Wave‐character preserving pre‐stack map migration using curvelets

Leading-order seismic imaging using curvelets

Geophysics ◽

10.1190/1.2785047 ◽

2007 ◽

Vol 72 (6) ◽

pp. S231-S248 ◽

Cited By ~ 59

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.

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The isochron ray in seismic modeling and imaging

Geophysics ◽

10.1190/1.1778248 ◽

2004 ◽

Vol 69 (4) ◽

pp. 1053-1070 ◽

Cited By ~ 22

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.

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Approximating subsurface structure in areas of severe subsea erosion—A nonprocessing technique

Geophysics ◽

10.1190/1.1442603 ◽

1989 ◽

Vol 54 (11) ◽

pp. 1397-1409

Author(s):

Fred W. Lishman ◽

Michael N. Christos

Keyword(s):

Seismic Data ◽

Seismic Reflection ◽

Velocity Model ◽

Subsurface Structure ◽

Submarine Canyons ◽

Arrival Times ◽

Sea Floor ◽

Static Corrections ◽

Map Migration ◽

Made In

Severe subsea erosion distorts seismic reflection times and velocity analyses and makes determining subsurface structure difficult. Although data reprocessing is the logical solution for removing these distortions, reprocessing can be expensive. We present a case history describing a nonprocessing depth‐conversion technique using a geologic erosional model. A grid of common‐midpoint seismic data located in and around several submarine canyons was used for this study. Establishing a geologic erosional model requires an accurate representation of the sea floor, which we obtain by map migration of the sea‐floor reflection. A velocity model was developed using only those analyses not adversely affected by sea‐floor erosion. To remove the effects of erosion from the arrival times of a mapped horizon, static corrections (velocity replacement and compaction) were developed. We replaced the water velocity in the eroded section with depth‐equivalent rock velocities from the velocity model. The compaction correction, which was derived empirically, is based on the assumption that porosity restoration occurred in the sediments beneath the canyons when erosion reduced the overlying pressure. Compaction correction in conjunction with velocity replacement produced structure maps (time and depth) that exhibit only minor effects of erosion. These results were further improved by applying dynamic corrections obtained by ray tracing a subsurface model to determine the traveltime through the water for the reflection from the mapped horizon. Our final structure maps demonstrate that a geologically reasonable structural interpretation in depth can be made in areas of severe subsea erosion without reprocessing the data.

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Post‐MAP migration of crosswell seismic data

10.1190/1.1885888 ◽

1997 ◽

Author(s):

Joongmoo Byun ◽

James W. Rector

Keyword(s):

Seismic Data ◽

Map Migration

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Nonlinear 3D tomographic least-squares inversion of residual moveout in Kirchhoff prestack-depth-migration common-image gathers

Geophysics ◽

10.1190/1.2956427 ◽

2008 ◽

Vol 73 (5) ◽

pp. VE13-VE23 ◽

Cited By ~ 16

Author(s):

Frank Adler ◽

Reda Baina ◽

Mohamed Amine Soudani ◽

Pierre Cardon ◽

Jean-Baptiste Richard

Keyword(s):

Inverse Problem ◽

Least Squares ◽

Nonlinear Least Squares ◽

Velocity Model ◽

Migration Velocity ◽

Prestack Depth Migration ◽

Data Set ◽

Depth Migration ◽

Layer Velocity ◽

Map Migration

Velocity-model estimation with seismic reflection tomography is a nonlinear inverse problem. We present a new method for solving the nonlinear tomographic inverse problem using 3D prestack-depth-migrated reflections as the input data, i.e., our method requires that prestack depth migration (PSDM) be performed before tomography. The method is applicable to any type of seismic data acquisition that permits seismic imaging with Kirchhoff PSDM. A fundamental concept of the method is that we dissociate the possibly incorrect initial migration velocity model from the tomographic velocity model. We take the initial migration velocity model and the residual moveout in the associated PSDM common-image gathers as the reference data. This allows us to consider the migrated depth of the initial PSDM as the invariant observation for the tomographic inverse problem. We can therefore formulate the inverse problem within the general framework of inverse theory as a nonlinear least-squares data fitting between observed and modeled migrated depth. The modeled migrated depth is calculated by ray tracing in the tomographic model, followed by a finite-offset map migration in the initial migration model. The inverse problem is solved iteratively with a Gauss-Newton algorithm. We applied the method to a North Sea data set to build an anisotropic layer velocity model.

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Extended isochron rays in prestack depth (map) migration

Geophysics ◽

10.1190/1.3459955 ◽

2010 ◽

Vol 75 (4) ◽

pp. S139-S150 ◽

Cited By ~ 14

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.

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Anisotropic parsimonious prestack depth migration

Geophysics ◽

10.1190/geo2011-0408.1 ◽

2013 ◽

Vol 78 (1) ◽

pp. S25-S36 ◽

Cited By ~ 4

Author(s):

Ernesto V. Oropeza ◽

George A. McMechan

Keyword(s):

Model Building ◽

Computing Time ◽

Synthetic Data ◽

Transversely Isotropic ◽

Velocity Model ◽

P Wave ◽

Prestack Depth Migration ◽

Depth Migration ◽

Tti Media ◽

Map Migration

An efficient Kirchhoff-style prestack depth migration, called “parsimonious” migration, was developed a decade ago for isotropic 2D and 3D media by using measured slownesses to reduce the amount of ray tracing by orders of magnitude. It is conceptually similar to “map” migration, but its implementation has some differences. We have extended this approach to 2D tilted transversely isotropic (TTI) media and illustrated it with synthetic P-wave data. Although the framework of isotropic parsimonious may be retained, the extension to TTI media requires redevelopment of each of the numerical components, calculation of the phase and group velocity for TTI media, development of a new two-point anisotropic ray tracer, and substitution of an initial-angle isotropic shooting ray-trace algorithm for an anisotropic one. The model parameterization consists of Thomsen’s parameters ([Formula: see text], [Formula: see text], [Formula: see text]) and the tilt angle of the symmetry axis of the TI medium. The parsimonious anisotropic migration algorithm is successfully applied to synthetic data from a TTI version of the Marmousi2 model. The quality of the image improves by weighting the impulse response by the calculation of the anisotropic Fresnel radius. The accuracy and speed of this migration makes it useful for anisotropic velocity model building. The elapsed computing time for 101 shots for the Marmousi2 TTI model is 35 s per shot (each with 501 traces) in 32 Opteron cores.

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