Finite-Frequency Resolution Limits of Traveltime Tomography for Smoothly Varying Velocity Models Jianming Sheng and Gerard T. Schuster University of Utah February, 2000

Outline ObjectiveObjective Inverse GRT and Resolution LimitsInverse GRT and Resolution Limits Numerical ExamplesNumerical Examples SummarySummary

Objective Estimates spatial resolution of traveltime tomogramsEstimates spatial resolution of traveltime tomograms Accounts for finite-frequency effectsAccounts for finite-frequency effects Is applicable for arbitrary velocity modelsIs applicable for arbitrary velocity models Develop a resolution method that

Outline ObjectiveObjective Inverse GRT and Resolution LimitsInverse GRT and Resolution Limits Numerical ExamplesNumerical Examples SummarySummary

Linearization Under Rytov approximationUnder Rytov approximation = TraveltimeResidual ObjectFunction Wavepath

Linearization = It is related to the causal generalized Radon transform ( Beylkin, 1985) Using geometrical approximationUsing geometrical approximation

Inverse GRT Partial Reconstruction Controls resolution and what model parts can be recovered

Inverse GRT Partial Reconstruction

Wavenumber rsrsrsrs rgrgrgrg Source Geophone r K

Spatial Resolution Limits Formula

source-receiver pairs where the wavepath visits r the wavepath visits r

Reflection Traveltime Tomography rsrsrsrs rgrgrgrg Source Geophone

Transmission Traveltime Tomography rsrsrsrs rgrgrgrg Source Geophone

Available Wavenumbers rsrsrsrs rgrgrgrg Transmission ReflectionSource Geophone

Outline ObjectiveObjective Inverse GRT and Resolution LimitsInverse GRT and Resolution Limits Numerical ExamplesNumerical Examples SummarySummary

Numerical Examples Crosswell ExperimentCrosswell Experiment Refraction TomographyRefraction Tomography Global TomographyGlobal Tomography

Crosswell Experiment X L (0, L/2) (X, L/2) (0, -L/2) (X, -L/2) r 0 (X/2, 0) Source Geophone

Crosswell Experiment A. Reflection Tomography the same as the migration-spatial-resolution the same as the migration-spatial-resolution limits for crosswell migration derived by limits for crosswell migration derived by Schuster (1996) in far-field approximation. Schuster (1996) in far-field approximation.

Crosswell Experiment B. Transmission Tomography The results are similar to Schuster (1996) for traveltime tomography in far-field approximation

Key Idea The velocity anomalies within the first- Fresnel zone or wavepath affect the traveltimeThe velocity anomalies within the first- Fresnel zone or wavepath affect the traveltime The intersection area of the wavepaths at the scatterer defines the spatial resolution limitsThe intersection area of the wavepaths at the scatterer defines the spatial resolution limits

Wavepath Intersection Transmission Example r s1 r g1 Fresnel Zone

Wavepath Intersection Transmission Example r s1 r g1 r g2 r s2

r s1 r g1 r s2 r g2 r s3 r g3 Wavepath Intersection Transmission Example

200 (m) 400 (m) 400 (m) 72m 44.7m C=3000 m/s f=300 Hz Wavepath Intersection Transmission Example

Numerical Examples Crosswell ExperimentCrosswell Experiment Refraction TomographyRefraction Tomography Global TomographyGlobal Tomography

Refraction TomographyS V1V1V1V1 V2V2V2V2 R The same as the result of Schuster (1995)

Numerical Examples Crosswell ExperimentCrosswell Experiment Refraction TomographyRefraction Tomography Global TomographyGlobal Tomography

1Hz Global Tomography (km) (km) (km/s) Core Mantle Scatterer Wavepath

Depth (km) Horizontal (km) 1Hz Global Tomography Resolution Limits (Depth=100km)

Depth (km) Horizontal (km) 1Hz Global Tomography Resolution Limits (Depth=300km)

Depth (km) Horizontal (km) 1Hz Global Tomography Resolution Limits (Depth=400km)

Depth (km) Horizontal (km) 1Hz Global Tomography Resolution Limits (Depth=800km)

Outline ObjectiveObjective Inverse GRT and Resolution LimitsInverse GRT and Resolution Limits Numerical ExamplesNumerical Examples SummarySummary

Summary Derived the inverse GRT and the spatial resolution formulasDerived the inverse GRT and the spatial resolution formulas We have Developed a practical means of estimating resolution limits for arbitrary velocity models and finite-frequency source dataDeveloped a practical means of estimating resolution limits for arbitrary velocity models and finite-frequency source data Reexamined whole-earth tomogramsReexamined whole-earth tomograms

Acknowledgment We thank the sponsors of the 1999 University of Utah Tomography and Modeling /Migration (UTAM) Consortium for their financial support.