Systematics of time‐migration errors

Geophysics ◽  
1994 ◽  
Vol 59 (9) ◽  
pp. 1419-1434 ◽  
Author(s):  
James L. Black ◽  
Matthew A. Brzostowski
Keyword(s):  
General Scheme ◽  
Velocity Variation ◽  
Lateral Velocity ◽  
Depth Migration ◽  
Time Migration ◽  
Simple Rules ◽  
Rules Of Thumb ◽  
Velocity Changes ◽  
Dip Angle ◽  
Depth Of Burial

Even if the correct velocity is used, time migration mispositions events whenever the velocity changes laterally. These errors increase with lateral velocity variation, depth of burial, and dip angle θ. Our analyses of two model types, one with an implicit gradient and one with an explicit gradient, yield simple “rules of thumb” for these errors to first order in the lateral gradient. The x error is [Formula: see text], and the z error is [Formula: see text], where the quantity A = A(x, z) contains the information about depth of burial and magnitude of lateral gradient. These rules can be used to determine when depth migration is needed. Further analysis also shows that the image‐ray correction to time migration is accurate only at small dip. For dipping events, the image‐ray correction must be supplemented by a shift in x of the form [Formula: see text] and a shift in z given by [Formula: see text]. These time‐migration corrections take the same form for both the models we have studied, suggesting a general scheme for correcting time migration, which we call “remedial migration.”

Evaluation of two‐pass three‐dimensional migration

Geophysics ◽  
1988 ◽  
Vol 53 (1) ◽  
pp. 32-49 ◽  
Author(s):  
John A. Dickinson
Keyword(s):  
Seismic Data ◽  
Field Data ◽  
Three Dimensional ◽  
Velocity Variation ◽  
Lateral Velocity ◽  
Data Manipulation ◽  
Depth Migration ◽  
Data Comparison ◽  
Time Migration ◽  
Order Of Magnitude

The theoretically correct way to perform a three‐dimensional (3-D) migration of seismic data requires large amounts of data manipulation on the computer. In order to alleviate this problem, a true, one‐pass 3-D migration is commonly replaced with an approximate technique in which a series of two‐dimensional (2-D) migrations is performed in orthogonal directions. This two‐pass algorithm produces the correct answer when the velocity is constant, both horizontally and vertically. Here I analyze the error due to this algorithm when the velocities vary vertically. The analysis has two parts: first, a theoretical analysis is performed in which a formula for the error is derived; and second, a field data comparison between one‐pass and two‐pass migrations is shown. My conclusion is that two‐pass 3-D migration is, in general, a very good approximation. Its errors are usually small, the exceptions being when both the reflector dip is large (in practice this typically means greater than about 25 to 40 degrees) and the orientation of the reflector is in neither the inline nor the crossline direction. Even then the error is the same order of magnitude as that due to the uncertainty in the migration velocities. These conclusions are still valid when there is lateral velocity variation, as long as this variation is accounted for by trace stretching. The analysis presented here deals with time migration; no claims are made regarding depth migration.

Separation and imaging of seismic diffractions using migrated dip-angle gathers

Geophysics ◽  
2012 ◽  
Vol 77 (6) ◽  
pp. S131-S143 ◽  
Author(s):  
Alexander Klokov ◽  
Sergey Fomel
Keyword(s):  
Radon Transform ◽  
Real Data ◽  
Separation Procedure ◽  
Shape Difference ◽  
Depth Migration ◽  
Time Migration ◽  
Reflection Angle ◽  
Radon Space ◽  
Dip Angle ◽  
Analytical Expressions

Common-reflection angle migration can produce migrated gathers either in the scattering-angle domain or in the dip-angle domain. The latter reveals a clear distinction between reflection and diffraction events. We derived analytical expressions for events in the dip-angle domain and found that the shape difference can be used for reflection/diffraction separation. We defined reflection and diffraction models in the Radon space. The Radon transform allowed us to isolate diffractions from reflections and noise. The separation procedure can be performed after either time migration or depth migration. Synthetic and real data examples confirmed the validity of this technique.

Velocity estimation in laterally varying media

Geophysics ◽  
1982 ◽  
Vol 47 (6) ◽  
pp. 884-897 ◽  
Author(s):  
Walter S. Lynn ◽  
Jon F. Claerbout
Keyword(s):  
Taylor Series Expansion ◽  
Velocity Estimation ◽  
Lateral Resolution ◽  
Velocity Variation ◽  
Lateral Velocity ◽  
Depth Migration ◽  
Derivative Term ◽  
Fourth Derivative ◽  
Zero Offset ◽  
Lateral Variations

In areas of large lateral variations in velocity, stacking velocities computed on the basis of hyperbolic moveout can differ substantially from the actual root mean square (rms) velocities. This paper addresses the problem of obtaining rms or migration velocities from stacking velocities in such areas. The first‐order difference between the stacking and the vertical rms velocities due to lateral variations in velocity are shown to be related to the second lateral derivative of the rms slowness [Formula: see text]. Approximations leading to this relation are straight raypaths and that the vertical rms slowness to a given interface can be expressed as a second‐order Taylor series expansion in the midpoint direction. Under these approximations, the effect of the first lateral derivative of the slowness on the traveltime is negligible. The linearization of the equation relating the stacking and true velocities results in a set of equations whose inversion is unstable. Stability is achieved, however, by adding a nonphysical fourth derivative term which affects only the higher spatial wavenumbers, those beyond the lateral resolution of the lateral derivative method (LDM). Thus, given the stacking velocities and the zero‐offset traveltime to a given event as a function of midpoint, the LDM provides an estimate of the true vertical rms velocity to that event with a lateral resolution of about two mute zones or cable lengths. The LDM is applicable when lateral variations of velocity greater than 2 percent occur over the mute zone. At variations of 30 percent or greater, the internal assumptions of the LDM begin to break down. Synthetic models designed to test the LDM when the different assumptions are violated show that, in all cases, the results are not seriously affected. A test of the LDM on field data having a lateral velocity variation caused by sea floor topography gives a result which is supported by depth migration.

Plumes: Response of time migration to lateral velocity variation

Geophysics ◽  
1995 ◽  
Vol 60 (4) ◽  
pp. 1118-1127 ◽  
Author(s):  
Dimitri Bevc ◽  
James L. Black ◽  
Gopal Palacharla
Keyword(s):  
Impulse Response ◽  
Synthetic Data ◽  
Migration Velocity ◽  
Velocity Variation ◽  
Velocity Dependence ◽  
Geometrical Construction ◽  
Lateral Velocity ◽  
Time Migration ◽  
Offset Time ◽  
Zero Offset

We analyze how time migration mispositions events in the presence of lateral velocity variation by examining the impulse response of depth modeling followed by time migration. By examining this impulse response, we lay the groundwork for the development of a remedial migration operator that links time and depth migration. A simple theory by Black and Brzostowski predicted that the response of zero‐offset time migration to a point diffractor in a v(x, z) medium would be a distinctive, cusp‐shaped curve called a plume. We have constructed these plumes by migrating synthetic data using several time‐migration methods. We have also computed the shape of the plumes by two geometrical construction methods. These two geometrical methods compare well and explain the observed migration results. The plume response is strongly influenced by migration velocity. We have studied this dependency by migrating synthetic data with different velocities. The observed velocity dependence is confirmed by geometrical construction. A simple first‐order theory qualitatively explains the behavior of zero‐offset time migration, but a more complete understanding of migration velocity dependence in a v(x, z) medium requires a higher order finite‐offset theory.

Enhancements to prestack frequency‐wavenumber (f-k) migration

Geophysics ◽  
1991 ◽  
Vol 56 (1) ◽  
pp. 27-40 ◽  
Author(s):  
Z. Li ◽  
W. Lynn ◽  
R. Chambers ◽  
Ken Larner ◽  
Ray Abma
Keyword(s):  
Computational Effort ◽  
Velocity Variation ◽  
Lateral Velocity ◽  
Time Migration ◽  
Migration Velocity Analysis ◽  
Prestack Time Migration ◽  
And Migration ◽  
Definition Of ◽  
Ray Bending ◽  
Excellent Source

Prestack frequency‐wavenumber (f-k) migration is a particularly efficient method of doing both full prestack time migration and migration velocity analysis. Conventional implementations of the method, however, can encounter several drawbacks: (1) poor resolution and spatial aliasing noise caused by insufficient sampling in the offset dimension, (2) poor definition of steep events caused by insufficient sampling in the velocity dimension, and (3) inadequate handling of ray bending for steep events. All three of these problems can be mitigated with modifications to the prestack f-k algorithm. The application of linear moveout (LMO) in the offset dimension prior to migration reduces event moveout and hence increases the bandwidth of non‐spatially aliased signals. To reduce problems of interpolation for steep events, the number of constant‐velocity migrations can be economically increased by performing residual poststack migrations. Finally, migration with a dip‐dependent imaging velocity addresses the issue of ray bending and thereby improves the positioning of steep events. None of these enhancements substantially increases the computational effort of f-k migration. Prestack f-k migration possesses a limitation for which no solution is readily available. Where lateral velocity variation is modest, steep events (such as fault‐plane reflections in sediments) may not be imaged as well as by other migration approaches. This shortcoming results from the restriction that, in the prestack f-k approach, a single velocity field must serve to perform two different functions: imaging and stacking. Nevertheless, in areas of strong velocity variation and gentle to moderate dip, the detailed velocity control afforded by the prestack f-k method is an excellent source of geologic information.

Response by Arthur E. Barnes to the reply by the authors

Geophysics ◽  
1995 ◽  
Vol 60 (6) ◽  
pp. 1947-1947 ◽  
Author(s):  
Arthur E. Barnes
Keyword(s):  
Lateral Velocity ◽  
Prestack Depth Migration ◽  
Depth Migration ◽  
Time Migration ◽  
Pulse Distortion ◽  
Velocity Variations ◽  
Zero Offset ◽  
Complex Geology

I appreciate the thoughtful and thorough response given by Tygel et al. They point out that even for a single dipping reflector imaged by a single non‐zero offset raypath, pulse distortion caused by “standard processing” (NM0 correction‐CMP sort‐stack‐time migration) and pulse distortion caused by prestack depth migration are not really the same, because the reflecting point is mispositioned in standard processing. Within a CMP gather, this mispositioning increases with offset, giving rise to “CMP smear.” CMP smear degrades the stack, introducing additional pulse distortion. Where i‐t is significant, and where lateral velocity variations or reflection curvature are large, such as for complex geology, the pulse distortion of standard processing can differ greatly from that of prestack depth migration.

Reverse time migration from topography

Geophysics ◽  
2014 ◽  
Vol 79 (4) ◽  
pp. S141-S152 ◽  
Author(s):  
Jeffrey Shragge
Keyword(s):  
Wave Propagation ◽  
Acoustic Wave ◽  
Data Acquisition ◽  
Velocity Variation ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Acoustic Wave Propagation ◽  
Lateral Velocity ◽  
Near Surface ◽  
Time Migration

Migration of seismic data from topography using methods based on finite-difference (FD) approximation to acoustic wave propagation commonly suffers from a number of imaging drawbacks due to the difficulty of applying FD stencils to irregular computational meshes. Altering the computational geometry from Cartesian to a topographic coordinate system conformal to the data acquisition surface can circumvent many of these issues. The coordinate transformation approach allows for acoustic wave propagation and the crosscorrelation and inverse-scattering imaging conditions to be posed and computed directly in topographic coordinates. Resulting reverse time migration (RTM) images may then be interpolated back to the Cartesian domain using the known inverse mapping. Orthogonal 2D topographic coordinates can be developed using known conformal mapping transforms and serve as the computational mesh for performing migration from topography. Impulse response tests demonstrate the accuracy of the 2D generalized acoustic wave propagation. RTM imaging examples show the efficacy of performing migration from topography directly from the data acquisition surface on topographic meshes and the ability to image complex near-surface structure even in the presence of strong lateral velocity variation.

Efficient migration through complex water‐bottom topography

Geophysics ◽  
1993 ◽  
Vol 58 (3) ◽  
pp. 393-398 ◽  
Author(s):  
Walt Lynn ◽  
Scott MacKay ◽  
Craig J. Beasley
Keyword(s):  
Bottom Topography ◽  
Migration Velocity ◽  
The Other ◽  
Lateral Velocity ◽  
Thin Lens ◽  
Simple Modification ◽  
Depth Migration ◽  
Time Migration ◽  
Sediment Interface ◽  
Water Bottom

An efficient means of imaging structures beneath complex water‐bottom topography is obtained using a conventional time‐migration algorithm with a simple modification to the migration‐velocity field. The process consists of two migration steps: one with the migration velocity set to zero below the water bottom and the other with the migration velocity set to zero above the water bottom. Between the two steps the data are vertically time shifted to account for the lateral velocity variations between the water‐sediment interface. The time shifts used are equivalent to the so‐called “thin‐lens” term used in depth‐migration algorithms. Efficiency is obtained by applying the thin‐lens term only once and by using computationally optimized time‐migration algorithms. Results obtained from this technique are nearly identical to more costly wave‐equation, layer‐replacement, and depth‐migration techniques.

Filtering coherent noise during prestack depth migration

Geophysics ◽  
1999 ◽  
Vol 64 (4) ◽  
pp. 1054-1066 ◽  
Author(s):  
Bertrand Duquet ◽  
Kurt J. Marfurt
Keyword(s):  
Least Squares ◽  
Oil And Gas ◽  
P Wave ◽  
Velocity Variation ◽  
Coherent Noise ◽  
Lateral Velocity ◽  
Converted Wave ◽  
Depth Migration ◽  
Short Period ◽  
Head Waves

We can often suppress short‐period multiples by predictive deconvolution. We can often suppress coherent noise with significantly different moveout by time‐invariant dip filtering on common‐shot, common‐receiver or NMO-corrected common‐midpoint gathers. Unfortunately, even time variant dip filtering on NMO-corrected data breaks down in the presence of strong lateral velocity variation where the underlying NMO correction breaks down. Underattenuated multiples, converted waves, and diffracted head waves can significantly impede and/or degrade prestack migration‐driven velocity analysis and amplitude variation with offset analysis as well as the quality of the final stacked image. Generalization of time‐variant dip filtering based on conventional NMO corrections of common‐midpoint gathers also breaks down for less conventional data processing situations where we wish to enhance data having nonhyperbolic moveout, such as converted wave energy or long‐offset P-wave reflections in structurally deformed anisotropic media. We present a methodology that defines a depth‐variant velocity filter based on an approximation to the true velocity/depth structure of the earth developed by the interpreter/processor during the normal course of their prestack imaging work flow. Velocity filtering in the depth domain requires the design and calibration of two new least‐squares transforms: a constrained least‐squares common offset Kirchhoff depth migration transform and a transform in residual migration‐velocity moveout space. Each of these new least‐squares transforms can be considered to be generalizations of the well‐known discrete Radon transform commonly used in the oil and gas exploration industry.

scholarly journals Wave-equation time migration

Geophysics ◽  
2020 ◽  
pp. 1-58
Author(s):  
Sergey Fomel ◽  
Harpreet Kaur
Keyword(s):  
Wave Equation ◽  
Time Domain ◽  
Field Data ◽  
Model Building ◽  
Lateral Velocity ◽  
Depth Migration ◽  
Time Migration ◽  
Expensive Model ◽  
Approximate Equations ◽  
Ray Coordinates

Time migration, as opposed to depth migration, suffers from two well-known shortcomings: (1)approximate equations are used for computing Green’s functions inside the imaging operator; (2) in case of lateral velocity variations, the transformation between the image ray coordinates andthe Cartesian coordinates is undefined in places where the image rays cross. We show that thefirst limitation can be removed entirely by formulating time migration through wave propagationin image-ray coordinates. The proposed approach constructs a time-migrated image without relyingon any kind of traveltime approximation by formulating an appropriate geometrically accurateacoustic wave equation in the time-migration domain. The advantage of this approach is that thepropagation velocity in image-ray coordinates does not require expensive model building and canbe approximated by quantities that are estimated in conventional time-domain processing. Synthetic and field data examples demonstrate the effectiveness of the proposed approach and show that theproposed imaging workflow leads to a significant uplift in terms of image quality and can bridge thegap between time and depth migrations. The image obtained by the proposed algorithm is correctlyfocused and mapped to depth coordinates it is comparable to the image obtained by depth migration.

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