# time vs depth migration insensitive to v(z) model sensitive to v(z) model time migration uses best...

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• Time vs Depth MigrationInsensitive to v(z) modelSensitive to v(z) modelTime migration uses bestfit hyperbolaDepth migration uses bestguess moveout curve Incoherent summationif guess is wrongCoherent summationTime MigrationDepth MigrationEasy calculation of v(z)Expensive calculation of t(x,y,z)False StructuresTrue Structures

• Time MigrationDepth MigrationStacked SectiontimeM(x,T)M(x,z)False structureLow-velocity zoneLow-velocity zone

• Depth Migration -> Time MigrationWe know 2z/c=T soDepth Migration: Maps data into function(x,z)Time Migration: Maps data into function(x,T)

• Prestack Time Migrationm(x,z) = g,sd (g,s, [(x-g)/c] + (z/c) 22Depth Migration: Maps data into function(x,z) [(x-s)/c] + T

• Time Migration for c(T)d (g, 4[(x-g)/c(T)] + T ) M(x,T) = 22Time Migration: Maps data into function(x,T)More generally, c(T) is a function of T!

• MATLAB ZO Depth Migrationfor ixtrace=1:ntrace; for ixs=istart:iend; for izs=1:nz; r = sqrt(4*(ixtrace*dx-ixs*dx )^2+(2*izs*dx)^2); time = 1 + round( r/c/dt ); mig(ixs,izs) = mig(ixs,izs)/r + data(ixtrace,time); end; end;end; TraveltimeTraveltime Table

• MATLAB ZO Time Migrationfor ixtrace=1:ntrace; for ixs=istart:iend; for iT=1:nT; time = sqrt(4*([ixtrace*dx-ixs*dx]/c(iT))^2+(iT*dt)^2); time = 1 + round( time/dt ); mig(ixs,iT) = mig(ixs,iT)/r + data(ixtrace,time); end; end;end; Traveltimed (g, 4[(x-g)/c(T)] + T ) Note: c(iT) or c(ixtrace,iT)for ixtrace=1:ntrace; for ixs=istart:iend; for izs=1:nz; r = sqrt(4*(ixtrace*dx-ixs*dx )^2+(2*izs*dx)^2); time = 1 + round( r/c/dt ); mig(ixs,izs) = mig(ixs,izs)/r + data(ixtrace,time); end; end;end; TraveltimeTraveltime Table

• MATLAB ZO Time Migrationfor ixtrace=1:ntrace; for ixs=istart:iend; for iT=1:nT; time = sqrt(4*([ixtrace*dx-ixs*dx]/c(iT))^2+(iT*dt)^2); time = 1 + round( time/dt ); mig(ixs,iT) = mig(ixs,iT)/r + data(ixtrace,time); end; end;end; Traveltimed (g, 4[(x-g)/c(T)] + T ) Note: c(iT) or c(ixtrace,iT)

• Time Migration vs Depth Migration

• Time Migration vs Depth MigrationInsensitive to v(z) modelTime migration uses bestfit hyperbola

• Time Migration vs Depth MigrationInsensitive to v(z) modelTime migration uses bestfit hyperbola

• Time Migration vs Depth MigrationInsensitive to v(z) modelTime migration uses bestfit hyperbola

• Summary: Time Migration vs Depth MigrationInsensitive to v(z) modelSensitive to v(z) modelTime migration uses bestfit hyperbolaDepth migration uses bestguess moveout curve d (g, t ) M(x,z) = gx Incoherent summationif guess is wrongCoherent summationCheaper: no ray tracingMore expensive: ray tracingUniform wavelet thicknessStretched wavelet thickness1/fBest focusing if v(x,z) correctBest focusing if v(x,z) really wrongGood focusing if v(x,z) smooth

• Depth Migration in Deep GOM is only Way to Go if V(x,y,z) CorrectTherefore, spend time to get v(x,y,z) Correct: Tomography, MVA, Waveform InversionConclusions

• NMO Velocity AnalysisCMPTimexTcC(T)

• Nyquist Sampling Frequency & AliasingData Aliasing Problem Dt > T/2Aliasing SolutionLow-pass filtering dataFancy InterpolationFiner time spacing

digital signalHigh freq. masqerade as low freq.

• Migration AliasingData Aliasing Problem Dx > lx/2Aliasing SolutionLow-pass filtering dataFancy InterpolationFiner rec-src spacing

2D dot product of migration Operator and d(g,t)No data aliasing: (dx/dt)min Tmin/2 > Dx

• Operator Aliasing (m(x)=dot product data by mig. Operator)2D dot product of migration Operator and d(g,t)(dx/dt)min Tmin/2 > Dx