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

Isotropic impulse response

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:

Implicit extrapolation with the 45 equation

Zq

q

A

Aq

DI

DIq

1

2

1

21

z

2Dwhere

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

2D filter 1D filter

Helical boundary conditions

• 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

zia

zia

z ee qLU

LUq

A

Aq

11

22

1

21

3-D implicit depth migration

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

3-D impulse response

Broad-band

Dip-limited

Cross-sections:

3-D impulse response

Time-slices:

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

3-D depth migration model

3-D depth migration results

Conclusions

• Shown how helical boundary conditions enable implicit 3-D wavefield extrapolation

• Lateral variations in velocity are handled by non-stationary inverse filtering

Conclusions

• Demonstrated 3-D depth migration with 45 wave equation

• Helical boundary conditions applicable for full range of implicit migration methods