Finite-Frequency Resolution Limits of Traveltime Tomography for Smoothly Varying Velocity Models...

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




• 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




• 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


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.


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



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



Refraction TomographySS

VV11

VV22

RR

2)max(

4

xKx 4/)cos(

)max(

41

zK

z

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


kmz

kmx

96

64



200200

400400

600600

Dep

th (

km

)D

epth

(k

m)

0 200 400 0 200 400


kmz

kmx

144

96



600600

800800

10001000

Dep

th (

km

)D

epth

(k

m)

0 200 400 0 200 400


kmz

kmx

148

164






• 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 .

Finite-Frequency Resolution Limits of Traveltime Tomography for Smoothly Varying Velocity Models...

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