implicit 3-d depth migration with helical boundary conditions james rickett, jon claerbout &...
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Implicit 3-D depth migration with helical boundary conditions
James Rickett, Jon Claerbout & Sergey Fomel
Stanford University
Implicit 3-D depth migration with helical boundary conditions
James Rickett, Jon Claerbout & Sergey Fomel
Stanford University
Implicit 3-D depth migrationwith helical boundary conditions
• Implicit extrapolation
• 45 equation
• 2D vs 3D
• Helical boundary conditions
• Lateral velocity variations
Wavefield extrapolation
zz Wqq 1
zkai
z e qq22
1
Ideally:
Explicit:
Implicit:
zia
z ke qAq )(1
zia
z k
ke q
A
Aq
)(
)(
1
21
• Advantages of implicit extrapolators– Unitary– More accurate for a given filter size
• BUT:– Need to inverse filter
Wavefield extrapolation
Implicit extrapolation with the 45 equation
Qi
vQ
zv 22
2
2
241
ZZ qDIqDI 211
Differential equation:
Matrix equation:
2-D implicit depth migration
• Matrix D is tridiagonal– easily invertible (cost N)
• 2-D implicit depth migration widely used
)121( d
3-D implicit depth migration
• Matrix D is blocked tridiagonal– NOT easily invertible– Splitting methods
• 3-D implicit not widely used– Explicit methods
1
141
1
d
• Rapid multi-D recursive inverse filtering:
1. Remap filter to 1-D
2. Factor 1-D filter into CCF of 2 minimum-phase filters
3. Divide by 2 minimum-phase filters
Helical boundary conditions
3-D implicit depth migration
• PROBLEM: 2-D inverse filtering
Non-causal 1-D filter
Causal/anti-causalfilter pair
LU decomposition
Helix2-D filter 1-D filter
Spectral factorization
Spectral factorization
• Estimate a minimum-phase function with a given spectrum
• Algorithm requirements:– Cross-spectra– Filter-size specified a priori
• Extension to cross-spectra
• BUT: Frequency domain– Non-zero coefficients cannot be specified
a priori
Kolmogoroff factorization
• Newton's iteration for square roots:
Wilson-Burg factorization
ttt a
saa
2
11
ttt
t
t
t
AA
S
A
A
A
A 111
• Generalized to polynomials (time series):
• Iterative– Quadratic convergence
• Cross-spectra
• Non-zero coefficients specified a priori
Wilson-Burg factorization
Lateral velocity variations
• Advantage of f-x vs f-k – Factor spatially variable filters– Non-stationary inverse filtering
• Rapid – Factors can be precomputed/tabulated
• Approximation – Similar to explicit methods
Lateral velocity variations
• Alternative method– Wilson-Burg factorization of non-stationary
filters– More accurate– More expensive
Conclusions
• Shown how helical boundary conditions enable implicit 3-D wavefield extrapolation
• Lateral variations in velocity are handled by non-stationary inverse filtering