3‐D migration to zero offset for a constant velocity gradient

Author(s):  
Wesley T. Perkins ◽  
William S. French
Geophysics ◽  
2010 ◽  
Vol 75 (1) ◽  
pp. T9-T14 ◽  
Author(s):  
David J. Monk

The concept of the Fresnel zone has been explored by many workers; most commonly, their work has involved examining the Fresnel zone in the limiting case of zero offset and constant velocity. I have examined the shape of the Fresnel zone for nonzero offset and in the situation of constant velocity gradient. Finite-offset Fresnel zones are not circular but are elliptical and may be many times larger than their zero-offset equivalents. My derivation takes a largely geometric approach, and I suggest a useful approximation for the dimension of the Fresnel zone parallel to the shot-receiver azimuth. The presence of a velocity gradient (velocity increasing with depth) in the subsurface leads to an expansion of the Fresnel zone to an area that is far larger than may be determined through a more usual straight-ray determination.


Geophysics ◽  
1992 ◽  
Vol 57 (3) ◽  
pp. 474-477 ◽  
Author(s):  
Mohammed Alfaraj ◽  
Ken Larner

The transformation to zero offset (TZO) of prestack seismic data for a constant‐velocity medium is well understood and is readily implemented when dealing with either P‐waves or S‐waves. TZO is achieved by inserting a dip moveout (DMO) process to correct data for the influence of dip, either before or after normal moveout (NMO) correction (Hale, 1984; Forel and Gardner, 1988). The TZO process transforms prestack seismic data in such a way that common‐midpoint (CMP) gathers are closer to being common reflection point gathers after the transformation.


Geophysics ◽  
1990 ◽  
Vol 55 (1) ◽  
pp. 10-19 ◽  
Author(s):  
Martin Karrenbach

Three‐dimensional migration of zero‐offset data using a velocity varying with depth can be performed in one pass using Fourier transforms of time slices. The migration process is carried out entirely in the two‐dimensional spatial Fourier domain. The algorithm consecutively filters and adds time slices of the 3-D data volume in a way that is equivalent to summing energy over the diffraction surface of a point scatterer. The partial energy being distributed along a circle in a time slice is properly added in each summation step. Time‐slice migration is based on an integral solution of the acoustic wave equation known as the “Kirchhoff integral.” The wavelet shape in a 3-D data volume is preserved throughout the entire migration process. The frequency characteristics are maintained by summing weighted differences between time slices instead of summing the time slices themselves. Automatic weighting is achieved by time slicing at equal increments of diffraction radius. Tapering the summation operator reduces effects introduced by limiting the summation window. Time‐slice migration preserves the frequency content of a 3-D data volume during summation in a natural way. Since the migration scheme assumes a constant velocity within the entire time slice, it is a local process in time which migrates a 3-D data volume with a constant velocity or with a velocity which varies with depth. The migration algorithm is applied to numerical and physical model data. This method is especially suitable for a migration of a targeted subset of the 3-D data volume.


Geophysics ◽  
1997 ◽  
Vol 62 (2) ◽  
pp. 589-597 ◽  
Author(s):  
Jörg Schleicher ◽  
Peter Hubral ◽  
German Höcht ◽  
Frank Liptow

When a seismic common midpoint (CMP) stack or zero‐offset (ZO) section is depth or time migrated with different (constant) migration velocities, different reflector images of the subsurface are obtained. If the migration velocity is changed continuously, the (kinematically) migrated image of a single point on the reflector, constructed for one particular seismic ZO reflection signal, moves along a circle at depth, which we call the Thales circle. It degenerates to a vertical line for a nondipping event. For all other dips, the dislocation as a function of migration velocity depends on the reflector dip. In particular for reflectors with dips larger than 45°, the reflection point moves upward for increasing velocity. The corresponding curves in a Time‐migrated section are parabolas. These formulas will provide the seismic interpreter with a better understanding of where a reflector image might move when the velocity model is changed. Moreover, in that case, the reflector image as a whole behaves to some extent like an ensemble of body waves, which we therefore call remigration image waves. In the same way as physical waves propagate as a function of time, these image waves propagate as a function of migration velocity. Different migrated images can thus be considered as snapshots of image waves at different instants of migration velocity. By some simple plane‐wave considerations, image‐wave equations can be derived that describe the propagation of image waves as a function of the migration velocity. The Thales circles and parabolas then turn out to be the characteristics or ray trajectories for these image‐wave equations.


Geophysics ◽  
2004 ◽  
Vol 69 (5) ◽  
pp. 1299-1310 ◽  
Author(s):  
Jörg Schleicher ◽  
Claudio Bagaini

Configuration transform operations such as dip moveout, migration to zero offset, and shot and offset continuation use seismic data recorded with a certain measurement configuration to simulate data as if recorded with other configurations. Common‐shot migration to zero offset (CS‐MZO), analyzed in this paper, transforms a common‐shot section into a zero‐offset section. It can be realized as a Kirchhoff‐type stacking operation for 3D wave propagation in a 2D laterally inhomogeneous medium. By application of suitable weight functions, amplitudes of the data are either preserved or transformed by replacing the geometrical‐spreading factor of the input reflections by the correct one of the output zero‐offset reflections. The necessary weight function can be computed via 2D dynamic ray tracing in a given macrovelocity model without any a priori knowledge regarding the dip or curvature of the reflectors. We derive the general expression of the weight function in the general 2.5D situation and specify its form for the particular case of constant velocity. A numerical example validates this expression and highlights the differences between amplitude preserving and true‐amplitude CS‐MZO.


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