jianming sheng and gerard t. schuster university of utah february, 2000

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

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

LinearizationLinearization

• Under Rytov approximationUnder Rytov approximation

),( gs rr =rO(r)

),,(

),,(),,(

0 gs

g0s0

rr

rrGrrG

G

i rd

Traveltime Traveltime ResidualResidual

ObjectObjectFunctionFunction

WavepathWavepath

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

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

Inverse Inverse GRTGRT

)(rOest KKOre Kn

di )(ˆ)2(

1

)(r

Partial ReconstructionPartial Reconstruction

)),(),(( gs rrrrK

Wavenumber Wavenumber )),(),(( gs rrrrK

rrss rrgg

SourceSourceGeophoneGeophone

rr

KK

Spatial Resolution Spatial Resolution Limits FormulaLimits Formula

)),(),(( gs rrrrK

iX

Spatial Resolution Spatial Resolution Limits FormulaLimits 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 ExperimentCrosswell Experiment

• RefractionRefraction Tomography Tomography

• Global TomographyGlobal Tomography

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


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.


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

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 (m)200 (m)

400

(m

)40

0 (m

)

72m 72m

44.7m44.7m

C=3000 m/sC=3000 m/sf=300 Hzf=300 Hz



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)

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

• 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

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 .

jianming sheng and gerard t. schuster university of utah february, 2000

Documents

spatial resolution formulas

horizontal kmdepth

kmoutlineobjectiveinverse

intersection area

velocity anomalies

reflection tomography

crosswell migration

result of schuster