Ling Chen and RuShan Wu

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A Comparative Study on Gabor-Daubechies Beamlet
Migration and Gaussian Beam Migration

• Ling Chen and Ru-Shan Wu
• Modeling and Imaging Laboratory
• University of California, Santa Cruz

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• Beamlet Updating (including coupling)

G-D frame Propagator Pjl,mn
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GB method has been well developed as variations
of both the asymptotic ray method and the
parabolic approximation method.
GB
a
Properties of a GB
GB
a
a
Final Solution

• A bundle of rays
• Gaussian decay of amplitude
• Parabolic approximation
• of phase field

GB


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Hills method (1990) Improvement to the ray
solutions in poststack depth migration

• GB decomposition of a plane wave
• where
• a space interval, a lt 2?
• xn initial points of GBs, xn n a, n
  0, , Nn-1
• p ray parameter of GBs, p sin? /v k
  /?
• For migration, a same set of ray parameters
  is used for all frequencies
• pm m?p sin?m /v
• (-Nm/2 ? m ? Nm/2 with Nm /2 ? ?p ? 1 /v)

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• GB decomposition of the original wave field
• where the amplitude Bmn of the GB uGB(xn, pm,?)
  accounts for the contributions by all plane waves
  within the region ?pm-1? k lt?pm1
• Downward propagating the Gaussian beams to z
• GB summation of the downward wave field

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• Point spreading response
• homogeneous medium
• layered medium
• laterally varying-layered medium

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beamlet method Nn16, Nm64
beamlet method Nn64, Nm16
GB method Nn19, Nm113
Point spreading responses for the homogeneous
medium
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beamlet method Nn16, Nm64
beamlet method Nn64, Nm16
GB method Nn23, Nm169
Point spreading responses for the layered medium
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G-D Frame decomposition Nn16, Nm64
G-D Frame decomposition Nn64, Nm16
G-D Frame decomposition Nn64, Nm64
Point spreading responses of beamlet propagation
with high -frequency asymptotic GB solutions for
the layered medium
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Laterally varying-layered medium
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beamlet method Nn16, Nm64
beamlet method Nn64, Nm16
GB method Nn13, Nm151
Point spreading responses for the laterally
varying-layered medium
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Table Numbers of frame atoms and GBs used in
the computations of point spreading
responses
The frame atom numbers listed here is only for
one of the two frame decompositions as described
in the context. For the other decomposition, Nn
and Nm are exchanged with each other.
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• Synthetic seismogram migration
• homogeneous medium
• high contrast velocity model

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beamlet migration
GB migration
Comparison between images by beamlet migration
and GB migration for the homogeneous medium
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High contrast velocity model
homogeneous background a high-velocity inclusion
low velocity background
z (20m)
high velocity inclusion
source
x (20m)
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Two velocity contrasts are tested
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beamlet migration 50
GB migration 50
beamlet migration 100
GB migration 100
Comparison between images by beamlet migration
and GB migration for the heterogeneous model
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Gaussian profile
in decomposition

Propagator localization properties
G-D beamlet migration
GB migration
Paraxial approximation
Local perturbation theory
High-frequency asymptotics
One-way wave theory