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 Easy calculation of v(z) Expensive calculation of t(x,y,z) False Structures True Structures

Time Migration Depth Migration Stacked Section time time M(x,T) M(x,z) False structure Low-velocity zone zoneLow-velocity

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

Prestack Time Migration m(x,z) =  g,s g,s d (g,s, [(x-g)/c] + (z/c) 2 2 Depth Migration: Maps data into function(x,z) [(x-s)/c] + (z/c) ) [(x-s)/c] + (z/c) ) 22+ [(x-s)/c] + T [(x-s)/c] + T d (g,s, [(x-g)/c] + T ) M(x,T) =  g,s g,s 2222+

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 T c(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 Traveltime Table

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) 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 Traveltime Table

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

Summary: 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 Cheaper: no ray tracing More 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 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 Therefore, spend time to get v(x,y,z) Correct: Tomography, MVA, Waveform Inversion Conclusions

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

Nyquist Sampling Frequency & Aliasing Data Aliasing Problem  t > T/2 Aliasing Solution Low-pass filtering data Fancy Interpolation Finer time spacing digital signal High freq. masqerade as low freq.

Migration Aliasing Data Aliasing Problem  x > x /2 Aliasing Solution Low-pass filtering data Fancy Interpolation Finer rec-src spacing 2D dot product of migration Operator and d(g,t) No data aliasing: (dx/dt) min T min /2 >  x x /2 x /2

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