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Center for Wave Phenomena
Department of GeophysicsColorado School of Mines
Golden, Colorado
CWP FacultyIlya Tsvankin
“We are committed to quality education.”
Roel SniederDave Hale
Ilya Tsvankin
Paul SavaNorm Bleistein
Ken Larner
What do we offer?• Full financial support
• Diverse and enthusiastic faculty
• State-of-the-art research
• Industry contacts and internships
Need more Information?
Prof. Ilya [email protected]
Prof. Roel Snieder [email protected]
www.cwp.mines.edu
Center for Wave PhenomenaColorado School of Mines
Golden, CO 80401
Problem with sub-salt imaging
reservoir
salt
What if we had virtual sources below the salt?
reservoir
salt
Response to random sources
Ar Br
Response to random sources
Ar Br
dttptpC B
T
AAB ),(),()(0
rrCorrelation:
Very Long Baseline Interferometry
http://www.lupus.gsfc.nasa.gov/brochure/bintro.html
Distance between USA and Germany
http://www.lupus.gsfc.nasa.gov/brochure/btoday2.html
1D example
)/()/(),( cxtLcxtRtxu
cxtR / cxtL /
0x Hx
Cross-correlation
)/()/()()0( cHtCcHtCHxuxu LR
t
cH / cH /
• sum of causal and acausal response• uncorrelated left- and rightgoing waves
Need to extend this to include:
- heterogeneous media
- more space dimensions
Derivation based on normal-modes
(Lobkis and Weaver, JASA, 110, 3011-3017, 2001)
Displacement response
m
mmmmm
m tbtau
tu
cossin)(
),(r
r
)(sin)'()(
),',( tHtuu
tGm
mm
mm
rrrr
Heaviside function
Velocity response
m
mmmmm tbtautv sincos)(),( rr
)(cos)'()(),',()( tHtuutGm
mmmv rrrr
Uncorrelated excitation
m
mmmmm tbtautv sincos)(),( rr
0
2
2
mn
nmmn
nmmn
ba
Fbb
Faa
Correlation
),(),()()( tvtvC BAvAB rr
m
mmmmBAmBA tbtautv sincos)(),( ,, rr
Correlation as sum over modes
ttbb
ttab
ttba
ttaa
uuC
mnmn
mnmn
mnmn
mnmn
mnBmAn
vAB
sinsin
cossin
sincos
coscos
)()()(,
)( rr
For uncorrelated modes
ttbb
ttab
ttba
ttaa
uuC
mnmn
mnmn
mnmn
mnmn
mnBmAn
vAB
sinsin
cossin
sincos
coscos
)()()(,
)( rr
nmF 2
For uncorrelated modes
ttbb
ttab
ttba
ttaa
uuC
mnmn
mnmn
mnmn
mnmn
mnBmAn
vAB
sinsin
cossin
sincos
coscos
)()()(,
)( rr
nmF 2
mcos
Correlation
m
mBmAmvAB uuFC cos)()()( 2)( rr
Green’s function
)(cos)()(),,()( HuuGm
mBmAmBAv rrrr
Correlation and Green’s function
),,(),,()( )()(2)( tGtGFtC BAv
BAvv
AB rrrr
- sum of causal and acausal Green’s function
- holds for arbitrary heterogeneity
(Weaver and Lobkis, Ultrasonics, 40, 435-439, 2002)
20 30 40 50 60 70 80 90 100 110
-4500
-4000
-3500
-3000
-2500
-2000
-1500
-1000
-500
0
500
1000
1500
2000
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
time (microseconds)
Pulse/echo
noiseautocorrelation
(Weaver and Lobkis, Ultrasonics, 40, 435-439, 2002)
Raindrop model
Sources can be:
- real sources
- secondary sources (scatterers)
Extract response by correlation
Ar Br
Sr
dttptpC B
T
AAB ),(),()(0
rrCorrelation:
Non-stationary source point
Ar Br
Non-stationary source point
Ar Br
Stationary source point
Ar Br
Stationary source point
Ar Br
Stationary phase regions
“anti-Fresnel zones”
Stationary phase integration
R
ikRe
R
ikReCAB
44
)(
)()()( * RGRGCAB
(Snieder, Phys. Rev. E, 69, 046610, 2004)
Four types of averaging
Computing synthetic seismograms
(Van Manen et al., Phys. Rev. Lett., 94, 164301,2005)
Field example of virtual sources
(Bakulin and Calvert, SEG expanded abstracts, 2477-2480, 2004)
reservoir
complicated overburden
Peace River 4D VSP
1 0.500.5
1x
10.5
00.51y
1 0.50
0.5
1
z
1 0.500.5
1x
10.5
00.51y
Component used,along-the-well (450)
Image from virtual sources
topbottom
Virtual source Surface
Surface waves
(Campillo and Paul, Science, 299, 547-549, 2003)
correlation Green’s tensor
Z/Z
Z/R
Z/T
Surface wave Green’s function
i
G
i
GC klklkl )(
(Snieder, Phys. Rev. E, 69, 046610, 2004)
Seismic interferometry in Millikan Library
(Snieder and Safak, Bull. Seismol. Soc. Am., 96, 586-598, 2006)
Deconvolution with top floor
Deconvolution with bottom floor
traveling waves normal modes
Four new wave states
(Snieder, Sheiman, and Calvert, Phys. Rev. E., 73, 066620, 2006)
building borehole
Advantage (1), virtual sources atnew locations
reservoir
salt
Advantage (1), virtual sources atnew locations
Advantage (2), virtual sources at“all” times
Advantage (3), use other type of data
(Shapiro et al., Science, 307, 1615-1618, 2005)
earthquake
correlation(1 year)
correlation(1 month)
5-10sec.
1010.10.010.01Frequency (Hz)
More information
www.mines.edu/~rsnieder
email [email protected]
www.mines.edu/~rsnieder/SPICE06_Snieder.ppt
