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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 UtahFebruary, 2000February, 2000
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OutlineOutline• ObjectiveObjective
• InverseInverse GRT and Resolution Limits GRT and Resolution Limits
• Numerical ExamplesNumerical Examples
• SummarySummary
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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
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OutlineOutline• ObjectiveObjective
• InverseInverse GRT and Resolution Limits GRT and Resolution Limits
• Numerical ExamplesNumerical Examples
• SummarySummary
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LinearizationLinearization
• Under Rytov approximationUnder Rytov approximation
),( gs rr =rO(r)
),,(
),,(),,(
0 gs
g0s0
rr
rrGrrG
G
i rd
Traveltime Traveltime ResidualResidual
ObjectObjectFunctionFunction
WavepathWavepath
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LinearizationLinearization
),( 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
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Inverse Inverse GRTGRT
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
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Inverse Inverse GRTGRT
)(rOest KKOre Kn
di )(ˆ)2(
1
)(r
Partial ReconstructionPartial Reconstruction
)),(),(( gs rrrrK
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Wavenumber Wavenumber )),(),(( gs rrrrK
rrss rrgg
SourceSourceGeophoneGeophone
rr
KK
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Spatial Resolution Spatial Resolution Limits FormulaLimits Formula
)),(),(( gs rrrrK
iX
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Spatial Resolution Spatial Resolution Limits FormulaLimits Formula
)(rsg
source-receiver pairs where source-receiver pairs where the wavepath visits the wavepath visits rr
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Reflection Traveltime Reflection Traveltime TomographyTomography
rrss rrgg
SourceSource GeophoneGeophone
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Transmission Traveltime Transmission Traveltime TomographyTomography
rrss rrgg
SourceSourceGeophoneGeophone
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Available WavenumbersAvailable Wavenumbers
rrss rrgg
TransmissionTransmission
ReflectionReflection
SourceSource GeophoneGeophone
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OutlineOutline• ObjectiveObjective
• InverseInverse GRT and Resolution Limits GRT and Resolution Limits
• Numerical ExamplesNumerical Examples
• SummarySummary
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Numerical ExamplesNumerical Examples
• Crosswell ExperimentCrosswell Experiment
• RefractionRefraction Tomography Tomography
• Global TomographyGlobal Tomography
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Crosswell ExperimentCrosswell Experiment
XX
LL
(0, L/2)(0, L/2) (X, L/2)(X, L/2)
(0, -L/2)(0, -L/2) (X, -L/2)(X, -L/2)
rr00(X/2, 0)(X/2, 0)SourceSource GeophoneGeophone
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Crosswell ExperimentCrosswell Experiment
A. Reflection TomographyA. Reflection 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
SchusterSchuster (1996) in far-field approximation. (1996) in far-field approximation.
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Crosswell ExperimentCrosswell Experiment
B. Transmission TomographyB. Transmission Tomography
x4 12
3 X Xz
The results are similar to The results are similar to SchusterSchuster (1996) (1996)
for traveltime tomography in far-field for traveltime tomography in far-field approximationapproximation
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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
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Wavepath IntersectionWavepath Intersection
Transmission ExampleTransmission Example
rrs1s1
rrg1g1
Fresnel ZoneFresnel Zone
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Wavepath IntersectionWavepath Intersection
Transmission ExampleTransmission Example
rrs1s1
rrg1g1
rrg2g2rrs2s2
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rrs1s1
rrg1g1
rrs2s2 rrg2g2
rrs3s3
rrg3g3
Wavepath IntersectionWavepath IntersectionTransmission ExampleTransmission Example
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200 (m)200 (m)
400
(m
)40
0 (m
)
72m 72m
44.7m44.7m
C=3000 m/sC=3000 m/sf=300 Hzf=300 Hz
Wavepath IntersectionWavepath Intersection
Transmission ExampleTransmission Example
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Numerical ExamplesNumerical Examples
• Crosswell ExperimentCrosswell Experiment
• RefractionRefraction Tomography Tomography
• Global TomographyGlobal Tomography
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Refraction TomographySS
VV11
VV22
RR
2)max(
2
xKx 4/)cos(
)max(
21
zK
z
The same as the result of The same as the result of SchusterSchuster (1995) (1995)
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Numerical ExamplesNumerical Examples
• Crosswell ExperimentCrosswell Experiment
• RefractionRefraction Tomography Tomography
• Global TomographyGlobal Tomography
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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
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-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)
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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)
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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)
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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)
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OutlineOutline• ObjectiveObjective
• InverseInverse GRT and Resolution Limits GRT and Resolution Limits
• Numerical ExamplesNumerical Examples
• SummarySummary
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SummarySummary
• Derived the Derived the inverseinverse GRT and the spatial GRT and the spatial resolution formulasresolution formulas
We haveWe have
• Developed a practical means of estimating Developed a practical means of estimating
resolution limits for arbitrary velocity resolution limits for arbitrary velocity
models and finite-frequency source datamodels and finite-frequency source data
• Reexamined whole-earth tomogramsReexamined whole-earth tomograms
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AcknowledgmentAcknowledgment
We thank the sponsors of the 1999 We thank the sponsors of the 1999 University of Utah Tomography and University of Utah Tomography and Modeling /Migration (UTAM) Consortium Modeling /Migration (UTAM) Consortium for their financial support .for their financial support .