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Riemannian Wavefield Migration:Imaging non-conventional wavepaths and geometries
Jeff Shragge
Geophysics Department
University Oral Qualification Exam
Agenda
• What is my problem? my solution?– Statement of Problem– Proposed Solution– Thesis Objectives– Potential Impact
Part I
Part II
• How am I going to solve it?– Riemannian Wavefield Extrapolation– Completed Work– Work to do
3-D Seismic Imaging
Seismic targets increasingly
complex
Improved3-D Seismic acquisition
Improved3-D seismic
imaging
Improved3-D
interpretation
3-D Seismic Imaging
Seismic targets increasingly
complex
Improved3-D Seismic acquisition
Improved3-D seismic
imaging
Improved3-D
interpretation
Wave-Equation Migration – Review
Downward continuation
WavefieldU(t,s,g,z = 0)
WavefieldU(t,s,g,z = nΔz)
Apply recursive filter e(-ikz Δz)
n times
Imaging Condition
3-D Imaging is Successful…
Distance – x axis
Dep
th
Dis
tan
ce –
y
axis
Distance – x axis
From: 3-DSI, Biondi (2004)
Distance
Dep
th…so why are we needed?
From: 3-DSI, Biondi (2004)
Seismic imaging science is good, but not perfect– Irregular and sparse data– Illumination limitations – physical and imaging procedure– Topographic surface complexity– …
Distance
Dep
th…so why are we needed?
From: 3-DSI, Biondi (2004)
Seismic imaging science is good, but not perfect– Irregular and sparse data– Illumination limitations – physical and imaging procedure– Topographic surface complexity– …
Distance
Dep
thIllumination
Steep Dip reflector
Weak reflector
Subsalt Imaging
Complex Structure
From: 3-DSI, Biondi (2004)
• Poor illumination of subsurface because of :– poor physical illumination (Acquisition)– incomplete imaging procedure (Processing)
Topographic surface complexity• How do we deal with topography directly in
wave-equation imaging?
Why do we have Imaging Limitations?
Imaging illumination limitations
Topographic surface limitations
Coordinate system not conformal to propagation direction or acquisition surface
Migration physics decoupled from geometry
Evidence – I
Problem • Non-conformal coordinate systems• Migration physics decoupled from geometry
Resulting Limitations • Inaccurate imaging of steep dips
– Downward continuation inaccurate at high angles– Overturning waves not used
• Extrapolation from acquisition surface topography– Hard to define extrapolation axis– Free-surface topography– Deviated well VSP geometry
Steep Dip Imaging
Accuracy of wavefield extrapolation decreases as propagating waves tend to horizontal
Extrap
olatio
nD
irection
Using Overturning Waves
Currently do not use potentially useful information provided by overturning waves
Extrap
olatio
nD
irection
Proposed Solution
Use coordinate system conformal with wavefield propagation
Extrapolation
Direction
Tilted Cartesian Example
ELF North-Sea Dataset
From:Shan and Biondi (2004)
0
1
2
3
4
5
2 4 6Distance [km]
Dep
th
[km]
08
Tilted Cartesian Example
Downward Continuation Reverse-time
Plane-wave w/ dipping Cartesian coordinates
From:Shan and Biondi (2004)
Evidence – II
Problem• Non-conformal coordinate systems• Migration physics decoupled from geometry Resulting Limitations • Inaccurate imaging of steep dips
– Downward continuation inaccurate at high angles– Overturning waves not used
• Extrapolation from acquisition surface topography– Hard to define extrapolation axis– Free-surface topography– Deviated well VSP geometry
Free Surface Topography
• How to define extrapolation surface orthogonal to free surface?
VSP Deviated Well Topography
• Receiver wavefield acquired in well deviated in 3-D• How to define wavefield extrapolation from borehole surface?
Summary of Problem
Imaging Illumination Limitations
Topographic Surface Limitations
Coordinate system not conformal to propagation direction or acquisition surface
Migration Physics decoupled from Geometry
DifficultSteep DipImaging
No use of Overturning
waves
Extrapolationfrom complex free-surface
Extrapolationfrom deviated
boreholes
Summary of Solution
Reduce Imaging Illumination Limitations
Enable W.E. imaging directly from Topographic Surfaces
Perform Migration on Coordinate systems conformal to propagation direction/acquisition surface
Couple Migration Physics with Geometry
ImproveSteep DipImaging
Use Overturning
waves
Eliminate need for free-surface
datuming
W.E. Imaging for massive 3-D VSP data
What am I going to do?
Handle multipathing/ triplication
Applicable to 3-D field
data
Handle arbitrary geometry
Handle large data volumes
Improvesteep dipImaging
Eliminate need for free-surface datuming
W.E. imaging for massive 3-D VSP data
Req
uirem
ent
s
AVA studies Physical property
analysis
Imp
act
3-D curvilinear coordinate wave-equation migration
method
End of Part I
Handle multipathing/ triplication
Applicable to 3-D field
data
Handle arbitrary geometry
Handle large data volumes
Improvesteep dipimaging
Eliminate need for free-surface datuming
W.E. imaging for massive 3-D VSP data
Req
uirem
ent
s
AVA studies Physical property
analysis
Imp
act
3-D curvilinear coordinate wave-equation migration
method
Part II – Agenda
• Riemannian Wavefield Extrapolation (RWE)
• Completed Work– Generating Coordinate systems– Extrapolation examples– Deal with or avoid Triplication
• Work to do– Tackling the Geophysical Imaging problem– Technical issues
Part II – Agenda
• Riemannian Wavefield Extrapolation (RWE)
• Completed Work– Generating Coordinate systems– Extrapolation examples– Deal with or avoid Triplication
• Work to do– Tackling the Geophysical Imaging problem– Technical issues
RWE in 2-D ray-coordinates
RiemannianCartesian
x
z Extrapolation Direction
Orthogonal Direction
z
x
RWE: Helmholtz equation
i j j
ij
i
UgU
g
g
1
(associated) metric tensor
UsU 22
Laplacian
)( kii x
Coordinate system
Sava and Fomel (2004)
RWE: (Semi)orthogonal coordinates
i j j
ij
i
UgU
g
g
1
200
0
0
GF
FE
gij
2
2
0
0
J
gij
1st order 2nd order2nd order 1st order
RWE: Helmholtz equation
UsU
J
U
JJ
UJ
J
U 222
2
22
2
2
1111
UsU
cU
cU
cU
c 222
2
2
2
Ray-coordinate Interpretationα = velocity function J = geometric spreading or Jacobian
RWE: Dispersion relationR
iem
anni
anC
arte
sian
2222 skckickickc
2222 skk xz 1
0
cc
cc
RWE: Dispersion relationR
iem
anni
anC
arte
sian
sk
cc
cc
o
1
0
22
2
k
c
ck
c
cik
c
cik o
222xz ksk
RWE: Wavefield extrapolationR
iem
anni
anC
arte
sian
sk
cc
cc
o
1
0
zikxx
ze ω)z,,U(k=ω)z,z,U(k ΔΔ
τΔγγ
τττΔτ ike ω),,U(k=ω),,U(k
Part II – Agenda
• Riemannian Wavefield Extrapolation (RWE)
• Completed Work– Generating coordinate systems– Extrapolation examples– Deal with or avoid Triplication
• Work to do– Tackling the Geophysical Imaging problem– Technical issues
Generating Coordinate Systems
Single Arrival
Multiple Arrival
Monochromaticray tracing
To what degree can wave propagation be modeled in a coordinate system?
Broad-band ray tracing
Cartesian Coordinates
(SEP-114)
RayCoordinates
(SEP-115)
Monochromatic ray tracing
• Use local velocity and WAVEFIELD PHASE information to calculate coordinate system
• Create a coordinate system conformal with wavefield propagation direction
Monochromatic ray tracing
0
Distance
Dep
th
Distance
Dep
th
NOTE: Gradient of monochromatic wavefield phase shows orientation of propagation direction
Calculating phase-rays
Cartesian
ray equations
z
Uy
Ux
U
U
1Im
z)dτy,v(x,
dz
dy
dx
φ
• Decoupled system of 1st order ODEs
• Ray solution: – Specify an initial point– Numerically integrate dx and dz – Output x and z coordinates of ray
• Rayfield explicitly dependent on frequency
Part II – Agenda
• Riemannian Wavefield Extrapolation (RWE)
• Completed Work– Generating Coordinate systems– Extrapolation examples– Deal with or avoid Triplication
• Work to do– Tackling the Geophysical Imaging problem– Technical issues
Adaptive phase-ray extrapolation
Calculate 1Phase-ray
step
Calculate1 wavefield
step
Bootstrapping procedure
Part II – Agenda
• Riemannian Wavefield Extrapolation (RWE)
• Completed Work– Generating Coordinate systems– Extrapolation examples– Deal with or avoid Triplication
• Work to do– Tackling the Geophysical Imaging problem– Technical issues
Coordinate System Triplication
SmoothVelocity
Ray-trace on single arrival
Eikonal solvers
Coordinate System Triplication:Avoid or Handle?
Avoid Triplication
Avoid Zero Division:
Add ε(Sava &
Fomel,2003)
Iterative coord.
updatingSEP-115
Handle Triplication
Numerically isolate triplication branches
SEP-115
Regularization through Inversion
Regularization through Inversion
0
Tim
e [s]
X-Coordinate
6000
4000
0 50-50
0
Tim
e [s]
3000
1000
0 50-50
Z-Coordinate0 4000
0
2000
4000
80000
0
00
0
1
2
0
1
2
Shooting Angle [deg] Shooting Angle [deg]
Regularization through Inversion
0
1
2
Tim
e [s]
X-Coordinate
6000
4000
0 50-50 00
1
2
Shooting Angle [deg]
0
Tim
e [s]
3000
1000
0 50-50
Z-Coordinate
0
0
1
2
Data: Xd and Zd coordinates from ray-tracing
Model parameters to be fit:Xm and Zm
Xm and Zm are triplication-free if they satisfy Laplace equation
ΔXm = 0 = ΔZm
Regularization through Inversion
0
1
2
Tim
e [s]
X-Coordinate
6000
4000
0 50-50 00
1
2
Shooting Angle [deg]
0
Tim
e [s]
3000
1000
0 50-50
Z-Coordinate
0
0
1
2
ΔXm = 0 = ΔZm
Geometric Regularization
Minimize Curvature of Xm and Zm
Fitting Goals
Equation Type X-Coordinate Z-Coordinate
Physical Data Fitting
W(J)(Xm-Xd) 0 W(J)(Zm-Zd) 0
Geometric Regularization
ΔXm 0 ΔZm 0
W(J) dependent on Jacobian of coordinate system, e.g.,n
0JJ
J)J(W
J0 = initial spreadingn = curve adjusting parameter
Part II – Agenda
• Riemannian Wavefield Extrapolation (RWE)
• Completed Work– Generating Coordinate systems– Extrapolation examples– Deal with or avoid Triplication
• Work to do– Tackling the geophysical imaging problem– Technical issues
Imaging – Overturning waves• 3-D surveys consist of 1000s of shot + receiver locations
– Shot-profile migration often prohibitively expensive
• Plane-wave migration (Liu et al., 2002)– Phase encoding scheme (Romero et al., 2000)
• independent of coordinate-system
Imaging – Overturning waves
• Source wavefield
• Receiver wavefield
• Imaging Condition
)xx(piN
1jjxall
0xe)0z,x,(S)p,0z,x,(S
ωωω
)xx(piN
1jjxall
0xe)0z,x,(R)p,0z,x,(R
ωωω
)p,,z,x(R)p,,z,x(S)p,z,x(I xallx*allxp ωω
ω
Imaging – Overturning waves
• Multiple plane-wave imaging
• 3-D Extension – use 2-D plane-wave filter
– Cylindrical wave migration (Duquet et al., 2001)
maxx
minx
p
pxxx p)p,z,x(I)p(f)z,x(I Δ
)]yy(p)xx(p[iyx
yxe)p,p,(g 00 ωω
Imaging – Massive 3-D VSPExperiment Setup
– 2-D areal surface source pattern
– 3-C downhole receivers
– Record forward- and backscattering wavefield
VSP Imaging – Reciprocity • Use reciprocity to create common receiver profiles
– Large computational savings
• Source wavefield – plane wave coordinates• Receiver wavefield – quasi-spherical coordinates
Source wavefield Receiver wavefield
VSP Imaging – Phase Encoding • Use algorithm similar to plane-wave
– Random time shifts
Source wavefield Receiver wavefield
)z,x(tie))z,x(t,(g ΔωΔω
VSP Imaging – Passive Seismic Mode• Back-propagate wavefield from borehole to source• Record P- and S-waves – multimode imaging
– Permutations of S + R velocity models, causal/acausal propagation– Examine 6 candidate scattering modes (Shragge and Artman, 2003)
Source wavefield Receiver wavefield
Part II – Agenda
• Riemannian Wavefield Extrapolation (RWE)
• Completed Work– Generating Coordinate systems– Extrapolation examples– Deal with or avoid Triplication
• Work to do– Tackling the Geophysical Imaging problem– Technical issues
Technical Issues - I• 3-D ray-coordinate system non-orthogonality
– May be problematic because have cross partial differential terms– Solution – Use a Nth order polynomial to approximate the coordinates– Analytical derivatives
0
1
2
Tim
e [s]
X-Coordinate
6000
4000
0 50-50 00
1
2
Shooting Angle [deg]
0
1
2
Tim
e [s]
3000
1000
0 50-50
Z-Coordinate
0
0
1
2
nx
N
1nn),(X εαγτ
nz
N
1nn),(Z εβγτ
Technical Issues - II• Code and test different extrapolation operators
– Ray-coordinate 15 degree equation used– Examine more accurate operators, e.g.,
• 45 degree, • split-step Fourier• Fourier Finite Differencing
– 2-D methods that approximate 3-D solutions, e.g.,• Splitting
– Examine stability of operators in 2-D and 3-D
Field Data Verification• Overturning Waves
– 3-D ELF North Sea synthetic + field data sets (@ SEP)– 3-D Exxon/Mobil data set (@ SEP)
• Massive 3-D VSP– Paulsson Geophysical Inc. has agreed to make available a data set
acquired in Long Beach, CA• 5 wells – straight to deviated in 3-D• 240 receivers, 1000s of shot • Adequate wavefield sampling ~ 50 feet/receiver
Timeline for Proposed Work
Activity 2004
Test 2-D code on synthetic/field data
Develop other extrapolation operators
Develop 2-D migration code
Test methods for handling triplications
Test non-orthogonality of 3-D coords.
3-D field tests – Overturning/VSP
Develop 3-D migration code
Graduate
3-D synthetic tests – Overturning/VSP
Write Thesis
2005 2006 2007
Internship
Summary of ProblemProblem• Coordinate system not conformal to wavefield propagation• Migration physics decoupled from geometry
Resulting Limitations• Inaccurate imaging of steeply dipping structure• Difficult to extrapolate from topographic surfaces
– free-surface topography, deviated well VSP
Extrapolation
Direction
Summary of SolutionProposed Solution• Define Migration on generalized coordinate systems
• Couple physics of migration to geometry– Curvilinear coordinate system more conformal with:
• orientation of propagating wavefield• topography of acquisition surface
Extrapolation
Direction
Potential Imaging Improvements• Use more wavepaths to improve steep dip imaging
– More information to contribute to imaging– Increase migration aperture
• Extrapolate direction from surfaces with topography
– Directly apply wave-equation migration– Eliminate need for datuming
Extrapolation
Direction
Potential Impact
Improved steep dip imaging• Better interpretation geologic structure• Assist downstream processing tasks
– AVA studies– Physical property analysis
Massive 3-D VSP imaging• W.E. imaging directly applicable even in deviated boreholes• Provide additional tools
– Angle-domain CIGs, etc.