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Wave-equation MVA by inversion of differential image perturbations Paul Sava & Biondo Biondi Stanford University SEP [email protected] Motivation [email protected] Wave-equation MVA (WEMVA) • Band-limited • Multi-pathing • Resolution • Born approximation – small anomaly • Rytov approximation – phase unwrapping [email protected] Wave-equation MVA (WEMVA) • WE tomography – data space • WE MVA – image space [email protected] Outline 1. 2. 3. 4. WEMVA overview Born image perturbation Differential image perturbation Example [email protected] A tomography problem min Δ s Δ q Ls q L Traveltime MVA Wave-equation tomography Wave-equation MVA t d R traveltime data image ray field wavefield wavefield [email protected] WEMVA: main idea Δ W W W0 s 0 Δs s0 W0 U e i s 0 W Ue i s 0 i s [email protected] Born approximation ΔW W0 e i Δs 1 ΔW W0 i Δs e e i Δ R Ls 1 i i [email protected] WEMVA: objective function Linear WEMVA operator slowness perturbation min Δ s Δ R Ls image perturbation (known) slowness perturbation (unknown) image perturbation [email protected] WEMVA: objective function min Δ s Δ R Ls Traveltime MVA Wave-equation tomography Wave-equation MVA t d R [email protected] Fat ray: GOM example [email protected] Outline 1. 2. 3. 4. WEMVA overview Born image perturbation Differential image perturbation Example [email protected] “Data” estimate min Δ s Δ R Ls Traveltime MVA Wave-equation tomography Wave-equation MVA t d R ray tracing data modeling residual migration [email protected] Prestack Stolt residual migration R S r R0 • Background image • Velocity ratio R0 r r R0 R [email protected] Prestack Stolt residual migration R R R0 • Image perturbation r R0 R [email protected] Born approximation [email protected] Residual migration: the problem Correct velocity Incorrect velocity Zero offset image Zero offset image Angle gathers Angle gathers [email protected] Born approximation [email protected] Outline 1. 2. 3. 4. WEMVA overview Born image perturbation Differential image perturbation Example [email protected] Differential image perturbation Image difference Image differential R S r R0 R0 Rˆ dS r dr R0 r r 1 Computed Measured [email protected] Differential image perturbation R Rr Rˆ dS r dr R0 r R S r R0 R0 r 1 R0 r r0 r [email protected] Phase perturbation f 3p 2p p 0 r p 2p [email protected] Differential image perturbation [email protected] Born approximation [email protected] Example: background image Zero offset image Background image Angle gathers [email protected] Example: differential image Zero offset image Differential image Angle gathers [email protected] Example: slowness inversion Image perturbation Slowness perturbation [email protected] Example: updated image Updated image Updated slowness [email protected] Example: correct image Correct image Correct slowness [email protected] Outline 1. 2. 3. 4. WEMVA overview Born image perturbation Differential image perturbation Example [email protected] Field data example • North Sea depth – Salt environment – Subset location – One non-linear iteration • • • • • Migration Residual migration Slowness inversion Slowness update Re-migration (background image) (image perturbation) (slowness perturbation) (updated slowness) (updated image) [email protected] depth depth location [email protected] depth velocity ratio velocity ratio [email protected] depth r 1 r 1 r 1 depth location [email protected] depth location location [email protected] depth location location [email protected] depth depth location [email protected] depth depth location [email protected] Summary • MVA – Wavefield extrapolation methods – Born linearization – Differential image perturbations • Key points – Band-limited (sharp velocity contrasts) – Multi-pathing (complicated wavefields) – Resolution (frequency redundancy) [email protected] [email protected] MVA information (a) Traveltime MVA • Wave-equation MVA • Offset focusing (flat ADCIG) Offset focusing (flat ADCIG) g x g x z z [email protected] MVA information (b) Traveltime MVA • Wave-equation MVA • • Offset focusing (flat ADCIG) Offset focusing (flat ADCIG) Spatial focusing g x g x z z [email protected] MVA information (c) Traveltime MVA • Wave-equation MVA • • • Offset focusing (flat ADCIG) Offset focusing (flat ADCIG) Spatial focusing Frequency redundancy wlow w g whigh w g [email protected] WEMVA cost reduction • Full image • Normal incidence image – Offset focusing – Spatial focusing – Frequency – Spatial focusing – “fat” rays wlow w g whigh w g [email protected] Another example [email protected] Example: correct model Zero offset image Angle gathers [email protected] Example: background model Zero offset image Angle gathers [email protected] Example: correct perturbation Zero offset image Angle gathers [email protected] Example: differential perturbation Zero offset image Angle gathers R dRr dr r r 1 [email protected] Example: perturbations comparison Correct Difference Differential [email protected] Example: differential perturbation Zero offset image Angle gathers [email protected] Example: difference perturbation Zero offset image Angle gathers [email protected] Example: updated model Zero offset image Angle gathers [email protected] Example: correct model Zero offset image Angle gathers [email protected]