ray parameter of a diffraction event to the offset and migration displacement.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2009 SEG Annual Meeting, October 25–30, 2009

Paper Number: SEG-2009-2383

... For constant-velocity demigration/remigration the DSR process is equivalent to the application of a mutli-dimensional shaping operator, applied after

**migration**. Here we extend the spectral shaping concept, to modify not just the**frequency**(temporal), but also the**wavenumber**(spatial) spectrum of the data...
Abstract

Summary Seismic inversion for acoustic impedance can be robustly and efficiently performed through the application of a spectral shaping filter combined with a -90o phase rotation. We show that this process, first published by Lancaster and Whitcombe (2000), is rigorously accurate and mathematically equivalent to all other methods of acoustic inversion based on a one-dimensional convolutional model. Yet, the one-dimensional convolutional model is inadequate for describing migrated traces, because it ignores the dip dependence of the wavelet after migration. Any seismic inversion method assuming such a model, will cause severe amplitude distortions when applied to migrated data, excessively amplifying dipping events, signal or noise. We describe here three methods to perform seismic inversion, that overcome the problem described above, namely: (i) applying spectral shaping before migration; (ii) applying spectral shaping to demigrated data, that are then remigrated; (iii) applying a multi-dimensional spectral shaping operator after migration. Introduction - Seismic Inversion by Spectral Shaping The term seismic inversion is commonly used to describe the estimation of acoustic impedance from seismic data. The starting point for most impedance inversions is the 1D convolutional model for the seismic trace, expressed as: where s(t) the seismic trace, w(t) the seismic wavelet and r(t) the reflectivity series. Lancaster and Whitcombe (2000) introduced the Coloured Inversion method as an efficient way to perfom seismic inversion. The method combines a phase rotation of -90o, to convert the input zero-phase seismic traces to quadrature-phase and a spectral shaping operation implemented as the application of a filter that reshapes the original seismic spectrum to make it similar to the average spectrum of impedance logs, recorded at wells in the area over which the seismic data have been collected. The concept is illustrated in Figure 1. As can be inferred from the figure, shaping filters necessary to perform inversion significantly amplify the energy in the low-frequency part of the seismic spectrum. It can actually be shown that this approach (although presented by Lancaster and Whitcombe as an empirical, quick and easy approach), is rigorously accurate, assuming a weak-scattering (Born-type) assumption holds, something that is generally the case in most situations. We sketch the proof of this statement in the Appendix. Inversion by spectral shaping of the type shown in Figure 1 is therefore mathematically equivalent to all other methods of 1D inversion that have equation (1) as their starting point. Figure 1. Impedance inversion can be performed by reshaping the original seismic spectrum (red) to make it similar to the average impedance log spectrum (blue) and rotating the phase by -90 o . The resulting frequency spectrum is shown in green. Yet, there is a significant problem with the use of equation (1) to describe migrated seismic traces. The key point is that, after migration, the wavelet is dip-dependent. Ignoring this effect can severely degrade inversion results. Having recognized this, we discuss here new approaches for performing seismic inversion. Although the discussion in this abstract is limited to the acoustic inversion case, the extension of the concepts to elastic inversion is straightforward.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the SEG International Exposition and Annual Meeting, September 15–20, 2019

Paper Number: SEG-2019-3200038

... laplacian filter exxonmobil upstream research company traditional laplacian filter dip-guided laplacian image filter seg international exposition low-

**wavenumber**noise noise artifact reservoir characterization effectiveness annual meeting equation**migration**reverse-time**migration**operator...
Abstract

ABSTRACT An efficient and more effective post-imaging noise filtering technique has been developed to filter low-wave number noise and other undesired artifacts simultaneously while preserving desired signals. This method is based on the combination of a Laplacian-like operator filter with desired image dip information. The filter operator is a combination of the traditional Laplacian operator filter and the image-dip-guided Laplacian filter with a weighting factor to control the relative contribution from each of the two components. More effective removal of both low-wavenumber noise and other undesired artifacts, in conjunction with the preservation of desired signals, has been achieved. Presentation Date: Monday, September 16, 2019 Session Start Time: 1:50 PM Presentation Start Time: 2:15 PM Location: 214C Presentation Type: Oral

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2005 SEG Annual Meeting, November 6–11, 2005

Paper Number: SEG-2005-1846

... table

**frequency**dataset phase shift spmi 2 application seg houston 2005 algorithm evanescent equation**migration****wavenumber**extrapolation compactly upstream oil & gas depth**migration**reservoir characterization explicit depth**migration**Optimizing explicit depth**migration**...
Abstract

ABSTRACT We present a new approach to the design of stable and accurate wavefield extrapolation operators needed for explicit depth migration. We split the theoretical operator into two component operators, one a forward operator that controls the phase accuracy and the other an inverse operator, designed as a Wiener filter that stabilizes the first operator. Both component operators are designed to have a specific fixed length and the final operator is formed as the convolution of the components. We utilize this operator design method to build an explicit, wavefield extrapolation method based on the migration of individual source records. Two other features of our method are the use of dual operator tables, with high and low levels of evanescent filtering, and frequency-dependent spatial down sampling. Both of these features improve the accuracy and efficiency of the overall method. Empirical testing shows that our method has a similar performance to the time-migration method called phase shift, meaning it scales as NlogN. We illustrate the method with tests on the Marmousi synthetic dataset. We call our method which is an acronym for .

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2001 SEG Annual Meeting, September 9–14, 2001

Paper Number: SEG-2001-0296

... inaccurate, amplitude responses. The amplitude problem is related to the fact that downward continuation

**migration**is the ad-joint of upward-continuation modeling, but it is a poor approximation of its inverse. We derive the weighting operators, diagonal in the**frequency**-**wavenumber**domain, that makes...
Abstract

ABSTRACT We present two methods to compute angle-domain common image gathers (ADCIG) by downward-continuation migration, and we analyze their amplitude response versus reflection angle (AVA). A straightforward implementation of the two methods leads to contradictory, and thus obviously inaccurate, amplitude responses. The amplitude problem is related to the fact that downward continuation migration is the ad-joint of upward-continuation modeling, but it is a poor approximation of its inverse. We derive the weighting operators, diagonal in the frequency-wavenumber domain, that makes migration a good approximation to the inverse of modeling. After weighting, the ADCIGs computed by the two methods become consistent. Other important factors degrading the accuracy of AVA in practical situation are the limited sampling and offset range, and the bandlimited nature of seismic data.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2010 SEG Annual Meeting, October 17–22, 2010

Paper Number: SEG-2010-3210

... by a phase shift of the form eikz z , where z is depth and kz is vertical

**wavenumber**, has been widely used previously as a basis for downwardcontinuation modeling and**migration**methods (e.g., Gazdag, 1978). RITE operators for scalar acoustic waves The square-root, exponential, and cosine...
Abstract

Summary We derive and compare a variety of algorithms for recursive time extrapolation of scalar waves using approximate operators derived from integral solutions of a wave equation. These methods fall into two categories: those based on combining or interpolating homogeneous solutions, and those based on series expansions of heterogeneous operators. The former suffer from oscillatory noise at large velocity discontinuities unless the time step is small, whereas the latter allow accurate extrapolation at large time steps, but are more costly. Introduction Reverse time migration requires high-accuracy time extrapolation of wavefields. Conventional explicit finitedifference or pseudospectral methods for time extrapolation of waves are usually limited to small time steps by numerical dispersion errors and instability. Recently, several alternative approaches have been proposed for solving scalar wave equations based on integral mixeddomain space / wavenumber time extrapolation methods (Zhang et al., 2007; Soubaras and Zhang, 2008; Wards et al., 2008; Zhang and Zhang, 2009; Etgen and Brandsberg- Dahl, 2009; Pestana and Stoffa, 2009; Stoffa and Pestana, 2009; Liu, et al., 2009; Du, et al., 2010). We refer here to this entire family of approaches generically as “recursive integral time extrapolation” (RITE), as they are all based on formal integral solutions of a wave equation. These integral solutions are then applied recursively to extrapolate the wavefield forward or backwards in discrete time steps. These RITE approaches attempt to achieve stable, dispersion-free extrapolation in heterogeneous media for large time steps, up to the Nyquist limit or beyond. We categorize and compare here a variety of RITE methods within a common derivation framework. RITE methods are conceptually based on a heuristic of time extrapolating a wavefield using multiplication by a phase shift of the form ei w t , where t is time and w has units of frequency and is derived from an appropriate dispersion relation. An analogous concept of spatial wavefield extrapolation using multiplication by a phase shift of the form eikz z , where z is depth and kz is vertical wavenumber, has been widely used previously as a basis for downwardcontinuation modeling and migration methods (e.g., Gazdag, 1978). RITE operators for scalar acoustic waves The square-root, exponential, and cosine functions of operators indicated in equations 4 and 5 can be given formal meaning by decomposition into eigenvectors and eigenvalues of the wavenumber-domain operator A defined in equation 3. This in principle gives integral solutions for free-space time extrapolation of the scalar wave equation 1 that will be accurate in heterogeneous media for arbitrarily large time steps. This is a time extrapolation analog of the eigen-decomposition downward-continuation approach discussed by Jacobs (1980), Pai (1985), and Kosloff and Kessler (1987). However, eigen-decomposition of the operator A is computationally too expensive for realistic three-dimensional models. The various approaches suggested for practical approximation of the heterogeneous RITE operator in equation 4 can be divided into two basic families: those based on combination or interpolation of locally homogeneous solutions, and those based on series expansion approximations of the heterogeneous operator.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2017 SEG International Exposition and Annual Meeting, September 24–29, 2017

Paper Number: SEG-2017-17731403

... gradient of reflection data. As shown by Mora (1989), reflection data produce two different components in the FWI gradient: the high-

**wavenumber**component, also known as the**migration**term, and the low-**wavenumber**component, also known as the tomographic term or rabbit ears (Figure 1). This tomographic...
Abstract

ABSTRACT We present a Reflection FWI (RFWI) workflow to update the velocity model using the low-wavenumber component of the FWI gradient of reflection data. This is achieved by alternately using high-wavenumber and low-wavenumber components to update density and velocity models, respectively. With synthetic examples, we discuss the limitations and requirements of this approach and propose possible ways to overcome some of the limitations. Finally, the method is applied to a deep-water survey in the Gulf of Mexico, where improvement is observed both in the migrated image and gathers. Presentation Date: Wednesday, September 27, 2017 Start Time: 2:15 PM Location: 361F Presentation Type: ORAL

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the SEG International Exposition and Annual Meeting, September 15–20, 2019

Paper Number: SEG-2019-3216654

... smoothing operator to

**migration**velocity model, which is followed by a standard RTM. The proposed imaging strategy suppresses the diving-wave imaging artifacts as well as the low-**frequency**noises. Meanwhile, it keeps the ability to imaging structures that are with high dip angle without any additional...
Abstract

ABSTRACT We analyze the wavenumber sampling characteristics of decomposed RTM images through different schemes, including wavenumber-domain distribution of decomposed images, analytical analysis of model wavenumber vector, illumination-based imaging resolution function analysis etc. Based on the physical formation mechanism of diving wavepaths, we illustrate and interpret the causes of the unique diving-wave imaging artifacts in RTM when strong gradient variations are included in the migration velocity. An effective artifacts removal strategy is established by introducing a wavelength-dependent smoothing operator to migration velocity model, which is followed by a standard RTM. The proposed imaging strategy suppresses the diving-wave imaging artifacts as well as the low-frequency noises. Meanwhile, it keeps the ability to imaging structures that are with high dip angle without any additional computational cost. Presentation Date: Wednesday, September 18, 2019 Session Start Time: 1:50 PM Presentation Start Time: 1:50 PM Location: 214C Presentation Type: Oral

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the SEG International Exposition and Annual Meeting, October 11–16, 2020

Paper Number: SEG-2020-3427923

... Seismic amplitudes can be biased by uneven illumination in the presence of a complex overburden. Iterative Least-Squares

**Migration**(LSM) can reduce the amplitude bias and improve the resolution of the images. We introduce a robust and**practical**iterative least-squares**migration**for the inversion...
Abstract

Seismic amplitudes can be biased by uneven illumination in the presence of a complex overburden. Iterative Least-Squares Migration (LSM) can reduce the amplitude bias and improve the resolution of the images. We introduce a robust and practical iterative least-squares migration for the inversion of angle domain common image gathers. The algorithm uses wave-equation migration and demigration in the extended subsurface offset domain followed by an offset to angle transformation. We demonstrate using the Sigsbee2b synthetic and field data from Santos Basin, offshore Brazil that iterative LSM provides high-resolution angle domain common image gathers, extending their usable angle range, and balancing their amplitudes. Presentation Date: Wednesday, October 14, 2020 Session Start Time: 8:30 AM Presentation Time: 10:35 AM Location: 361A Presentation Type: Oral

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2018 SEG International Exposition and Annual Meeting, October 14–19, 2018

Paper Number: SEG-2018-2995236

... became a

**practical**method for the inverting the subsurface velocity. Even many studies of FWI, recovering the full spatial-**frequency**(**wavenumber**) of the model is difficult because of the insufficient low**frequency**component and limitation of the offset range. These limitations make the FWI invert only...
Abstract

ABSTRACT Full waveform inversion (FWI) is a suitable algorithm to invert the subsurface velocity with high accuracy. The low frequency components and long offset seismic data are necessary for the FWI to invert the long wavelength velocity. To invert the long wavelength velocity from the reflection-dominant short offset seismic data, the reflection-based full waveform inversion (RFWI) which decomposed the gradient of the FWI into high and low wavenumber components was introduced. However, the conventional RFWI still contained the high wavenumber components which were generated between the primary wavefield and the scattered wavefield. Moreover, the true amplitude migration which needed additional computation was necessary. In this study, the new frequency-domain RFWI algorithm is introduced to overcome the drawbacks of the conventional RFWI. The new RFWI algorithm uses the up/down-going wavefield separation to remove the high wavenumber component and the two-step approach to enhance the computational efficiency. The effectiveness of the proposed algorithm is verified by the synthetic numerical test using short-offset Marmousi seismic data. Presentation Date: Tuesday, October 16, 2018 Start Time: 1:50:00 PM Location: 207C (Anaheim Convention Center) Presentation Type: Oral

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the SEG International Exposition and Annual Meeting, September 15–20, 2019

Paper Number: SEG-2019-3215798

... ) produced by

**migration**and FWI as a dot-product of slowness vectors associated to two local-plane waves. Thus the low instantaneous**wavenumbers**K are obtained through low**frequency**and wide scattering angle. In Chavent et al. (1994a), and Bunks et al. (1995) the low**wavenumbers**are obtained first, from...
Abstract

ABSTRACT The propagative wavenumbers are carried by the First Fresnel zone of the velocity gradient of RFWI and divingwaves FWI. Noting that the parameter-dependent radiation patterns follow iso-angle curves, we compare the ellipse of the First Fresnel with the iso-angles curves. The latter are arcs of circles in homogeneous media. We show that the propagative part of the First Fresnel Zone can be precisely carved by a local angle filtering, more precisely that the implicit angle filtering carried by the velocity/impedance radiation pattern. For a given central frequency, the explicit angle filter requires only the knowledge of the local scattering angle and of the clock time of the wavefield snapshots. Presentation Date: Tuesday, September 17, 2019 Session Start Time: 8:30 AM Presentation Start Time: 10:35 AM Location: 217B Presentation Type: Oral

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 1990 SEG Annual Meeting, September 23–27, 1990

Paper Number: SEG-1990-1301

... that this accuracy is rarely achieved in

**practice**. Over the wide rauge of normalized**frequencies**likely to be encountered in**practice**, stable explicit extrapolators outperform implicit ones. REFERENCES Blat ui&re, G., Debe e, H. W. J., Wapenaar, C. P. A., 8 an Berkhout, A. 3 ., 1989, 3D table-driven**migration**...
Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2015 SEG Annual Meeting, October 18–23, 2015

Paper Number: SEG-2015-5829692

... Summary I describe a new two-way wave equation

**migration**that combines the efficiency of one-way wave equation**migration**with the imaging capability of reverse time**migration**. This approach extrapolates the wavefield in the**frequency**-**wavenumber**domain along the depth axis. The key...
Abstract

Summary I describe a new two-way wave equation migration that combines the efficiency of one-way wave equation migration with the imaging capability of reverse time migration. This approach extrapolates the wavefield in the frequency-wavenumber domain along the depth axis. The key is to create an extrapolator to perform the upward continuation from the bottom depth to compute P P z / -- the ratio of the depth derivative of the wavefield and the wavefield itself -- using an absorbing boundary condition at the bottom. Then the wavefield is downward-continued from the surface using P z / P . Thus the image at any subsurface point is dependent on the entire wavefield. Steeply dipping events and turning waves can all be handled. The computational cost is one and a half times that of the one-way wave equation migration, far cheaper than the reverse-time migration whose cost is proportional to the fourth power of the frequency. Synthetic data examples show that results are comparable to those obtained from reverse time migration. Introduction Over the last a few decades, wave equation migration (WEM) has been increasingly preferred over Kirchhoff-type migrations for its ability to handle multiple ray paths. Standard (that is, one-way) WEM splits the two-way wave equation into two one-way wave equations, namely the down-going wave equation and the up-going wave equation, which are then solved in either the frequency-space or the frequency-wavenumber domain. For prestack depth migration, one-way WEM extrapolates the down-going wavefield (the source wavefield) and the up-going wavefield (the receiver wavefield) depth slice by depth slice for each frequency component. The operator for extrapolating the up-going wavefield is the complex conjugate of the down-going operator so that, in effect, the one-way WEM downward-continues both source and receiver wavefields from the surface using the same extrapolator. As a result, the image at any subsurface point only depends on the wavefield above this point. This makes one-way WEM efficient but dip-limited and unable to migrate turning waves.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2016 SEG International Exposition and Annual Meeting, October 16–21, 2016

Paper Number: SEG-2016-13872562

... associated with shallow gas pockets, and enabled us to mitigate imaging distortions at the target intervals. In addition, FWI uncovered a scale of sub-surface information typically unavailable on standard

**migrated**sections, in effect filling-in the low**wavenumbers**that correspond to**frequencies**below...
Abstract

ABSTRACT The use of acoustic FWI as a velocity model building tool has by now been well-documented in numerous publications. The vast majority of the published studies share two characteristics: (i) FWI is only using refracted arrivals and very low frequencies, typically less than 10 Hz; (ii) velocity models generated with FWI are only used to produce better migrations, but do not possess the resolution to be directly interpretable. We present here an FWI field data application, that differs from common practice in two substantial ways: (i) the input seismic data set is dominated by reflections, with no appreciable amounts of refracted energy; (ii) FWI products were generated with sufficient high-frequency bandwidth (40 Hz), to be directly interpretable. We found that FWI provided significant benefits for imaging and sub-surface interpretation. The FWI velocity model resolved the velocity complexity associated with shallow gas pockets, and enabled us to mitigate imaging distortions at the target intervals. In addition, FWI uncovered a scale of sub-surface information typically unavailable on standard migrated sections, in effect filling-in the low wavenumbers that correspond to frequencies below the traditional seismic bandwidth. Presentation Date: Tuesday, October 18, 2016 Start Time: 3:20:00 PM Location: 162/164 Presentation Type: ORAL

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2018 SEG International Exposition and Annual Meeting, October 14–19, 2018

Paper Number: SEG-2018-2996390

... for one-way wave equation

**migration**(though exactly how to smooth and damp are serious**practical**issues). Those methods naturally work one (or a few) (1) (4) (2) (3) 10.1190/segam2018-2996390.1 Page 4372 © 2018 SEG SEG International Exposition and 88th Annual Meeting Deconvolution ISIC**frequencies**...
Abstract

ABSTRACT We describe a new imaging condition for reverse-time migration (RTM). Specifically, we extend the inverse scattering imaging condition (ISIC) to deconvolution in the temporal frequency domain, and apply it in an RTM version of Separated Wavefield Imaging (SWIM). Tests on both synthetic and field data show this method successfully removes low-wavenumber artifacts associated with backscattered energy and has similar strengths and challenges to deconvolution-based SWIM with one-way wave equation migration. Presentation Date: Wednesday, October 17, 2018 Start Time: 1:50:00 PM Location: 207A (Anaheim Convention Center) Presentation Type: Oral

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2009 SEG Annual Meeting, October 25–30, 2009

Paper Number: SEG-2009-2552

... size. )2( 22 2 11 ]1cos t kFtv v Then, we compute phase velocity as a function of

**wavenumber**by dividing temporal**frequency**by the magnitude of the**wavenumber**vector. )3( 22 2 11 ]1cos )( kt kFtv kv phase v v Written this way, we find the numerical phase velocity of an acoustic wave...
Abstract

Summary We generalize the pseudo-spectral method for the acoustic wave equation to create analytical solutions to the constant velocity acoustic wave equation in an arbitrary number of space dimensions. We accomplish this by modifying the Fourier Transform of the Laplacian operator so that it compensates exactly for the error due to the second-order finite-difference time marching scheme used in the conventional pseudo-spectral method. Of more practical interest, we show that this modified or pseudo-Laplacian is a smoothly varying function of the parameters of the acoustic wave equation (velocity most importantly) and thus can be further generalized to create near-analytically-accurate solutions for the variable velocity case. We call this new method the pseudo-analytical method. We further show that by applying this approach to the concept of acoustic anisotropic wave propagation, we can create scalar-mode VTI and TTI wave equations that overcome the disadvantages of previously published methods for acoustic anisotropic wave propagation. These methods should be ideal for forward modeling and reverse time migration applications. Introduction The pseudo-spectral method (Reshef et al., 1988) is generally considered an accurate method for solving equations such as the acoustic or elastic wave equations. However, it still suffers from errors, namely grid dispersion, due to the fact that second-order (or sometimes higher-order) finite differences are applied on the time axis. Etgen (2007) describes a technique built upon the work of Holberg (1987) that includes the effect of second-order time discretization with finite time-step size to partially compensate errors due to space discretization, thus creating a globally optimized finite-difference scheme. However, optimized finite-difference techniques still must use some degree of oversampling compared to the Nyquist limit. Any technique that strives for “perfect” accuracy either has to have no error in both the time and space discretizations, or have errors in both that cancel each other exactly. Tal- Ezer et al. (1987) described an approach based on the former by using pseudo-spectral space derivatives coupled with an orthogonal polynomial expansion in time. While this method is accurate, it is somewhat cumbersome to code and has seen little industrial use to our knowledge. Our approach is of the later type; we use our freedom to modify the wavenumber response of the Laplacian, or other spatial derivatives that we need, directly in the wavenumber domain portion of a pseudo-spectral-like method to cancel the error caused by second-order finite-difference time marching. The pseudo-analytical method The accuracy of a numerical wave propagation scheme can be determined in the constant wave-speed case by finding the expression of the Fourier Transform of the method and solving that expression for temporal frequency as a function of all the other variables. Equation 1 gives an expression for the Fourier Transform of a second-order time-marching solution to the acoustic wave equation where we’ve left the details of the spatial difference method generic: We rearrange this expression to give temporal frequency as a function of velocity, the spatial Fourier Transform of the spatial differential operator and the time step size.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 1995 SEG Annual Meeting, October 8–13, 1995

Paper Number: SEG-1995-1377

... ABSTRACT No preview is available for this paper. geophysics amplitude time

**migration**diffraction prestack time**migration**reservoir characterization ekren**frequency**-**wavenumber**constant-offset**migration****migrated**data nmo correction equation time**migration**algorithm data...
Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2008 SEG Annual Meeting, November 9–14, 2008

Paper Number: SEG-2008-2381

... in the

**frequency**domain (commonly called ''Shot profile wave equation**migration**''), or ''two-way'' 3D finite difference modeling (commonly called ''Reverse time**migration**''). When analyzing the sampling issues, we should consider both those within a single supershot, in addition to the related issue...
Abstract

Summary Sampling theory is one of the basic foundations of geophysics. The spatial sampling criteria for classical marine seismic acquisitions are well known, and can be considered a solved problem. Modern acquisition practice uses very fine receiver group spacing and dense coverage, and spatial aliasing issues can practically be neglected. However, with the advent of wide-azimuth marine seismic acquisitions (Mitchell et al., 2006, Threadgold et al., 2006), and their rapidly increasing popularity, the classical spatial sampling considerations once again become pertinent. This paper investigates the effects of, primarily, coarse sampling in crossline receivers, in addition to the related issue of coarse shot spacing. Both of these are typical characteristics of wide-azimuth marine acquisitions. We also assess the role of data preprocessing in alleviating theproblems created by these spatial sampling issues. Introduction Much of the design of wide-azimuth towed streamer (WATS) surveys is dictated by acquisition cost, particularly in an exploration context. The fundamental parametersgoverning spatial sampling density are: the number of vessels to be used; the number of streamers that each receiver vessel can tow; the maximum width of spread that can be towed by each vessel; the minimum safe distances between streamers of the same group and between vessels; and the time and budget available to cover a given area. In order to achieve wide-azimuth coverage with towed streamers, it is necessary to make multiple passes with the source vessel(s), with differing relative positions of the receivers. These multiple shot gathers can then be combined in preprocessing to yield supershot gathers, assuming that the physical shot locations fall within a certain tolerated distance of each other (which is, indeed, usually the case thanks to the accuracy of modern navigation). The shot, or supershot, domain is a natural one in which to perform typical preprocessing tasks, and is also required for wave equation migration techniques, whether those making use of one-way propagators in the frequency domain (commonly called ''Shot profile wave equation migration''), or ''two-way'' 3D finite difference modeling (commonly called ''Reverse time migration''). When analyzing the sampling issues, we should consider both those within a single supershot, in addition to the related issue of the sampling density of the supershot locations themselves. The link between these two is not only theoretical-in practice, there is a trade-off between the two. For example, we may densify the crossline receiver spacing by doubling the number of passes the vessel makes, but at the expense of only acquiring half the number of supershots per unit area, or per unit of acquisition time. Wide-azimuth acquisition feasibility studies are often carried out by performing dense 3D finite difference modeling (e.g. Regone, 2006), in order to allow tests to be carried out on candidate acquisition geometries. Due to the cost of 3D finite difference modeling, the maximum frequency of such synthetic data is limited, and many of the aliasing issues are therefore neglected. However, in real data, particularly as we seek to use wide-azimuth seismic as a tool for development, as well as exploration, the frequencies we seek to migrate will increase, along with the aliasing problems they entail.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2007 SEG Annual Meeting, September 23–28, 2007

Paper Number: SEG-2007-2240

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**frequency**and**wavenumber**dependent, phase shift. The Gabor transform is implemented as a windowed discrete Fourier transform where the windows are confined to form a partition of unity (POU), meaning that they sum to one. For efficiency, an adaptive partitioning scheme that relates window width...
Abstract

SUMMARY We present a new seismic depth migration algorithm using the Gabor transform, also termed as the windowed Fourier transform, over the lateral spatial coordinates and the discrete Fourier transform over time. These transforms enable a wavefield depth extrapolation by laterally variable, frequency and wavenumber dependent, phase shift. The Gabor transform is implemented as a windowed discrete Fourier transform where the windows are confined to form a partition of unity (POU), meaning that they sum to one. For efficiency, an adaptive partitioning scheme that relates window width to the lateral velocity variation is developed, and defines an adaptive Gabor imaging scheme. Within each window, the Gabor method uses the familiar split-step Fourier technique. The construction of the adaptive partition of unity is guided by an accuracy threshold that constrains the spatial positioning error, for each depth step. The spatial positioning error is estimated by comparing the Gabor method to a nonstationary phase shift that changes according to the local velocity at each position. We present the details of building the adaptive POU for both 2D and 3D imaging. The performance of Gabor depth imaging using this partitioning algorithm is illustrated with imaging results from prestack depth migration of the Marmousi dataset. INTRODUCTION Seismic migration by phase shift (also referred to as wavefield extrapolation with phase shift) was proposed as an accurate and efficient method that is considered to be unconditionally stable provided that there are no lateral velocity variations. An approximate extension to lateral velocity variations, called PSPI (phase-shift-plus-interpolation) was presented by Gazdag and Sguazzero (1984), who proposed spatial interpolation between a set of constant velocity reference extrapolated wavefields. This method has proven popular and, although it is no longer unconditionally stable (e.g., Etgen, 1994), it is more stable than a typical explicit space-frequency method (Margrave et al., 2006). Other ways of extending the concept of a phase-shift include the split-step Fourier method (Stoffa et al., 1990), the various phase-screen methods (Wu and Huang, 1992; Roberts et al., 1997; Rousseau and de Hoop, 2001; Jin et al., 2002), the generalized phase-shift-plus-interpolation (GPSPI) and nonstationary phase-shift methods (NSPS) (Margrave and Ferguson, 1999), and the Gabor method (Grossman et al., 2002; Ma and Margrave, 2005). Margrave and Ferguson (1999) showed that the GPSPI Fourier integral is the limit of PSPI in the extreme case of using a distinct reference velocity for each output location. Writing outside the typical seismic literature, Fishman and McCoy (1985) derived the GPSPI formula as a high frequency approximation to the exact wavefield extrapolator for a laterally variable medium. They refer to the GPSPI formula as the locally homogeneous approximation (LHA) and we adopt that nomenclature. While the LHA is called a high frequency approximation it is still much more accurate than raytracing because the approximation is done at a different place in the theoretical development (see Fishman and McCoy for more discussion). In fact, it can be shown that virtually all explicit depth-stepping methods in practice today, including those mentioned above, are approximations to the LHA formula.

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