Transcript Elastic Wavefield Extrapolators

Elastic Wavefield
Extrapolation in HTI Media
Richard A. Bale and
Gary F. Margrave
Outline
Introduction
• Elastic wavefield extrapolation in
layered HTI media
• Extension to laterally heterogeneous
media
• Conclusions and ongoing work
Introduction
• Recursive wavefield extrapolation
– Given u at depth z, generate u at z+Dz
• One-way wave equation
– Avoids multiple scattered energy
– Robust to velocity errors
• Elastic anisotropic wave equation
– Proper handling of polarizations, mode
conversions , shear-wave splitting
• Related to Alford rotation
– Avoids normal incidence assumption
• Extensions to lateral heterogeneity
– Making parameters function of x and kx
– PSPI, NSPS, AGPS, FFLA
Previous Work
• Etgen (1988)
– Stolt scalar migration, after wavefield
decomposition using div and curl operators in
wavenumber domain
– Dellinger and Etgen (1990) generalized wavefield
decomposition to anisotropy using Christoffel eqn.
• Zhe and Greenhalgh (1997)
– Applied wavefield separation after RTM by a few
steps, then split-step scalar migration
• Hou and Marfurt (2003)
– Extrapolate each component with all velocities,
apply separation as part of imaging condition
VTI and HTI: decks of Cards
VTI: Vertical
symmetry axis
E.g. Shales
HTI: Horizontal
symmetry axis,
with azimuthal angle
f
Strong (fast)
direction
Weak (slow)
direction
E.g. Fractured
carbonates
Outline
• Introduction
Elastic wavefield extrapolation in
layered HTI media
• Extension to laterally heterogeneous
media
• Conclusions and ongoing work
Elastic Wave-equation Migration
b continuous
at z, z+Dz
vM discontinuous
at z, z+Dz
vvSP12kkxxx,,zz,,

bk x , z,  
vvSSP21kkxx,,zz DDzz,,

bk x , z  Dz,  
b = displacement-traction vector
vM = magnitude of mode M
+ some kind of imaging conditions
A Generic Extrapolation Step
• Decomposition of displacement-traction into P
and S waves (SV, SH or S1, S2), at depth z
• Scalar extrapolation using 3 different dispersion
relations
• Recomposition to displacement-traction at depth
z+Dz
Extrapolations Compared
Theoretical model
Decomposition
Extrapolation
Acoustic, isotropic
None
Single scalar w.e.:
operator derived
analytically (square-root)
Elastic, isotropic
Helmholtz decomposition
1 P-wave  s (x-free)
2 S-waves  s (.-free)
P, SV and SH (=SV)
scalar w.e.s: operator
derived analytically
Elastic, anisotropic
P, S1 and S2 scalar
De/Re-composition
w.e.s: operator from
matrix from eigenvectors
eigenvalues of Christoffel
of Christoffel equation
equation
Slowness Curves: Isotropic
Slowness Curves: HTI (45°)
S2
S2
S1
S1
Isotropic Extrapolator:
Impulse (on Z)
Isotropic Extrapolator:
Impulse (on Z), Decomposed to P-SV
SV
Isotropic Extrapolator:
Impulse (on Z) Response, P-SV
SV
Isotropic Extrapolator:
Impulse (on Z) Response, X-Z
Isotropic Extrapolator, X-Z:
FK Amplitude Spectrum
Evanescent P and S Propagating P and S
Propagating S,
Evanescent P
Isotropic Extrapolator:
Downward and Upward, X-Z
Elastic Extrapolation Example
Pseudospectral Modelled
Wavefield at A (HTI - 45°)
After Wavefield Decomposition
After Extrapolation and
Recomposition at B
Pseudospectral Modelled
Wavefield at B (HTI - 45°)
Elastic Extrapolation Example
Wavefield from C, after Extrapolation
and Recomposition at B
Pseudospectral Modelled
Wavefield at B (HTI - 45°)
Outline
• Introduction
• Elastic wavefield extrapolation in
layered HTI media
Extension to laterally heterogeneous
media
• Conclusions and ongoing work
PSPI Elastic Extrapolation
z
z+Dz
Cij(x1)
Cij(x2)
Cij(x3)
vP
vS1
vS2
f:
NSPS Elastic Extrapolation
z
z+Dz
Cij(x1)
Cij(x2)
Cij(x3)
vP
vS1
vS2
f:
Heterogeneous HTI Model
f = 45°
f = 0°
400m
0m
1280m
Reference Models:
f = 0°
f = 45°
2560m
Homogeneous HTI, f =0°
V(z) Elastic Extrapolation
Heterogeneous HTI,
PSPI Elastic Extrapolation
Homogeneous HTI, f =45°
V(z) Elastic Extrapolation
Heterogeneous HTI,
NSPS Elastic Extrapolation
Homogeneous HTI, f =0°
V(z) Elastic Extrapolation
PSPI downward 400m
PSPI upward 400m
PSPI downward 400m
NSPS upward 400m
Outline
• Introduction
• Elastic wavefield extrapolation in
layered HTI media
• Extension to laterally heterogeneous
media
Conclusions and ongoing work
Conclusions
• Anisotropic elastic extrapolators are
constructed from eigen-solutions of the
Christoffel matrix
• Based on continuity of displacement-stress
vector
• Extrapolation includes decomposition /
recomposition + scalar extrapolation
– Decomposition operator derived from eigenvectors
(polarizations)
– Scalar extrapolation operator derived from
eigenvalues (vertical slownesses)
Conclusions
• Different evanescent cutoffs for each mode
– Cause of residual terms after inverse extrapolation
• V(z) algorithm tested with data from
pseudospectral elastic modeling
• Extension to lateral heterogeneity via elastic
versions of PSPI or NSPS algorithms
• Elastic NSPS natural adjoint (pseudo-inverse)
of Elastic PSPI and vice versa
Ongoing Work
• Elastic extrapolation with interfacepropagators
– Equivalent to decomposition/recomposition for
V(z) medium
– Accuracy of PSPI/NSPS form?
• Shot record migration: paper in report (Bale &
Margrave)
– V(z) version coded so far: tested on model with
density anomaly
– Imaging of P-P, P-S1, P-S2 etc.
– Will extend with PSPI or NSPS elastic
extrapolators
Acknowledgements
• Sponsors of CREWES
• Sponsors of POTSI: Pseudodifferential
Operator Theory and Seismic Imaging
– MITACS: Mathematics of Information Technology
and Complex Systems
– PIMS: Pacific Institute of the Mathematical
Sciences
– NSERC: Natural Sciences and Engineering
Research Council of Canada
• Ed Krebes, Jeff Grossman, Hugh Geiger