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.

Wave-equation migration velocity analysis by focusing diffractions and reflections

Geophysics ◽  
2005 ◽  
Vol 70 (3) ◽  
pp. U19-U27 ◽  
Author(s):  
Paul C. Sava ◽  
Biondo Biondi ◽  
John Etgen
Keyword(s):  
Wave Equation ◽  
Migration Velocity ◽  
Velocity Estimation ◽  
Velocity Analysis ◽  
Depth Migration ◽  
Interval Velocity ◽  
Migration Velocity Analysis ◽  
Inversion Procedure ◽  
Velocity Information ◽  
Zero Offset

We propose a method for estimating interval velocity using the kinematic information in defocused diffractions and reflections. We extract velocity information from defocused migrated events by analyzing their residual focusing in physical space (depth and midpoint) using prestack residual migration. The results of this residual-focusing analysis are fed to a linearized inversion procedure that produces interval velocity updates. Our inversion procedure uses a wavefield-continuation operator linking perturbations of interval velocities to perturbations of migrated images, based on the principles of wave-equation migration velocity analysis introduced in recent years. We measure the accuracy of the migration velocity using a diffraction-focusing criterion instead of the criterion of flatness of migrated common-image gathers that is commonly used in migration velocity analysis. This new criterion enables us to extract velocity information from events that would be challenging to use with conventional velocity analysis methods; thus, our method is a powerful complement to those conventional techniques. We demonstrate the effectiveness of the proposed methodology using two examples. In the first example, we estimate interval velocity above a rugose salt top interface by using only the information contained in defocused diffracted and reflected events present in zero-offset data. By comparing the results of full prestack depth migration before and after the velocity updating, we confirm that our analysis of the diffracted events improves the velocity model. In the second example, we estimate the migration velocity function for a 2D, zero-offset, ground-penetrating radar data set. Depth migration after the velocity estimation improves the continuity of reflectors while focusing the diffracted energy.

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.

An analytical approach to migration velocity analysis

Geophysics ◽  
1997 ◽  
Vol 62 (4) ◽  
pp. 1238-1249 ◽  
Author(s):  
Zhenyue Liu
Keyword(s):  
Synthetic Data ◽  
Migration Velocity ◽  
Velocity Estimation ◽  
Velocity Analysis ◽  
Lateral Velocity ◽  
Depth Migration ◽  
Curvature Analysis ◽  
Derivative Function ◽  
Migration Velocity Analysis ◽  
Residual Curvature

Prestack depth migration provides a powerful tool for velocity analysis in complex media. Both prominent approaches to velocity analysis—depth‐focusing analysis and residual‐curvature analysis, rely on approximate formulas to update velocity. Generally, these formulas are derived under the assumptions of horizontal reflector, lateral velocity homogeneity, or small offset. Therefore, the conventional methods for updating velocity lack accuracy and computational efficiency when velocity has large, lateral variations. Here, based on ray theory, I find the analytic representation for the derivative of imaged depths with respect to migration velocity. This derivative function characterizes a general relationship between residual moveout and residual velocity. Using the derivative function and the perturbation method, I derive a new formula to update velocity from residual moveout. In the derivation, I impose no limitation on offset, dip, or velocity distribution. Consequently, I revise the residual‐curvature‐analysis method for velocity estimation in the postmigrated domain. Furthermore, my formula provides sensitivity and error estimation for migration‐based velocity analysis, which is helpful in quantifying the reliability of the estimated velocity. The theory and methodology in this paper have been tested on synthetic data (including the Marmousi data).

Description of a scanless method of stacking velocity analysis

Geophysics ◽  
1993 ◽  
Vol 58 (11) ◽  
pp. 1596-1606 ◽  
Author(s):  
Hans J. Tieman
Keyword(s):  
Empirical Studies ◽  
Real Data ◽  
Velocity Variation ◽  
Velocity Analysis ◽  
Velocity Control ◽  
Lateral Velocity ◽  
Interval Velocity ◽  
Offset Time ◽  
Zero Offset ◽  
Traditional Approaches

Stacking velocities can be directly estimated from seismic data without recourse to a multivelocity stack and subsequent search techniques that many current procedures use. This is done as follows: (1) apply NMO to the data (over a window, for a particular common midpoint) using initial estimates for zero offset time and velocity; (2) produce two stacks by summing the data over offset after applying different weighting functions; (3) cross correlate the two stacks; and (4) translate the lag into velocity and time updates. The procedure is iterated until convergence has occurred. Referred to as ARAMVEL (U.S. Patent No. 4,813,027), the method is best implemented as an interactive continuous velocity analysis. Although very simple, both empirical studies and theoretical analysis have shown that it determines velocities more accurately than more traditional approaches based on a scan approach. Convergence is fast, with only one or two iterations usually necessary. The method is robust, as only approximate information is necessary initially. Results with real data show that the method can economically give the detailed velocity control necessary for processing data from areas with considerable lateral velocity variation, as well as provide traveltime information that can be used for sophisticated inversion into interval velocity and depth.

scholarly journals Fast time-to-depth conversion and interval velocity estimation in the case of weak lateral variations

Geophysics ◽  
2018 ◽  
Vol 83 (3) ◽  
pp. S227-S235 ◽  
Author(s):  
Yanadet Sripanich ◽  
Sergey Fomel
Keyword(s):  
Velocity Model ◽  
Velocity Estimation ◽  
Fast Time ◽  
Lateral Velocity ◽  
First Order ◽  
Interval Velocity ◽  
Order Of Magnitude ◽  
Efficient Alternative ◽  
Homogeneous Media ◽  
Lateral Variations

Time-domain processing has a long history in seismic imaging and has always been a powerful workhorse that is routinely used. It generally leads to an expeditious construction of the subsurface velocity model in time, which can later be expressed in the Cartesian depth coordinates via a subsequent time-to-depth conversion. The conventional practice of such a conversion is done using Dix inversion, which is exact in the case of laterally homogeneous media. For other media with lateral heterogeneity, the time-to-depth conversion involves solving a more complex system of partial differential equations (PDEs). We have developed an efficient alternative for time-to-depth conversion and interval velocity estimation based on the assumption of weak lateral velocity variations. By considering only first-order perturbative effects from lateral variations, the exact system of PDEs required to accomplish the exact conversion reduces to a simpler system that can be solved efficiently in a layer-stripping (downward-stepping) fashion. Numerical synthetic and field data examples show that our method can achieve reasonable accuracy and is significantly more efficient than previously proposed methods with a speedup by an order of magnitude.

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.

3-D moveout inversion in azimuthally anisotropic media with lateral velocity variation: Theory and a case study

Geophysics ◽  
1999 ◽  
Vol 64 (4) ◽  
pp. 1202-1218 ◽  
Author(s):  
Vladimir Grechka ◽  
Ilya Tsvankin
Keyword(s):  
Correction Term ◽  
Anisotropic Media ◽  
Azimuthal Anisotropy ◽  
P Wave ◽  
Velocity Variation ◽  
Variation Theory ◽  
Fractured Reservoir ◽  
Lateral Velocity ◽  
Data Set ◽  
Zero Offset

Reflection moveout recorded over an azimuthally anisotropic medium (e.g., caused by vertical or dipping fractures) varies with the azimuth of the source‐receiver line. Normal‐moveout (NMO) velocity, responsible for the reflection traveltimes on conventional‐length spreads, forms an elliptical curve in the horizontal plane. While this result remains valid in the presence of arbitrary anisotropy and heterogeneity, the inversion of the NMO ellipse for the medium parameters has been discussed so far only for horizontally homogeneous models above a horizontal or dipping reflector. Here, we develop an analytic moveout correction for weak lateral velocity variation in horizontally layered azimuthally anisotropic media. The correction term is proportional to the curvature of the zero‐offset traveltime surface at the common midpoint and, therefore, can be estimated from surface seismic data. After the influence of lateral velocity variation on the effective NMO ellipses has been stripped, the generalized Dix equation can be used to compute the interval ellipses and evaluate the magnitude of azimuthal anisotropy (measured by P-wave NMO velocity) within the layer of interest. This methodology was applied to a 3-D “wide‐azimuth” data set acquired over a fractured reservoir in the Powder River Basin, Wyoming. The processing sequence included 3-D semblance analysis (based on the elliptical NMO equation) for a grid of common‐midpoint “supergathers,” spatial smoothing of the effective NMO ellipses and zero‐offset traveltimes, correction for lateral velocity variation, and generalized Dix differentiation. Our estimates of depth‐varying fracture trends in the survey area, based on the interval P-wave NMO ellipses, are in good agreement with the results of outcrop and borehole measurements and the rotational analysis of four‐ component S-wave data.

Modal expansion of one‐way operators in laterally varying media

Geophysics ◽  
1998 ◽  
Vol 63 (3) ◽  
pp. 995-1005 ◽  
Author(s):  
Joris L. T. Grimbergen ◽  
Frank J. Dessing ◽  
Kees Wapenaar
Keyword(s):  
Orthogonal Basis ◽  
Lateral Resolution ◽  
Acoustic Parameters ◽  
Prestack Depth Migration ◽  
Modal Expansion ◽  
Helmholtz Operator ◽  
Depth Migration ◽  
Depth Direction ◽  
Lateral Variations ◽  
Wave Modes

One of the main benefits of prestack depth migration in seismic processing is its ability to handle complicated medium configurations. When considerable lateral variations in the acoustic parameters are present in the subsurface, prestack depth migration is necessary for optimal lateral resolution. However, most migration algorithms still deal with lateral variations in an approximate manner because these variations are in many cases moderate compared to the profound variations in the depth direction. From other areas of science (e.g., optics, oceanography, and seismology), it is known that lateral variations can be dealt with by a decomposition of the wavefield into wave modes. In this paper, we explore the possibility of applying this concept to the construction of one‐way wavefield operators for depth migration. We expand the Helmholtz operator on an orthogonal basis of wave modes and obtain one‐way wavefield operators that are unconditionally stable and significantly increase the lateral resolution of the result.

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.

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