Gary Margrave Saleh Al-Saleh Hugh Geiger Michael Lamoureux ... · Saleh Al-Saleh Hugh Geiger...
Transcript of Gary Margrave Saleh Al-Saleh Hugh Geiger Michael Lamoureux ... · Saleh Al-Saleh Hugh Geiger...
The FOCIThe FOCI™™ method method of depth migrationof depth migration
Gary MargraveSaleh Al-SalehHugh Geiger
Michael Lamoureux
POTSI
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Outline
Wavefield extrapolators and stabilityA stabilizing Wiener filter
Dual operator tablesSpatial resamplingPost stack testingPre stack testing
FOCI
Wavefield Extrapolators
( )( )
( )111, , , 0,
2z T T
nik z ik x
T T Tnx z k z e e dkψ ω ϕ ωπ −
−−
⎡ ⎤= = ⎢ ⎥⎣ ⎦∫ i
The phase-shift extrapolation expression
( ), ,Tx zψ ω output wavefield
( ), 0,Tk zϕ ω= Fourier transform of input wavefield
Wavefield Extrapolators
( )ˆzik zne W z=
The phase-shift operator
wavelike
evanescent
2 22 2
2 2
2 22 2
2 2
,
,
T T
z
T T
k kv vk
i k kv v
ω ω
ω ω
⎧⎪⎪⎪ − >⎪⎪⎪=⎨⎪⎪⎪ − >⎪⎪⎪⎩
( ) ( )
( ) ( )
2 22 2
2 2
2 22 2
2 2
,
,
T TT T
z
T TT T
k kv x v x
k
i k kv x v x
ω ω
ω ω
⎧⎪⎪⎪ − >⎪⎪⎪⎪=⎨⎪⎪⎪ − >⎪⎪⎪⎪⎩
Wavefield Extrapolators
( )ˆzik zne W z=
The PSPI extension
Wavefield Extrapolators
( )2W z
wavenumbervω+
vω−
Wavefield Extrapolators
wavenumbervω+
vω−
( )2ˆphase W z⎡ ⎤
⎢ ⎥⎣ ⎦
Wavefield Extrapolators
In the space-frequency domain
( ) ( ) ( )1ˆ ˆ ˆ, , , 0, , , ,
nT T n T T Tx z x z W x x z v dxψ ω ψ ω ω−
= = −∫
where
( )( )
( ) ( )1
ˆ1
1 ˆˆ , , , ,2
T T Tn
ik x xn T T n T TnW x x z W k z e dkω ω
π −
− −−− = ∫ i
imaginary
real
meters
Wavefield ExtrapolatorsIn the space-frequency domain
Wavefield ExtrapolatorsBack to the wavenumber domain
wavenumber
( )2F WΩ2W
Wavefield Extrapolators
wavenumber
Back to the wavenumber domain
Stabilization by Wiener FilterTwo useful properties
( )ˆ ˆ ˆ, , , , , ,2 2n T n T n Tz zW k z W k W kω ω ω
⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜= ⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠Product of two half-steps make a whole step.
( ) ( )1 * 2 2ˆ ˆ, , , , ,n T n T xW k z W k z k kω ω− = >The inverse is equal to the conjugate
in the wavelike region.
Stabilization by Wiener Filter
A windowed forward operator for a half-step
( ) ( )2 2n nW z W z=Ω
( ) ( )1 ˆ2 2n n nW z WI F W zη− ⎡ ⎤• = ⎢ ⎥⎢ ⎥⎣ ⎦
Solve by least squares
0 2η≤ ≤
Stabilization by Wiener Filter
is a band-limited inverse for nWI ( )/ 2nW zBoth have compact support
( ) ( ) ( )* 2nF n n nW z WI W z W z= • ≈Form the FOCI™ approximate operator by
FOCI™ is an acronym for
Forward Operator with Conjugate Inverse.
Linear System to Solve
2,0 2,1 2, 2,0
2,1 2,0 2,1 2,12,0
2,1 2,0 2, 2,22,1
2, 2,1
2, 2,1
2,2,0
2,
0
0
0
0
p
p
p
p
m
m p
W W W DW W W DWI
W W W DWI
W W
W WWIW
D +
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ = ⎢⎢ ⎥⎢ ⎥ ⎢⎢ ⎥⎢ ⎥ ⎢⎢ ⎥⎢ ⎥ ⎢⎢ ⎥⎢ ⎥ ⎢⎢ ⎥⎣ ⎦⎢ ⎥ ⎢⎢ ⎥ ⎢⎢ ⎥ ⎣ ⎦⎣ ⎦
⎥⎥⎥⎥⎥⎥⎥
Properties of FOCI operator
( )inv nn length WI= ( )( )/ 2for nn length W z=
( )( ) 1nF op for invlength W z n n n= = + −Then
Let
Properties of FOCI operator
invn determines stability.
forn determines phase accuracy.
1.5inv forn n≈Empirical observation:
Properties of FOCI operator
Amount of evanescent filtering is inversely related to stability
0 no evanescent filtering ( 1000 steps)1 half evanescent filtering ( 100 steps)2 full evanescent filtering ( 50 steps)
η⎧⎪⎪⎪⎪= ⎨⎪⎪⎪⎪⎩
∼∼∼
Operator tablesSince the operator is purely numerical, migration proceeds by construction of operator tables.
minmin
maxk
vω=
( )minnFW kmink( )minnFW k k+∆mink k+∆
( )maxnFW kmaxk
maxmax
mink
vω=
( )k
mean vω∆∆ =
( )min 2nFW k k+ ∆min 2k k+ ∆
Operator tablesWe construct two operator tables for small and large η. The small η table is used most of the time, with the large η being invoked only every nth step.
0 no evanescent filtering ( 1000 steps)1 half evanescent filtering ( 100 steps)2 full evanescent filtering ( 50 steps)
η⎧⎪⎪⎪⎪= ⎨⎪⎪⎪⎪⎩
∼∼∼
Operator Design Example
Improved Operator Design
Spatial Resampling
( ) 12 x −∆( ) 12 x −− ∆ Wavenumber
Freq
uenc
y
Wavelike
minxk vω =minxk vω =−
maxω
Spatial Resampling
( ) 12 x −∆( ) 12 x −− ∆ Wavenumber
Freq
uenc
y
In red are the wavenumbers of a 7 point filter
Spatial Resampling
( ) 12 x −∆( ) 12 x −− ∆
Freq
uenc
y
Downsampling for the lower frequencies uses the filter more effectively
( ) 12 x −′∆( ) 12 x −′− ∆
Spatial Resampling
( ) 12 x −∆( ) 12 x −− ∆
Freq
uenc
y
Spatial resampling is done in frequency “chunks”.
Operator in Space
Improved Operator in Space
Run Time Experiment
PS slope 1.07FOCI slope 1.03
FOCI slope 1.05
Slopes for last three points
Run Times log-log Scale
Phase Shift Impulse Response
FOCI Impulse Responsenfor=7, ninv=15, (21 pt), no spatial resampling
FOCI Impulse Responsenfor=7, ninv=15, (21 pt), with spatial resampling
Marmousi Velocity Model
Exploding Reflector Seismogram
FOCI Post-Stack Migrationnfor=21, ninv=31, nwin=0
FOCI Post-Stack Migrationnfor=21, ninv=31, nwin=51
FOCI Post-Stack Migrationnfor=21, ninv=31, nwin=21
FOCI Post-Stack Migrationnfor=7, ninv=15, nwin=0
FOCI Post-Stack Migrationnfor=7, ninv=15, nwin=7
Marmousi Velocity Model
FOCI Pre-Stack Migrationnfor=21, ninv=31, nwin=0, deconvolution imaging condition
FOCI Pre-Stack Migrationnfor=21, ninv=31, nwin=0, deconvolution imaging condition
FOCI Pre-Stack MigrationShot 30
FOCI Pre-Stack MigrationShot 30
FOCI Pre-Stack MigrationStack +50*Shot 30
Detail of Pre-Stack Migration
Marmousi Reflectivity Detail
Marmousi Velocity Model
FOCI Pre-Stack MigrationVelocity model convolved with 200m smoother
FOCI Pre-Stack MigrationVelocity model convolved with 400m smoother
FOCI Pre-Stack MigrationExact Velocities
FOCI Pre-Stack MigrationExact Velocities
FOCI Pre-Stack MigrationVelocities smoothed over 200m
FOCI Pre-Stack MigrationVelocities smoothed over 400m
Detail of Pre-Stack MigrationExact Velocities
Detail of Pre-Stack MigrationVelocities smoothed over 200m
Detail of Pre-Stack MigrationVelocities smoothed over 400m
Run times
Full prestack depth migrations of Marmousi on a single 2.5GHz PC using Matlab code.
20 hours for the best result
1 hour for a usable result
ConclusionsExplicit wavefield extrapolators can be made local and
stable using Wiener filter theory.
The FOCI method designs an unstable forward operator that captures the phase accuracy and
stabilizes this with a band-limited inverse operator.
Reducing evanescent filtering increases stability.
Spatial resampling increases stability, improves operator accuracy, and reduces runtime.
The final method appears to be ~O(NlogN).
Very good images of Marmousi have been obtained.
Research DirectionsBetter phase accuracy.
Extension to 3D.
Extension to more accurate wavefield extrapolation schemes.
Migration velocity analysis.
Development of C/Fortran code on imaging engine.
Acknowledgements
Sponsors of CREWES
Sponsors of POTSI
NSERC
MITACS
PIMSA US patent application has been made for the FOCI process.
We thank:
Detail of Pre-Stack Migration
FOCI Pre-Stack Migrationnfor=21, ninv=31, nwin=0, deconvolution imaging condition
FOCI Pre-Stack Migrationnfor=21, ninv=31, nwin=0, deconvolution imaging condition