Explicit expressions for prestack map time migration in isotropic and VTI media and the applicability of map depth migration in heterogeneous anisotropic media

Geophysics ◽  
2006 ◽  
Vol 71 (1) ◽  
pp. S13-S28 ◽  
Author(s):  
Huub Douma ◽  
Maarten V. de Hoop
Keyword(s):  
Closed Form ◽  
Anisotropic Media ◽  
Velocity Model ◽  
Pure Mode ◽  
Anisotropic Parameter ◽  
Depth Migration ◽  
Time Migration ◽  
Isotropic Media ◽  
The Common ◽  
Vti Media

We present 3D prestack map time migration in closed form for qP-, qSV-, and mode-converted waves in homogeneous transversely isotropic media with a vertical symmetry axis (VTI). As far as prestack time demigration is concerned, we present closed-form expressions for mapping in homogeneous isotropic media, while for homogeneous VTI media we present a system of four nonlinear equations with four unknowns to solve numerically. The expressions for prestack map time migration in VTI homogeneous media are directly applicable to the problem of anisotropic parameter estimation (i.e., the anellipticity parameter η) in the context of time-migration velocity analysis. In addition, we present closed-form expressions for both prestack map time migration and demigration in the common-offset domain for pure-mode (P-P or S-S) waves in homogeneous isotropic media that use only the slope in the common-offset domain as opposed to slopes in both the common-shot and common-receiver (or equivalently the common-offset and common-midpoint) domains. All time-migration and demigration equations presented can be used in media with mild lateral and vertical velocity variations, provided the velocity is replaced with the local rms velocity. Finally, we discuss the condition for applicability of prestack map depth migration and demigration in heterogeneous anisotropic media that allows the formation of caustics and explain that this condition is satisfied if, given a velocity model and acquisition geometry, one can map-depth-migrate without ambiguity in either the migrated location or the migrated orientation of reflectors in the image.

Time-migration velocity analysis by image-wave propagation of common-image gathers

Geophysics ◽  
2008 ◽  
Vol 73 (5) ◽  
pp. VE161-VE171 ◽  
Author(s):  
J. Schleicher ◽  
J. C. Costa ◽  
A. Novais
Keyword(s):  
Wave Propagation ◽  
Seismic Event ◽  
Velocity Model ◽  
Migration Velocity ◽  
Velocity Analysis ◽  
Reference Velocity ◽  
Time Migration ◽  
Migration Velocity Analysis ◽  
Velocity Image ◽  
The Common

Image-wave propagation or velocity continuation describes the variation of the migrated position of a seismic event as a function of migration velocity. Image-wave propagation in the common-image gather (CIG) domain can be combined with residual-moveout analysis for iterative migration velocity analysis (MVA). Velocity continuation of CIGs leads to a detection of those velocities in which events flatten. Although image-wave continuation is based on the assumption of a constant migration velocity, the procedure can be applied in inhomogeneous media. For this purpose, CIGs obtained by migration with an inhomogeneous macrovelocity model are continued starting from a constant reference velocity. The interpretation of continued CIGs, as if they were obtained from residual migrations, leads to a correction formula that translates residual flattening velocities into absolute time-migration velocities. In this way, the migration velocity model can be improved iteratively until a satisfactory result is reached. With a numerical example, we found that MVA with iterative image continuation applied exclusively to selected CIGs can construct a reasonable migration velocity model from scratch, without the need to build an initial model from a previous conventional normal-moveout/dip-moveout velocity analysis.

Velocity-estimation improvements and migration/demigration using the common-reflection surface with continuing deconvolution in the time domain

Geophysics ◽  
2019 ◽  
Vol 84 (4) ◽  
pp. S229-S238 ◽  
Author(s):  
Martina Glöckner ◽  
Sergius Dell ◽  
Benjamin Schwarz ◽  
Claudia Vanelle ◽  
Dirk Gajewski
Keyword(s):  
Model Building ◽  
Velocity Model ◽  
Velocity Estimation ◽  
Initial Time ◽  
Model Errors ◽  
Imaging Methods ◽  
Time Migration ◽  
Common Reflection Surface ◽  
The Common ◽  
And Migration

To obtain an image of the earth’s subsurface, time-imaging methods can be applied because they are reasonably fast, are less sensitive to velocity model errors than depth-imaging methods, and are usually easy to parallelize. A powerful tool for time imaging consists of a series of prestack time migrations and demigrations. We have applied multiparameter stacking techniques to obtain an initial time-migration velocity model. The velocity model building proposed here is based on the kinematic wavefield attributes of the common-reflection surface (CRS) method. A subsequent refinement of the velocities uses a coherence filter that is based on a predetermined threshold, followed by an interpolation and smoothing. Then, we perform a migration deconvolution to obtain the final time-migrated image. The migration deconvolution consists of one iteration of least-squares migration with an estimated Hessian. We estimate the Hessian by nonstationary matching filters, i.e., in a data-driven fashion. The model building uses the framework of the CRS, and the migration deconvolution is fully automated. Therefore, minimal user interaction is required to carry out the velocity model refinement and the image update. We apply the velocity refinement and migration deconvolution approaches to complex synthetic and field data.

Depth migration by the Gaussian beam summation method

Geophysics ◽  
2010 ◽  
Vol 75 (2) ◽  
pp. S81-S93 ◽  
Author(s):  
Mikhail M. Popov ◽  
Nikolay M. Semtchenok ◽  
Peter M. Popov ◽  
Arie R. Verdel
Keyword(s):  
Wave Equation ◽  
Model Building ◽  
Velocity Structure ◽  
Velocity Model ◽  
Summation Method ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Depth Migration ◽  
Time Migration ◽  
Ray Methods

Seismic depth migration aims to produce an image of seismic reflection interfaces. Ray methods are suitable for subsurface target-oriented imaging and are less costly compared to two-way wave-equation-based migration, but break down in cases when a complex velocity structure gives rise to the appearance of caustics. Ray methods also have difficulties in correctly handling the different branches of the wavefront that result from wave propagation through a caustic. On the other hand, migration methods based on the two-way wave equation, referred to as reverse-time migration, are known to be capable of dealing with these problems. However, they are very expensive, especially in the 3D case. It can be prohibitive if many iterations are needed, such as for velocity-model building. Our method relies on the calculation of the Green functions for the classical wave equation by per-forming a summation of Gaussian beams for the direct and back-propagated wavefields. The subsurface image is obtained by cal-culating the coherence between the direct and backpropagated wavefields. To a large extent, our method combines the advantages of the high computational speed of ray-based migration with the high accuracy of reverse-time wave-equation migration because it can overcome problems with caustics, handle all arrivals, yield good images of steep flanks, and is readily extendible to target-oriented implementation. We have demonstrated the quality of our method with several state-of-the-art benchmark subsurface models, which have velocity variations up to a high degree of complexity. Our algorithm is especially suited for efficient imaging of selected subsurface subdomains, which is a large advantage particularly for 3D imaging and velocity-model refinement applications such as subsalt velocity-model improvement. Because our method is also capable of providing highly accurate migration results in structurally complex subsurface settings, we have also included the concept of true-amplitude imaging in our migration technique.

Kirchhoff time migration for transversely isotropic media: An application to Trinidad data

Geophysics ◽  
2006 ◽  
Vol 71 (1) ◽  
pp. S29-S35 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Symmetry Axis ◽  
Transversely Isotropic ◽  
Data Set ◽  
Time Migration ◽  
Transversely Isotropic Media ◽  
Higher Order Terms ◽  
Isotropic Media ◽  
Migration Method ◽  
Vti Media ◽  
Vertical Inhomogeneity

Using a newly developed nonhyperbolic offset-mid-point traveltime equation for prestack Kirchhoff time migration, instead of the conventional double-square-root (DSR) equation, results in overall better images from anisotropic data. Specifically, prestack Kirchhoff time migration for transversely isotropic media with a vertical symmetry axis (VTI media) is implemented using an analytical offset-midpoint traveltime equation that represents the equivalent of Cheop's pyramid for VTI media. It includes higher-order terms necessary to better handle anisotropy as well as vertical inhomogeneity. Application of this enhanced Kirchhoff time-migration method to the anisotropic Marmousi data set demonstrates the effectiveness of the approach. Further application of the method to field data from Trinidad results in sharper reflectivity images of the subsurface, with the faults better focused and positioned than with images obtained using isotropic methods. The superiority of the anisotropic time migration is evident in the flatness of the image gathers.

3-D prestack Kirchhoff depth migration: From prototype to production in a massively parallel processor environment

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 546-556 ◽  
Author(s):  
Herman Chang ◽  
John P. VanDyke ◽  
Marcelo Solano ◽  
George A. McMechan ◽  
Duryodhan Epili
Keyword(s):  
Model Building ◽  
Velocity Model ◽  
Production Scale ◽  
Data Set ◽  
Depth Migration ◽  
Prestack Migration ◽  
Prototype Development ◽  
Time Migration ◽  
Intel Paragon ◽  
Kirchhoff Depth Migration

Portable, production‐scale 3-D prestack Kirchhoff depth migration software capable of full‐volume imaging has been successfully implemented and applied to a six‐million trace (46.9 Gbyte) marine data set from a salt/subsalt play in the Gulf of Mexico. Velocity model building and updates use an image‐driven strategy and were performed in a Sun Sparc environment. Images obtained by 3-D prestack migration after three velocity iterations are substantially better focused and reveal drilling targets that were not visible in images obtained from conventional 3-D poststack time migration. Amplitudes are well preserved, so anomalies associated with known reservoirs conform to the petrophysical predictions. Prototype development was on an 8-node Intel iPSC860 computer; the production version was run on an 1824-node Intel Paragon computer. The code has been successfully ported to CRAY (T3D) and Unix workstation (PVM) environments.

Noise removal by migration of time-shift images

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. S105-S111 ◽  
Author(s):  
Sheng Xu ◽  
Feng Chen ◽  
Bing Tang ◽  
Gilles Lambare
Keyword(s):  
Time Lag ◽  
Real Data ◽  
Velocity Model ◽  
Noise Removal ◽  
Time Shift ◽  
Reverse Time ◽  
Coherent Noise ◽  
Data Set ◽  
Depth Migration ◽  
Time Migration

When using seismic data to image complex structures, the reverse time migration (RTM) algorithm generally provides the best results when the velocity model is accurate. With an inexact model, moveouts appear in common image gathers (CIGs), which are either in the surface offset domain or in subsurface angle domain; thus, the stacked image is not well focused. In extended image gathers, the strongest energy of a seismic event may occur at non-zero-lag in time-shift or offset-shift gathers. Based on the operation of RTM images produced by the time-shift imaging condition, the non-zero-lag time-shift images exhibit a spatial shift; we propose an approach to correct them by a second pass of migration similar to zero-offset depth migration; the proposed approach is based on the local poststack depth migration assumption. After the proposed second-pass migration, the time-shift CIGs appear to be flat and can be stacked. The stack enhances the energy of seismic events that are defocused at zero time lag due to the inaccuracy of the model, even though the new focused events stay at the previous positions, which might deviate from the true positions of seismic reflection. With the stack, our proposed approach is also able to attenuate the long-wavelength RTM artifacts. In the case of tilted transverse isotropic migration, we propose a scheme to defocus the coherent noise, such as migration artifacts from residual multiples, by applying the original migration velocity model along the symmetry axis but with different anisotropic parameters in the second pass of migration. We demonstrate that our approach is effective to attenuate the coherent noise at subsalt area with two synthetic data sets and one real data set from the Gulf of Mexico.

Processing of anisotropic data in the τ‐p domain: I—Geometric spreading and moveout corrections

Geophysics ◽  
2004 ◽  
Vol 69 (3) ◽  
pp. 719-730 ◽  
Author(s):  
Mirko van der Baan
Keyword(s):  
Plane Waves ◽  
Vertical Axis ◽  
Pure Mode ◽  
Spherical Waves ◽  
Time Migration ◽  
Head Waves ◽  
Geometric Spreading ◽  
Isotropic Media ◽  
P Domain ◽  
Time Offset

Stacking of seismic data is conventionally done in the time‐offset domain. This has the disadvantage that geometric spreading must be removed before true‐amplitude processing can be attempted. This inconvenience arises since wave motion in the time‐offset domain is determined by spherical waves. Plane waves in layered media, on the other hand, are not subject to geometric spreading. Hence, processing of both isotropic and anisotropic data in such media benefits from first applying a plane‐wave decomposition such as a proper τ‐p transform. The resulting τ‐p gathers can be flattened and stacked over slowness. Subsequent time differentiation is needed to counter the loss of high frequencies during stacking. This approach has the advantage that the geometric spreading is removed without prior knowledge of the actual (an)isotropic velocity field and without any need to pick traveltimes or moveout velocities. Subsequent moveout corrections naturally require knowledge of the velocityfield. The proposed methodology is exact for 3D data volumes and arbitrary anisotropy in laterally homogeneous media or for 2D acquisition lines over 1D, isotropic media or over 1D, transversely isotropic media with vertical axis of symmetry (VTI). It relies on the same principles as more conventional geometric spreading corrections and time‐offset stacking. In many respects, it is even more flexible. For instance, geometric spreading has been correctly removed for all present wave modes and types simultaneously (primary, multiple, pure‐mode, and converted waves), and nonhyperbolic moveout resulting from isotropic layering is also taken into account. In addition, head waves may now contribute constructively to the stacked section. Moreover, both multiple elimination and predictive deconvolution are straightforward and known to yield very good results in the τ‐p domain. The resulting stacked section can then be used for any poststack processing such as time migration.

Migration of seismic data from inhomogeneous media

Geophysics ◽  
1981 ◽  
Vol 46 (5) ◽  
pp. 751-767 ◽  
Author(s):  
Les Hatton ◽  
Ken Larner ◽  
Bruce S. Gibson
Keyword(s):  
Theoretical Approach ◽  
Synthetic Data ◽  
Velocity Model ◽  
Implicit Assumption ◽  
Geologic Structure ◽  
Structural Shape ◽  
Depth Migration ◽  
Position Errors ◽  
Time Migration ◽  
Imaging Results

Because conventional time‐migration algorithms are founded on the implicit assumption of locally lateral homogeneity, they leave events mispositioned when overburden velocity varies laterally. The ray‐theoretical depth migration procedure of Hubral often can provide adequate first‐order corrections for such position errors. Complex geologic structure, however, can so severely distort wavefronts that resulting time‐migrated sections may be barely interpretable and thus not readily correctable. A more accurate, wave‐theoretical approach to depth migration then becomes essential to image the subsurface properly. This approach, which transforms an unmigrated time section directly into migrated depth, more completely honors the wave equation for a medium in which variations in interval velocity and details of structural shape govern wave propagation. Where geologic structure is complicated, however, we usually lack an accurate velocity model. It is important, therefore, to understand the sensitivity of depth migration to velocity errors and, in particular, to assess whether it is justified to go to the added effort of doing depth migration. We show a synthetic data example in which the wave‐theoretical approach to depth migration properly images deep reflections that are poorly resolved and left distorted by either time migration or ray‐theoretical depth migration. These imaging results are, moreover, surprisingly insensitive to errors introduced into the velocity model. Application to one field data example demonstrates the superior treatment of amplitude and waveform by wave‐theoretical depth migration. In a second data example, deep reflections are so influenced by anomalous overburden structure that the only valid alternative to performing wave‐theoretical depth migration is simply to convert the unmigrated data to depth. When the overburden is laterally variable, conventional time migration of unstacked data can be as destructive to steeply dipping reflections as is CDP stacking prior to migration. A schematic example illustrates that when migration of unstacked data is judged necessary, it should normally be performed as a depth migration.

Prestack time migration by common-migrated-reflector-element stacking

Geophysics ◽  
2012 ◽  
Vol 77 (3) ◽  
pp. S73-S82 ◽  
Author(s):  
Sergius Dell ◽  
Dirk Gajewski ◽  
Claudia Vanelle
Keyword(s):  
Initial Time ◽  
Depth Migration ◽  
Time Migration ◽  
Migration Method ◽  
Prestack Time Migration ◽  
The Common ◽  
A New Technique ◽  
New Formulation ◽  
Time Image ◽  
Migration Velocities

Time migration is an attractive tool to produce a subsurface image because it is faster and less sensitive to velocities errors than depth migration. However, a highly focused time image is only achievable with well-determined time-migration velocities. Therefore, a refinement of the initial time-migration velocities often is required. We introduced a new technique for prestack time migration, based on the common-migrated-reflector-element stack of common scatterpoint gathers, including an automatic update of time-migration velocities. The common scatterpoint gathers are generated using a new formulation of the double-square-root equation that is parametrized with the common-offset apex time. The common-migrated-reflector-element stack is a multiparameter stacking technique based on the Taylor expansion of traveltimes of time-migrated reflections in the paraxial vicinity of the image ray. Our 2D synthetic and field data examples demonstrated that the proposed method provides updated time-migration velocities that are more robust and have higher resolution compared with the initial time-migration velocities. The prestack time migration method also showed a clear improvement of the focusing of reflections for such geologic features as faults and salt structures.

scholarly journals The offset‐midpoint traveltime pyramid in transversely isotropic media

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1316-1325 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Anisotropy Parameter ◽  
Transversely Isotropic ◽  
Fourth Power ◽  
Weak Anisotropy ◽  
Time Migration ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Equivalent Equation ◽  
Two Parameters ◽  
Vti Media

Prestack Kirchhoff time migration for transversely isotropic media with a vertical symmetry axis (VTI media) is implemented using an offset‐midpoint traveltime equation, Cheop’s pyramid equivalent equation for VTI media. The derivation of such an equation for VTI media requires approximations that pertain to high frequency and weak anisotropy. Yet the resultant offset‐midpoint traveltime equation for VTI media is highly accurate for even strong anisotropy. It is also strictly dependent on two parameters: NMO velocity and the anisotropy parameter, η. It reduces to the exact offset‐midpoint traveltime equation for isotropic media when η = 0. In vertically inhomogeneous media, the NMO velocity and η parameters in the offset‐midpoint traveltime equation are replaced by their effective values: the velocity is replaced by the rms velocity and η is given by a more complicated equation that includes summation of the fourth power of velocity.

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