Extended isochron rays in prestack depth (map) migration

Geophysics ◽  
2010 ◽  
Vol 75 (4) ◽  
pp. S139-S150 ◽  
Author(s):  
Anton A. Duchkov ◽  
Maarten V. de Hoop
Keyword(s):  
Evolution Equation ◽  
Evolution Equations ◽  
Dimensional Space ◽  
Depth Map ◽  
Constructive Proof ◽  
Natural Manner ◽  
Depth Migration ◽  
Surface Data ◽  
Map Migration ◽  
Zero Offset

Many processes in seismic data analysis and seismic imaging can be identified with solution operators of evolution equations. These include data downward continuation and velocity continuation. We have addressed the question of whether isochrons defined by imaging operators can be identified with wavefronts of solutions to an evolution equation. Rays associated with this equation then would provide a natural way of implementing prestack map migration. Assuming absence of caustics, we have developed constructive proof of the existence of a Hamiltonian describing propagation of isochrons in the context of common-offset depth migration. In the presence of caustics, one should recast to a sinking-survey migration framework. By manipulating the double-square-root operator, we obtain an evolution equation that describes sinking-survey migration as a propagation in two-way time with surface data being a source function. This formulation can be viewed as an extension of the exploding reflector concept from zero-offset to sinking-survey migration. The corresponding Hamiltonian describes propagation of extended isochrons (fronts with constant two-way time) connected by extended isochron rays. The term extended reflects the fact that two-way time propagation now takes place in high-dimensional space with the following coordinates: subsurface midpoint, subsurface offset, and depth. Extended isochron rays can be used in a natural manner for implementing sinking-survey migration in a map-migration fashion.

Solutions of evolution equations associated to infinite-dimensional Laplacian

2016 ◽  
Vol 14 (04) ◽  
pp. 1640018 ◽  
Author(s):  
Habib Ouerdiane
Keyword(s):  
Evolution Equation ◽  
Evolution Equations ◽  
Dimensional Space ◽  
Generalized Function ◽  
Initial Condition ◽  
Distribution Space ◽  
One Dimensional ◽  
Infinite Dimensional ◽  
Main Technique ◽  
The One

We study an evolution equation associated with the integer power of the Gross Laplacian [Formula: see text] and a potential function V on an infinite-dimensional space. The initial condition is a generalized function. The main technique we use is the representation of the Gross Laplacian as a convolution operator. This representation enables us to apply the convolution calculus on a suitable distribution space to obtain the explicit solution of the perturbed evolution equation. Our results generalize those previously obtained by Hochberg [K. J. Hochberg, Ann. Probab. 6 (1978) 433.] in the one-dimensional case with [Formula: see text], as well as by Barhoumi–Kuo–Ouerdiane for the case [Formula: see text] (See Ref. [A. Barhoumi, H. H. Kuo and H. Ouerdiane, Soochow J. Math. 32 (2006) 113.]).

Finite‐element prestack reverse‐time migration for elastic waves

Geophysics ◽  
1989 ◽  
Vol 54 (9) ◽  
pp. 1204-1208 ◽  
Author(s):  
Yu‐chiung Teng ◽  
Ting‐fang Dai
Keyword(s):  
Acoustic Waves ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Initial Value ◽  
Depth Migration ◽  
Time Migration ◽  
Surface Data ◽  
Zero Offset ◽  
Dip Angle ◽  
Well Posed

Reverse‐time migration of zero‐offset data for acoustic waves has been successfully implemented by Whitmore (1983), Baysal et al. (1983), McMechan (1983), and Loewenthal and Mufti (1983). In reverse‐time migration, data recorded on the surface are used as the boundary condition and are extrapolated backward in time (Whitmore, 1983; Levin, 1984). Reverse‐time migration is mathematically a well‐posed problem. This is in contrast to conventional depth‐extrapolation‐migration schemes, in which the surface data are initial‐value conditions for solving the wave equation. Reverse‐time migration may offer improvements over conventional depth migration due to its freedom from dip‐angle limitations.

Solvability of the abstract evolution equations in L s -spaces with critical temporal weights

2021 ◽  
Vol 19 (1) ◽  
pp. 111-120
Author(s):  
Qinghua Zhang ◽  
Zhizhong Tan
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Evolution Equation ◽  
Nonlinear Operator ◽  
Evolution Equations ◽  
Bounded Function ◽  
Inhomogeneous Term ◽  
Linear Evolution ◽  
Abstract Evolution Equations ◽  
Linear Evolution Equation

Abstract This paper deals with the abstract evolution equations in L s {L}^{s} -spaces with critical temporal weights. First, embedding and interpolation properties of the critical L s {L}^{s} -spaces with different exponents s s are investigated, then solvability of the linear evolution equation, attached to which the inhomogeneous term f f and its average Φ f \Phi f both lie in an L 1 / s s {L}_{1\hspace{-0.08em}\text{/}\hspace{-0.08em}s}^{s} -space, is established. Based on these results, Cauchy problem of the semi-linear evolution equation is treated, where the nonlinear operator F ( t , u ) F\left(t,u) has a growth number ρ ≥ s + 1 \rho \ge s+1 , and its asymptotic behavior acts like α ( t ) / t \alpha \left(t)\hspace{-0.1em}\text{/}\hspace{-0.1em}t as t → 0 t\to 0 for some bounded function α ( t ) \alpha \left(t) like ( − log t ) − p {\left(-\log t)}^{-p} with 2 ≤ p < ∞ 2\le p\lt \infty .

Leading-order seismic imaging using curvelets

Geophysics ◽  
2007 ◽  
Vol 72 (6) ◽  
pp. S231-S248 ◽  
Author(s):  
Huub Douma ◽  
Maarten V. de Hoop
Keyword(s):  
Seismic Data ◽  
Order Approximation ◽  
Angular Frequency ◽  
Leading Order ◽  
Numerical Examples ◽  
Depth Migration ◽  
Map Migration ◽  
Kirchhoff Depth Migration ◽  
Homogeneous Media ◽  
Spatial Support

Curvelets are plausible candidates for simultaneous compression of seismic data, their images, and the imaging operator itself. We show that with curvelets, the leading-order approximation (in angular frequency, horizontal wavenumber, and migrated location) to common-offset (CO) Kirchhoff depth migration becomes a simple transformation of coordinates of curvelets in the data, combined with amplitude scaling. This transformation is calculated using map migration, which employs the local slopes from the curvelet decomposition of the data. Because the data can be compressed using curvelets, the transformation needs to be calculated for relatively few curvelets only. Numerical examples for homogeneous media show that using the leading-order approximation only provides a good approximation to CO migration for moderate propagation times. As the traveltime increases and rays diverge beyond the spatial support of a curvelet; however, the leading-order approximation is no longer accurate enough. This shows the need for correction beyond leading order, even for homogeneous media.

PROBABILITY DENSITY EVOLUTION EQUATIONS — A HISTORICAL INVESTIGATION

2009 ◽  
Vol 03 (03) ◽  
pp. 209-226 ◽  
Author(s):  
LI JIE ◽  
CHEN JIANBING
Keyword(s):  
Evolution Equation ◽  
Probability Density ◽  
Evolution Equations ◽  
Planck Equation ◽  
Point Of View ◽  
Lagrangian Description ◽  
Density Evolution ◽  
Physical Point ◽  
Probability Density Evolution ◽  
Generalized Probability

The paper aims at clarifying the essential relationship between traditional probability density evolution equations and the generalized probability density evolution equation which is developed by the authors in recent years. Using the principle of preservation of probability as a uniform fundamental, the probability density evolution equations, including the Liouville equation, Fokker–Planck equation and the Dostupov–Pugachev equation, are derived from the physical point of view. It is pointed out that combining with Eulerian or Lagrangian description of the associated dynamical system will lead to different probability density evolution equations. Particularly, when both the principle and dynamical systems are viewed from Lagrangian description, we are led to the generalized probability density evolution equation.

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.

scholarly journals Asymptotic properties of mild solutions of nonautonomous evolution equations with applications to retarded differential equations

1999 ◽  
Vol 4 (3) ◽  
pp. 169-194 ◽  
Author(s):  
Gabriele Gühring ◽  
Frank Räbiger
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Evolution Equation ◽  
Unique Solution ◽  
Evolution Equations ◽  
Bounded Solution ◽  
Mild Solutions ◽  
Asymptotic Properties ◽  
Yosida Operator ◽  
Retarded Differential Equations

We investigate the asymptotic properties of the inhomogeneous nonautonomous evolution equation(d/dt)u(t)=Au(t)+B(t)u(t)+f(t),t∈ℝ, where(A,D(A))is a Hille-Yosida operator on a Banach spaceX,B(t),t∈ℝ, is a family of operators inℒ(D(A)¯,X)satisfying certain boundedness and measurability conditions andf∈L loc 1(ℝ,X). The solutions of the corresponding homogeneous equations are represented by an evolution family(UB(t,s))t≥s. For various function spacesℱwe show conditions on(UB(t,s))t≥sandfwhich ensure the existence of a unique solution contained inℱ. In particular, if(UB(t,s))t≥sisp-periodic there exists a unique bounded solutionusubject to certain spectral assumptions onUB(p,0),fandu. We apply the results to nonautonomous semilinear retarded differential equations. For certainp-periodic retarded differential equations we derive a characteristic equation which is used to determine the spectrum of(UB(t,s))t≥s.

The isochron ray in seismic modeling and imaging

Geophysics ◽  
2004 ◽  
Vol 69 (4) ◽  
pp. 1053-1070 ◽  
Author(s):  
Einar Iversen
Keyword(s):  
Seismic Imaging ◽  
Model Updating ◽  
Velocity Model ◽  
Seismic Modeling ◽  
Prestack Depth Migration ◽  
Depth Migration ◽  
Geometric Spreading ◽  
Model Independent ◽  
Map Migration ◽  
Upper Level

The isochron, the name given to a surface of equal two‐way time, has a profound position in seismic imaging. In this paper, I introduce a framework for construction of isochrons for a given velocity model. The basic idea is to let trajectories called isochron rays be associated with iso chrons in an way analogous to the association of conventional rays with wavefronts. In the context of prestack depth migration, an isochron ray based on conventional ray theory represents a simultaneous downward continuation from both source and receiver. The isochron ray is a generalization of the normal ray for poststack map migration. I have organized the process with systems of ordinary differential equations appearing on two levels. The upper level is model‐independent, and the lower level consists of conventional one‐way ray tracing. An advantage of the new method is that interpolation in a ray domain using isochron rays is able to treat triplications (multiarrivals) accurately, as opposed to interpolation in the depth domain based on one‐way traveltime tables. Another nice property is that the Beylkin determinant, an important correction factor in amplitude‐preserving seismic imaging, is closely related to the geometric spreading of isochron rays. For these reasons, the isochron ray has the potential to become a core part of future implementations of prestack depth migration. In addition, isochron rays can be applied in many contexts of forward and inverse seismic modeling, e.g., generation of Fresnel volumes, map migration of prestack traveltime events, and generation of a depth‐domain–based cost function for velocity model updating.

Depth‐focusing analysis using a wavefront‐curvature criterion

Geophysics ◽  
1993 ◽  
Vol 58 (8) ◽  
pp. 1148-1156 ◽  
Author(s):  
Scott MacKay ◽  
Ray Abma
Keyword(s):  
Seismic Energy ◽  
Radius Of Curvature ◽  
Large Radius ◽  
Data Set ◽  
Depth Migration ◽  
Amplitude Data ◽  
Zero Radius ◽  
Inverse Radius ◽  
Interpretation Process ◽  
Zero Offset

Depth‐focusing analysis (DFA), a method of refining velocities for prestack depth migration, relies on amplitude buildups at zero offset to determine the extrapolation depths that best focus the migrated data. Unfortunately, seismic energy from dipping interfaces, diffractions, and noise often produce spurious amplitude indications of focusing. To reduce possible ambiguity in the DFA interpretation process, we introduce a new attribute for determining focusing that is relatively independent of amplitude. Our approach is based on estimates of the radius of wavefront curvature. The estimates are derived from normal moveout analysis of nonzero‐offset data saved during migration. By relating steeper moveout to smaller radius of wavefront curvature, focusing is defined by a wavefront curvature of zero radius. Additionally, we show that applying inverse‐radius weights to the amplitude data attenuates nonfocused events due to their large radius of curvature. Using the Marmousi data set, our weighting scheme resulted in reduced spurious focusing and enhanced velocity resolution in DFA.

Reply by the authors to Arthur E. Barnes

Geophysics ◽  
1995 ◽  
Vol 60 (6) ◽  
pp. 1944-1946
Author(s):  
M. Tygel ◽  
J. Schleicher ◽  
P. Hubral
Keyword(s):  
Destructive Interference ◽  
Depth Migration ◽  
Time Migration ◽  
Pulse Distortion ◽  
Nmo Correction ◽  
Important Processing ◽  
Offset Time ◽  
Zero Offset ◽  
Seismic Reflections ◽  
Processing Steps

We highly appreciate the useful remarks of Dr. Barnes relating our work to well‐known practical seismic processing effects. This is of particular interest as normal‐moveout (NMO) correction and post‐stack time migration are still two very important processing steps. Most exploration geophysicists know about the significance of pulse distortions known as “NM0 stretch” and “frequency shifting due to zero‐offset time migration.” As a result of the discussion of Dr. Barnes, it should now be possible to better appreciate the importance of our very general formulas (27) describing the pulse distortion of seismic reflections from an arbitrarily curved subsurface reflector when subjected to a prestack depth migration in 3‐D laterally inhomogeneous media. This discussion thus relates in particular to such important questions as how to correctly sample signals in the time or depth domain in order to avoid spatial aliasing, or how to stack seismic data without loss of information due to destructive interference of wavelets of different lengths.

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