Transcript Anatomy of CSP gathers formed during equivalent offset

AVO, migration apertures
Fresnel zones, stacking, Q,
anisotropy, and fast inversions
John C. Bancroft and grad students
University of Calgary
CREWES 2003
NSERC
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Grad students
•
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•
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Shuang Sun
Pavan Elapavuluri
Kun Liu (and Hugh Geiger)
John Millar
Xiang Du
Zhihong (Nancy) Cao
Chunyan (Mary) Xiao
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Outline
• Differentiators
• Fresnel zones AVO migration apertures
• Q
• Wavefield extrapolators
• Multiples and radon transforms
• Inversion
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Digital differentiator
• Accuracy
• Speed
• Beauty
• (1, -1)
• 1, 0, -1) better
• Theoretical shape?
4
Inverse FFT of (j)
Close up of true differential operator from jw, t=0 in center
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
5
0
5
10
15
20
25
30
35
40
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Derivative assumes:
Close up of true differential operator from jw, t=0 in center
1
0.8
0.6
0.4
0.2
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•
•
•
Maximum bandwidth
Ideal interpolator
Sinx/x operator
Derivative of sinx/x
0
-0.2
-0.4
-0.6
-0.8
-1
0
5
10
15
20
25
30
35
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sin( x)
d
x  cos x  sin x
2
dx
x
x
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SincFn.m
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Shuang Sun
• Stacking specula energy
• Fresnel zone for offset and dipping reflectors
• Balanced AVO in areas of poor geometry
• Caution: model based migration
What is the basic unit of laryngitis?
One hoarsepower
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What size hole can we see with seismic?
8
What happens at the edge of a reflector?
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Fresnel zone
Fresnel Size:
• frequency
• depth
• velocity
Fresnel zone
v
R
2
Claerbout
kt
f
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Depends on the size of the Fresnel radius
R

v
2
kt
f
1 k  2
Only for stacked data!!!
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Analytic function for zero-offset case
(no wavelet)
Solution to the acoustic wave equation for zero-offset, the
diffraction response can be written as:
pn (t )  f t   D0 t U t  te 
Diffraction operator D0:
D0  t ' , e  
te
cos e

t
te2
'
 te 
2


d 
arctan 
' 

dt




t '  t '  2te   


te sin  e



: minimum two-way travel time to the edge of the reflector;
t’=t-te: time measured after onset time te ;
e
: angle between the normal to the reflector and raypath of the
minimum travel time to the edge of the reflector.
Berryhill, 1977
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Diffraction amplitude
Amplitude
Time
Angle
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Diffraction amplitude
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Scaled amplitude (evaluate phase)
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Fresnel zone with Ricker wavelet
source
Source signature with dominant
frequency 50hz
Reflected signal
amplitude
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Fresnel zone with fixed bandwidth
Frequency from 0-150 hz
Frequency from 0-120 hz
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Reflector smaller than Fresnel zone:
width 50m
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Reflector width: 1000m
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Amplitude (no wavelet)
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Diffraction modelling
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Valid area for AVO
?
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Comments
• Locations for AVO analysis before migration should be at least
a Fresnel radius away from the target edge.
• The amplitudes will be in error if the target size is smaller than
or equal to the Fresnel zone.
• Wavelet will contribute to the size of contamination area.
• Only zero offset was considered
• Should consider a prestack migration to perform AVO analysis.
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Offset traveltime
CSP
2h
2h
t
z
T 2  x  h
T  0 
2
V
 4
2



1
2
T 2  x  h
 0 
2
V
 4
2



1
2
DSR eqn.
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Cheop’s pyramid - DSR eqn.
2
Hyperbola
CMP at T0
Zero off.
1
Offset h 0
-1
-2
0
Time t
-1
DSR eqn.
Const. Off
CMP gath.
-2
-3
-2
-1
Displacement x
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0
1
2
Prestack data, flat…
h
7 Cheop’s .
h
x
x
t
t
Specula
energy
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Reflections from dipping event
h
Midpoint ?
h
CMP

h=0
h=0
Reflecting
Scatterpoint
element
Offset reflection
moves up dip


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Prestack data, dipping... Cheop’s p.
h
h
x
x
t
Reflection
t
Reflector
Reflector


 2 4 x cos  
T   T0 

2
V
RMS


2
1
2
Defines exact shape of surface,
specula energy
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Cheop’s summation: specula energy
h
x
t
Fresnel
zones
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Prestack Fresnel zone
horizontal layers
S
R
Zero offset
Offset
Fresnel zone
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P. S. Fresnel zone for dipping event
S
R
Zero offset
Fresnel zone
Offset
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Contours : equal angle of incidence
Angle 2
Scatterpoint
Offset increases with x
Angle 1
Scatterpoint
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Constant offset migration
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What about dipping specula energy?
Smear of dips
Still OK ???
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CMP gather
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Limited aperture EO gather
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Comments …
1. Prestack specula energy can be identified,
• horizontal and dipping
2. Limited aperture
• faster, better SNR, may preserve amplitudes
3. Optimum size of a super-CMP gather can be
defined using Fresnel zones
4. Should use prestack migration gathers for AVO
5. Use model based migration with caution
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Comments
6. Constant offset migration
•
smears AVO energy (OK)
•
requires velocity information
7. EOM
•
also smears AVO energy (OK)
8. Stacking within the Fresnel zone may reduce
acquisition geometry artifacts
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Pavan Elapavuluri
•Q
What is one millionth of a mouth wash?
One microscope
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Wavelets with varying Q and Time
• Build Q wavelet table: [Q, time]
–
–
Q: 30, 50, 70, 90, 110, …
T: 0 to 6.0 sec
• Cross-correlate (coherence) trace with all wavelets
• At a given time, peak corresponds to Q
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9 wavelets
T=1000
T=3000
T=5000
Q=90
Q=70
Q=50
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3 Spikes on trace Q = 70
T=1000
T=3000
T=5000
Q=90
Q=70
Q=50
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Noise on trace Q = 70
T=1000
T=3000
T=5000
Q=90
Q=70
Q=50
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Kun Liu (and Hugh Geiger)
• Wavefield extrapolation
What is 2000 pounds of Chinese soup?
Won ton
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Input
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200m depth extrapolation
…with adaptive taper
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Xiang Du
• Finite element method integrated with
• Finite difference migration
What is half a large intestine?
A semicolon
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Comparison: FE-FDM with FX-FDM
Result of FE-FDM
Result of FX-FDM (SU)
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Trace comparisons
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8
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6
Amplitude
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(3) FKFD
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3
2
(2) FEDM
1
0
-1
(1) Source
-2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Time(s)
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Steep oblique model
Velocity model
Seismic section
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Comparison of methods
Result of FE-FDM
Result of FX-FDM (SU)
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Runtime and memory requirements
• FE-FD
• FK-FD
26 secs
31 secs
2.2 meg
1.3 meg.
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Zhihong (Nancy) Cao
Chunyan (Mary) Xiao
• Tutorials
• Review of Multiple suppression techniques
• Multiple attenuation using the Radon transform
What is one thousand aches?
A kilohurtz
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after Kabir and Marfurt, 1999
Semblance plot is a Radon transform
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Lorraine Bloom
Larry Lines
David Henley
The model
and inverse
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Modelling
process
Inversion
expert
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The
Model
(Not the
inverse)
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The
Inverse
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The artifacts (of inversion)
Before modelling
After inversion
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Of course, …
some things are
easier to model
and invert.
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Inversion
• Mathematical process
–Used in many geophysical applications:
– decon., statics, tomography, tau-p, …
• Geophysical terminology
–Estimation of rock properties
– Modelling <-> Inversion
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Millar time
• Multigrid inversion
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•
•
•
•
Borrowed from fluid dynamics
Recursive
Variable grid size
Solutions are frequency dependent
Fast
What is the ratio of an igloo’s circumference to its diameter?
Eskimo
pi
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ax1  bx2  cx3  y1
lx1  mx2  nx3  y2
px1  qx2  rx3  y3
 a b c   x1  y1
 i m n  x   y
2

 2
 p q r   x3  y3
1
xA y
x   A A A y
T
1
T
Ax  y
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ax1  bx2  cx3  y1
lx1  mx2  nx3  y2
px1  qx2  rx3  y3
x1  a  l  0   y1  y2  y3   bx2  mx2  qx2  cx3  nx3  rx3 
x2  b  m  q   y1  y2  y3   ax1  lx1  px1  cx3  nx3  rx3 
x3  l  m  n   y1  y2  y3   ax1  lx1  px1  bx2  mx2  qx2 
Iterative solution
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Laplace’s equation
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Variable grid size
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Analogous to FFT ???
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Sorting a 2-D array into a vector
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Extractions, smoothings and
interpolations
S = Solution with one iterative step
33x33
17x17
9x9
9x9
5x5
5x5
3x3
3x3
Pass defects down and errors up
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Performance
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Conclusions:
• Migration apertures
speed, SNR, balanced amplitudes
• CMP AVO: consider Fresnel zones
• Finite element / Finite difference
migration
• Multiples and Radon transforms
• Q and differential operators
• Multi grid inversions
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The end
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