d(r)d(r) = = m(x)m(x)G(G(ss|x)|x) G(x|G(x|rr))[[ ]]****

T=0T=0

Reverse Time MigrationReverse Time MigrationGeneralized Kirch. MigrationGeneralized Kirch. MigrationCalc. Green’s Func. By FD solvesCalc. Green’s Func. By FD solves

= dot product data with hyperbola= dot product data with hyperbolaGeneralized Kirchhoff kernelGeneralized Kirchhoff kernel

Convolution of G(Convolution of G(ss|x) with G(x||x) with G(x|rr))

QED:QED: RTM can now enjoy: RTM can now enjoy:

Anti-aliasing filterAnti-aliasing filter

Obliquity factorObliquity factor

Angle GathersAngle Gathers

UD SeparationUD Separation

Decomplexify back&forwardDecomplexify back&forward

felds according 2 tastefelds according 2 taste

Etc. etc.Etc. etc.

[[GG((ss|x)G(x|g)]* d(s|g) |x)G(x|g)]* d(s|g) s,g

1. RTM: 1. RTM: [{[{ } {} { } ]* d(s|g) } ]* d(s|g)

GG((ss|x)|x) GG((ss|x)|x) GG(x|g) (x|g) GG(x|g)(x|g)++ ++s,g==

dd (x) = (x) =

GG((ss|x)|x) GG(x|g)(x|g){ }s,g~~ ** d(s|g) d(s|g)

Super-wide Angle PhaseSuper-wide Angle Phase

Shift MigrationShift Migration

First Arrival FilterFirst Arrival Filter

Single Arrival KirchhoffSingle Arrival Kirchhoffw/o high-freq. appoxw/o high-freq. appox

Early Arrival FilterEarly Arrival Filter

Multiple Arrival KirchhoffMultiple Arrival Kirchhoffw/o high-freq. appoxw/o high-freq. appox

(or Super beam migration)(or Super beam migration)

Frechet DerivativeFrechet Derivative

True RTMTrue RTM

Phase Shift, Beam, Kirchhoff Migrations Phase Shift, Beam, Kirchhoff Migrations

are Special Cases of True RTMare Special Cases of True RTM

dsds

First Arrival FilterFirst Arrival Filter

& U p+Down filter& U p+Down filter

Example Example (Min Zhou, 2003)(Min Zhou, 2003)

Standard FD Wavefront G(s|x)Standard FD Wavefront G(s|x) Early Arrival FD Wavefront G(s|x)Early Arrival FD Wavefront G(s|x)

Standard RTM vs Early Arrival RTMStandard RTM vs Early Arrival RTM

Is Superresolution by RTM Achievable?

Tucson, Arizona Test

T u n n el a t lev e l 2(3 0 m fro m g ro u n d )

T u n n el a t lev e l 3(4 5 m fro m g ro u n d )

Sh

aft G rou n d S u rface3 0 m

1 5 m

S h o t lo ca tio n

R ece iv e r lo ca tio n

S h aft en tran ce

60 m

Poststack MigrationPoststack Migration

1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0X (m )

-0 .2

0

0 .2

0 .4

0 .6

0 .8

1

Nor

mal

ized

Am

pli

tud

e

D ire c t - F u ll A p ra tu reD ire c t - H a lf A p ra tu reS c a tte re rs - F u ll A p ra tu reS c a tte re rs - H a lf A p ra tu re

S u p er-reso lu tio n T est - P o in t # 1 4 a t X = 2 4 m

~Kirchhoff Mig.

~Scattered RTM

This is highest fruit on the tree..who dares pick it?This is highest fruit on the tree..who dares pick it?

(Hanafy et al., 2008)

Can Scatterers Beat the Resolution Limit?

Recorded Green’s functions G(s|x) divided into:

- Shot gathers with direct arrivals only

- Shot gathers with scattered arrivals only

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0

D ista n ce (m )

0 .5

0 .4

0 .3

0 .2

0 .1

0

Tim

e (m

s)

G reen 's F u n ctio nA fter b an d -p a ss filter

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0

D ista n ce (m )

0 .5

0 .4

0 .3

0 .2

0 .1

0

Tim

e (m

s)

G reen 's F u n ctio nA fter b an d -p a ss filter

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0

D ista n ce (m )

0 .5

0 .4

0 .3

0 .2

0 .1

0

Tim

e (m

s)

G reen 's F u n ctio nA fter b an d -p a ss filter