Post on 19-Dec-2015
Finite-Frequency Resolution Limits Finite-Frequency Resolution Limits of Traveltime Tomography of Traveltime Tomography
for Smoothly Varying for Smoothly Varying Velocity ModelsVelocity Models
Jianming Sheng and Gerard T. SchusterJianming Sheng and Gerard T. Schuster
University of UtahUniversity of Utah
OutlineOutline• ObjectiveObjective
• InverseInverse GRT and Resolution Limits GRT and Resolution Limits
• Numerical ExamplesNumerical Examples
• SummarySummary
ObjectiveObjective
• Estimates spatial resolution of traveltime Estimates spatial resolution of traveltime
tomogramstomograms
• Accounts for finite-frequency effectsAccounts for finite-frequency effects
• Is applicable for arbitrary velocity Is applicable for arbitrary velocity
modelsmodels
Develop a resolution method thatDevelop a resolution method that
OutlineOutline• ObjectiveObjective
• InverseInverse GRT and Resolution Limits GRT and Resolution Limits
• Numerical ExamplesNumerical Examples
• SummarySummary
Traveltime LinearizationTraveltime Linearization
• Under Rytov approximationUnder Rytov approximation
),( gs rr =rO(r)
),,(
),,(),,(
0 gs
g0s0
rr
rrGrrG
G
i rd
Traveltime Traveltime ResidualResidual
ObjectObjectFunctionFunction
Wavepath Wavepath (Woodward, 1992)(Woodward, 1992)
Traveltime LinearizationTraveltime Linearization
),( gs rr =rO(r) rd in
i Ae2
1
)(
It is related to the causal generalized It is related to the causal generalized Radon transform (Radon transform (BeylkinBeylkin, 1985), 1985)
• Using geometrical approximationUsing geometrical approximation
Partial ReconstructionPartial Reconstruction
)(rOestKKOre K
ndi )(ˆ
)2(
1
)(r
Controls resolution and Controls resolution and what model parts can be recoveredwhat model parts can be recovered
Inverse Traveltime Inverse Traveltime GRTGRT
Inverse Traveltime Inverse Traveltime GRTGRT
)(rOest
Partial ReconstructionPartial Reconstruction
)),(),(( gs rrrrK
)(rKKOre K
ndi )(ˆ
)2(
1
Depth migration (Lecomte, 1998 SEG)Depth migration (Lecomte, 1998 SEG)
Wavenumber Wavenumber )),(),(( gs rrrrK
rrss rrgg
SourceSourceGeophoneGeophone
rr
KK
Spatial Traveltime Resolution Spatial Traveltime Resolution Limit FormulaLimit Formula
)),(),(( gs rrrrK
iX
Spatial Traveltime Resolution Spatial Traveltime Resolution Limit FormulaLimit Formula
)(rsg
source-receiver pairs where source-receiver pairs where the wavepath visits the wavepath visits rr
Reflection Traveltime Reflection Traveltime TomographyTomography
rrss rrgg
SourceSource GeophoneGeophone
Transmission Traveltime Transmission Traveltime TomographyTomography
rrss rrgg
SourceSourceGeophoneGeophone
Available WavenumbersAvailable Wavenumbers
rrss rrgg
TransmissionTransmission
ReflectionReflection
SourceSource GeophoneGeophone
OutlineOutline• ObjectiveObjective
• InverseInverse GRT and Resolution Limits GRT and Resolution Limits
• Numerical ExamplesNumerical Examples
• SummarySummary
Numerical ExamplesNumerical Examples
• Crosswell Traveltime TomographyCrosswell Traveltime Tomography
• RefractionRefraction Traveltime Tomography Traveltime Tomography
• Global TomographyGlobal Tomography
Crosswell Traveltime Crosswell Traveltime TomographyTomography
XX
LL
(0, (0, L/2)L/2) ((X, L/2)X, L/2)
(0, -(0, -L/2)L/2) ((X, -L/2)X, -L/2)
rr00(X/2, 0)(X/2, 0)SourceSource GeophoneGeophone
Crosswell Traveltime Crosswell Traveltime TomographyTomography
A. Reflection Traveltime TomographyA. Reflection Traveltime Tomography2
2
4
L
Xx
L
Xz
the same as the migration-spatial-resolutionthe same as the migration-spatial-resolution
limits for crosswell migration derived bylimits for crosswell migration derived by
Schuster (1996, GJI) in far-field approximation.Schuster (1996, GJI) in far-field approximation.
Crosswell Traveltime Crosswell Traveltime TomographyTomography
B. Transmission TomographyB. Transmission Tomography
x4 12
3 X Xz
The results are similar to SchusterThe results are similar to Schuster (1996) for (1996) for
traveltime tomography in far-field traveltime tomography in far-field approximationapproximation
Key IdeaKey Idea
• The velocity anomalies within the The velocity anomalies within the first-first-Fresnel zone or wavepathFresnel zone or wavepath affect the affect the traveltime traveltime
• The intersection area of the wavepathsThe intersection area of the wavepaths at the at the
scatterer scatterer defines the spatial resolution limitsdefines the spatial resolution limits
Wavepath IntersectionWavepath Intersection
Transmission ExampleTransmission Example
rrs1s1
rrg1g1
Fresnel ZoneFresnel Zone
Wavepath IntersectionWavepath Intersection
Transmission ExampleTransmission Example
rrs1s1
rrg1g1
rrg2g2rrs2s2
rrs1s1
rrg1g1
rrs2s2 rrg2g2
rrs3s3
rrg3g3
Wavepath IntersectionWavepath IntersectionTransmission ExampleTransmission Example
200 (200 (m)m)
400
(40
0 ( m
)m
)
7272m m
44.744.7mm
C=3000 m/sC=3000 m/sf=300 Hzf=300 Hz
Wavepath IntersectionWavepath Intersection
Transmission ExampleTransmission Example
Numerical ExamplesNumerical Examples
• Crosswell Traveltime TomographyCrosswell Traveltime Tomography
• RefractionRefraction Traveltime Tomography Traveltime Tomography
• Global TomographyGlobal Tomography
Refraction TomographySS
VV11
VV22
RR
2)max(
4
xKx 4/)cos(
)max(
41
zK
z
Numerical ExamplesNumerical Examples
• Crosswell Traveltime TomographyCrosswell Traveltime Tomography
• RefractionRefraction Traveltime Tomography Traveltime Tomography
• Global TomographyGlobal Tomography
1Hz Global Tomography00 60006000 1200012000
00
60006000
1200012000
((km)km)
((km)km)
13.7213.72
10.2910.29
6.8586.858
3.4293.429
00((km/s)km/s)
CoreCore
MantleMantle
ScattererScatterer
WavepathWavepath
-100-100
100100
300300
Dep
th (
km
)D
epth
(k
m)
0 200 400 0 200 400
Horizontal (km)Horizontal (km)
kmz
kmx
48
64
1Hz Global TomographyResolution Limits
((Depth=100km)Depth=100km)
100100
300300
500500
Dep
th (
km
)D
epth
(k
m)
0 200 400 0 200 400
Horizontal (km)Horizontal (km)
kmz
kmx
96
64
1Hz Global TomographyResolution Limits
((Depth=300km)Depth=300km)
200200
400400
600600
Dep
th (
km
)D
epth
(k
m)
0 200 400 0 200 400
Horizontal (km)Horizontal (km)
kmz
kmx
144
96
1Hz Global TomographyResolution Limits
((Depth=400km)Depth=400km)
600600
800800
10001000
Dep
th (
km
)D
epth
(k
m)
0 200 400 0 200 400
Horizontal (km)Horizontal (km)
kmz
kmx
148
164
1Hz Global TomographyResolution Limits
((Depth=800km)Depth=800km)
OutlineOutline• ObjectiveObjective
• InverseInverse GRT and Resolution Limits GRT and Resolution Limits
• Numerical ExamplesNumerical Examples
• SummarySummary
SummarySummary
• Used the Used the inverseinverse GRT to get the spatial GRT to get the spatial traveltime resolution formulastraveltime resolution formulas
We haveWe have
• Developed a practical means of estimating Developed a practical means of estimating
traveltime resolution limits for arbitrary velocity traveltime resolution limits for arbitrary velocity
models and finite-frequency source datamodels and finite-frequency source data
• Obtained resolution limits of global tomo.Obtained resolution limits of global tomo.
AcknowledgmentAcknowledgment
We thank the sponsors of the University of We thank the sponsors of the University of Utah Tomography and Modeling Utah Tomography and Modeling /Migration (UTAM) Consortium for their /Migration (UTAM) Consortium for their financial support .financial support .