Virtual Source Imaging vs Interferometric Imaging Gerard T. Schuster, Andrey Bakulin and Rodney Calvert

Outline Data Illustration Summary Theory

D(s’|s) s U(g’|s’) s’g’ U(g’|s) = G(g’|s’) D(s’|s)ds’ Greens Thm: Every pt along well acts as a secondary source Greens Function for s & g in well u = Dg u =[D*D] D*g G(g’|s’) ~ U(g’|s)D(s’|s)*ds Redatumed data for s & g in well W D(s’|s) g D*u (g, M,s)= d( g |s) d(M|s)* s,s

Outline Data Illustration Summary Theory

Time (s) Depth (ft) Raw Data(CRG15) D(s’|s) U(g’|s) G(g’|s’) ~ U(g’|s)D(s’|s)*ds G(g’|s’)

Outline Data Illustration Summary Theory

Summary VSP Direct VSP Ghost u = Dg U(g’|s) = G(g’|s’) D(s’|s)ds’ 1. CDP Refl. g =[D*D] D*u 2. G(g’|s’) [D*D] ~ 1/W Interferometry Redatuming [D*D] ~ 1/D Virtual Source Readtuming

Time (s) Depth (ft) Ghosts

Time (s) Depth (ft) Exxon Raw Data(CRG15)

Time (s) Depth (ft) Ghosts (Exxon) D(g’|s)

Time (s) Depth (ft) Primary(Exxon)

524 Trace No. Time (s) xcorr data (muted) Time (s) Trace No. Exxon CSG 25 Raw data (muted) Master trace

Depth (ft) X (ft) Xcorr. mig

Uninteresting Part of Medium of Medium Specular Reflection Time Fermat’s Principle: T+T – T >0 T – T > -T ghostdiffraction s min(T – T) = -T s Diffraction T TT

Uninteresting Part of Medium of Medium Specular Reflection Time Fermat’s Principle: T+T – T >0 T – T > -T ghostdiffraction s min(T – T) = -T s Diffraction T TT

Uninteresting Part of Medium of Medium Specular Reflection Time Diffraction Fermat’s Principle: T+T – T >0 T – T > -T ghostdiffraction min(T – T) = -T s s

Uninteresting Part of Medium of Medium Specular Reflection Time Diffraction Fermat’s Principle: T+T – T >0 T – T > -T ghostdiffraction min(T – T) = -T s s

Uninteresting Part of Medium of Medium Time Minimum Time Difference = Specular Reflection Time Fermat’s Principle: T+T – T >0 T – T > -T ghostdiffraction min(T – T) = -T s

Uninteresting Part of Medium of Medium Time Fermat’s Principle: T+T – T >0 T – T > -T ghostdiffraction min(T – T) = -T s

Summary1. Fermat’s Principle: min(T – T) = -T 2. No need to know velocity model or source location. 3. VSP Data  CDP Data + Tomography+Mig. 4. Redatums buried sources to surface

Outline Target Oriented Fermat’s Interferometric Principle Fermat’s Interferometric Principle Numerical Example Interferometry Background

Uninteresting Part of Medium of Medium VSP Tomostatics Problem: From VSP Data Find Full Coverage Velocity Tomogram

Uninteresting Part of Medium of Medium VSP Tomostatics Problem: From VSP Data Find Full Coverage Velocity Tomogram

Fermat’s Principle: T - T > 0 multmultsgsg = When is there equality? Fermat’s Interferometric Principle For VSP Multiple Tomography m 0 m 3.5 km/s 1.4 km/s Weathering zone VSP Data: 120 shots on surface, 120 receivers in well Goal: Determine weathering zone velocity by interferometric tomography

Fermat’s Principle: T - T > 0 multmultsgsg = When is there equality? Fermat’s Interferometric Principle For Multiple Tomography m 0 m 3.5 km/s 1.4 km/s Weathering zone min(T – T) = T TT

Fermat’s Principle: T - T > 0 multmultsgsg = When is there equality? Fermat’s Interferometric Principle For Multiple Tomography m 0 s s Time (s) Ghost Reflection min(T – T) = T Depth (m)

Fermat’s Principle: T - T > 0 multmultsgsg = When is there equality? Fermat’s Interferometric Principle For Multiple Tomography m 0 s s Time (s) Ghost Reflection Primary Reflection min(T – T) = T = Depth (m) X (m)

Fermat’s Principle: T - T > 0 multmultsgsg = When is there equality? Fermat’s Interferometric Principle For Multiple Tomography m 0 s s 3.5 km/s 1.4 km/s Time (s) Ghost Reflection Primary Reflection Interfer. Reflection min(T – T) = T Depth (m) X (m)

Fermat’s Principle: T - T > 0 multmultsgsg = When is there equality? Fermat’s Interferometric Principle For Multiple Tomography m 1.35 km/s Velocity (km/s) 1.65 km/s Actual Interferometric min(T – T) = T

VSP Summary TT direct prim gggggggg sgsgsgsg - T > - mult sg min( ) g VSP  CDP traveltimes 3.Removes rec. statics & multiple tomo. Tcdp(s,g) =min[min( Tdirect(g,:) - Tmult(s,:) )] s g TTTg g

Outline Target Oriented Fermat’s Interferometric Principle Fermat’s Interferometric Principle Numerical Example Interferometry Background

Uninteresting Part of Medium of Medium Specular Reflection Fermat’s Principle: T+T – T >0 ghostdiffraction s Diffraction

Uninteresting Part of Medium of Medium Specular Reflection Fermat’s Principle: T+T – T >0 reflectiondiffraction s Diffraction Target Oriented Interferometric Tomography

Acknowledgments UTAM sponsors Exxon for 2-D field data J. Claerbout + J. Rickett II evolved from daylight imaging

Earthquake Data typically Use Direct Waves => Tomography Uninteresting Part of Medium of Medium Time Direct Direct Problem: Uninteresting Parts of Medium Distort Tomogram Goal: Transform Traveltimes into Primary Reflection Traveltimes Specular Reflection Basin

Uninteresting Part of Medium of Medium Direct Specular Reflection Time Goal: Transform Traveltimes into Primary Reflection Traveltimes

Fermat’s Principle: T - T > 0 multmultsgsg = When is there equality? Fermat’s Interferometric Principle For Multiple Tomography m 0 s s 3.5 km/s 1.4 km/s Time (s) Ghost Reflection Primary Reflection Interfer. Reflection min(T – T) = T

CDP Summary TT prim prim sgsgsgsg gggggggg - T > - mult sg min( ) s Extend CDP sources laterally 3.Removes 1/3 statics & multiple tomo. Tdatum(gp,g) =min[min( Tprim(:,gp) + Tmult(:,g) )] s g g TTT

Uninteresting Part of Medium of Medium Specular Reflection Fermat’s Principle: T+T – T >0 reflectiondiffraction s Diffraction