Time vs Depth Migration Insensitive to v(z) model Sensitive to v(z) model Time migration uses best fit hyperbola Depth migration uses best guess moveout curve Incoherent summation if guess is wrong Coherent summation Time Migration Depth Migration CMP Gather time time

Depth Migration -> Time Migration We know 2z/c=T so m(x,z) = g d (g, 4[(x-g)/c] + (2z/c) ) 22 z 2-way vertical traveltime d (g, 4[(x-g)/c] + T ) M(x,T) = g 22 Depth Migration: Maps data into function(x,z) Time Migration: Maps data into function(x,T) m(x,z(T)) = g d (g, 4[(x-g)/c] + T ) 22

Time Migration for c(T) d (g, 4[(x-g)/c(T)] + T ) M(x,T) = g 2 2 Time Migration: Maps data into function(x,T) v1 v2 v3 v4 v5 v6 Tc(T) More generally, c(T) is a function of T!

MATLAB ZO Depth Migration d (g, ) xgxgxgxg m(x,z) = g for ixtrace=1:ntrace; for ixtrace=1:ntrace; for ixs=istart:iend; for ixs=istart:iend; for izs=1:nz; for izs=1:nz; r = sqrt(4*(ixtrace*dx-ixs*dx )^2+(2*izs*dx)^2); r = sqrt(4*(ixtrace*dx-ixs*dx )^2+(2*izs*dx)^2); time = 1 + round( r/c/dt ); time = 1 + round( r/c/dt ); mig(ixs,izs) = mig(ixs,izs)/r + data(ixtrace,time); mig(ixs,izs) = mig(ixs,izs)/r + data(ixtrace,time); end; end; end; end; Traveltime Loop over x in model Loop over z in model

MATLAB ZO Time Migration for ixtrace=1:ntrace; for ixtrace=1:ntrace; for ixs=istart:iend; for ixs=istart:iend; for iT=1:nT; for iT=1:nT; time = sqrt(4*([ixtrace*dx-ixs*dx]/c(iT))^2+(iT*dt)^2); time = sqrt(4*([ixtrace*dx-ixs*dx]/c(iT))^2+(iT*dt)^2); time = 1 + round( time/dt ); time = 1 + round( time/dt ); mig(ixs,iT) = mig(ixs,iT)/r + data(ixtrace,time); mig(ixs,iT) = mig(ixs,iT)/r + data(ixtrace,time); end; end; end; end; Traveltime Loop over x in model Loop over iT in model M(x,T) = g 22 d (g, 4[(x-g)/c(T)] + T ) Note: c(iT) or c(ixtrace,iT)

Time Migration vs Depth Migration Insensitive to c(z) model Time migration uses best fit hyperbola d (g, 4[(x-g)/c] + T ) M(x,T) = 2 2

Time Migration vs Depth Migration Insensitive to v(z) model Time migration uses best fit hyperbola d (g, 4[(x-g)/c] + T ) M(x,T) = 2 2

Time Migration vs Depth Migration Insensitive to v(z) model Time migration uses best fit hyperbola d (g, 4[(x-g)/c] + T ) M(x,T) = 2 2

Time Migration vs Depth Migration Insensitive to v(z) model Time migration uses best fit hyperbola d (g, 4[(x-g)/c] + T ) M(x,T) = 2 2

Time Migration vs Depth Migration Insensitive to v(z) model Sensitive to v(z) model Time migration uses best fit hyperbola Depth migration uses best guess moveout curve d (g, 4[(x-g)/c] + T ) M(x,T) = 2 2 d (g, t ) M(x,z) = gxgxgxgx Incoherent summation if guess is wrong Coherent summation Cheap: no ray tracing Expensive: ray tracing Uniform wavelet thickness Stretched wavelet thickness =c/f1/f Best focusing if v(x,z) correct Best focusing if v(x,z) really wrong Good focusing if v(x,z) smooth

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

Velocity Analysis CMP Time x T c d (g, 4[(x-g)/c] + T ) M(x,T) = 2 2 C(T)