Gary Margrave Saleh Al-Saleh Hugh Geiger Michael Lamoureux ... · Saleh Al-Saleh Hugh Geiger Michael Lamoureux POTSI a. Outline Wavefield extrapolators and stability A stabilizing

The FOCIThe FOCI™™ method method of depth migrationof depth migration

Gary MargraveSaleh Al-SalehHugh Geiger

Michael Lamoureux

POTSI

aa

Outline

Wavefield extrapolators and stabilityA stabilizing Wiener filter

Dual operator tablesSpatial resamplingPost stack testingPre stack testing

FOCI

Wavefield Extrapolators

( )( )

( )111, , , 0,

2z T T

nik z ik x

T T Tnx z k z e e dkψ ω ϕ ωπ −

−−

⎡ ⎤= = ⎢ ⎥⎣ ⎦∫ i

The phase-shift extrapolation expression

( ), ,Tx zψ ω output wavefield

( ), 0,Tk zϕ ω= Fourier transform of input wavefield

Wavefield Extrapolators

( )ˆzik zne W z=

The phase-shift operator

wavelike

evanescent

2 22 2

2 2

2 22 2

2 2

,

,

T T

z

T T

k kv vk

i k kv v

ω ω

ω ω

⎧⎪⎪⎪ − >⎪⎪⎪=⎨⎪⎪⎪ − >⎪⎪⎪⎩

( ) ( )

( ) ( )

2 22 2

2 2

2 22 2

2 2

,

,

T TT T

z

T TT T

k kv x v x

k

i k kv x v x

ω ω

ω ω

⎧⎪⎪⎪ − >⎪⎪⎪⎪=⎨⎪⎪⎪ − >⎪⎪⎪⎪⎩

Wavefield Extrapolators

( )ˆzik zne W z=

The PSPI extension

Wavefield Extrapolators

( )2W z

wavenumbervω+

vω−

Wavefield Extrapolators

wavenumbervω+

vω−

( )2ˆphase W z⎡ ⎤

⎢ ⎥⎣ ⎦

Wavefield Extrapolators

In the space-frequency domain

( ) ( ) ( )1ˆ ˆ ˆ, , , 0, , , ,

nT T n T T Tx z x z W x x z v dxψ ω ψ ω ω−

= = −∫

where

( )( )

( ) ( )1

ˆ1

1 ˆˆ , , , ,2

T T Tn

ik x xn T T n T TnW x x z W k z e dkω ω

π −

− −−− = ∫ i

imaginary

real

meters

Wavefield ExtrapolatorsIn the space-frequency domain

Wavefield ExtrapolatorsBack to the wavenumber domain

wavenumber

( )2F WΩ2W

Wavefield Extrapolators

wavenumber

Back to the wavenumber domain

Stabilization by Wiener FilterTwo useful properties

( )ˆ ˆ ˆ, , , , , ,2 2n T n T n Tz zW k z W k W kω ω ω

⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜= ⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠Product of two half-steps make a whole step.

( ) ( )1 * 2 2ˆ ˆ, , , , ,n T n T xW k z W k z k kω ω− = >The inverse is equal to the conjugate

in the wavelike region.

Stabilization by Wiener Filter

A windowed forward operator for a half-step

( ) ( )2 2n nW z W z=Ω

( ) ( )1 ˆ2 2n n nW z WI F W zη− ⎡ ⎤• = ⎢ ⎥⎢ ⎥⎣ ⎦

Solve by least squares

0 2η≤ ≤

Stabilization by Wiener Filter

is a band-limited inverse for nWI ( )/ 2nW zBoth have compact support

( ) ( ) ( )* 2nF n n nW z WI W z W z= • ≈Form the FOCI™ approximate operator by

FOCI™ is an acronym for

Forward Operator with Conjugate Inverse.

Linear System to Solve

2,0 2,1 2, 2,0

2,1 2,0 2,1 2,12,0

2,1 2,0 2, 2,22,1

2, 2,1

2, 2,1

2,2,0

2,

0

0

0

0

p

p

p

p

m

m p

W W W DW W W DWI

W W W DWI

W W

W WWIW

D +

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ = ⎢⎢ ⎥⎢ ⎥ ⎢⎢ ⎥⎢ ⎥ ⎢⎢ ⎥⎢ ⎥ ⎢⎢ ⎥⎢ ⎥ ⎢⎢ ⎥⎣ ⎦⎢ ⎥ ⎢⎢ ⎥ ⎢⎢ ⎥ ⎣ ⎦⎣ ⎦

⎥⎥⎥⎥⎥⎥⎥

Properties of FOCI operator

( )inv nn length WI= ( )( )/ 2for nn length W z=

( )( ) 1nF op for invlength W z n n n= = + −Then

Let

Properties of FOCI operator

invn determines stability.

forn determines phase accuracy.

1.5inv forn n≈Empirical observation:

Properties of FOCI operator

Amount of evanescent filtering is inversely related to stability

0 no evanescent filtering ( 1000 steps)1 half evanescent filtering ( 100 steps)2 full evanescent filtering ( 50 steps)

η⎧⎪⎪⎪⎪= ⎨⎪⎪⎪⎪⎩

∼∼∼

Operator tablesSince the operator is purely numerical, migration proceeds by construction of operator tables.

minmin

maxk

vω=

( )minnFW kmink( )minnFW k k+∆mink k+∆

( )maxnFW kmaxk

maxmax

mink

vω=

( )k

mean vω∆∆ =

( )min 2nFW k k+ ∆min 2k k+ ∆

Operator tablesWe construct two operator tables for small and large η. The small η table is used most of the time, with the large η being invoked only every nth step.

0 no evanescent filtering ( 1000 steps)1 half evanescent filtering ( 100 steps)2 full evanescent filtering ( 50 steps)

η⎧⎪⎪⎪⎪= ⎨⎪⎪⎪⎪⎩

∼∼∼

Operator Design Example

Improved Operator Design

Spatial Resampling

( ) 12 x −∆( ) 12 x −− ∆ Wavenumber

Freq

uenc

y

Wavelike

minxk vω =minxk vω =−

maxω

Spatial Resampling

( ) 12 x −∆( ) 12 x −− ∆ Wavenumber

Freq

uenc

y

In red are the wavenumbers of a 7 point filter

Spatial Resampling

( ) 12 x −∆( ) 12 x −− ∆

Freq

uenc

y

Downsampling for the lower frequencies uses the filter more effectively

( ) 12 x −′∆( ) 12 x −′− ∆

Spatial Resampling

( ) 12 x −∆( ) 12 x −− ∆

Freq

uenc

y

Spatial resampling is done in frequency “chunks”.

Operator in Space

Improved Operator in Space

Run Time Experiment

PS slope 1.07FOCI slope 1.03

FOCI slope 1.05

Slopes for last three points

Run Times log-log Scale

Phase Shift Impulse Response

FOCI Impulse Responsenfor=7, ninv=15, (21 pt), no spatial resampling

FOCI Impulse Responsenfor=7, ninv=15, (21 pt), with spatial resampling

Marmousi Velocity Model

Exploding Reflector Seismogram

FOCI Post-Stack Migrationnfor=21, ninv=31, nwin=0

FOCI Post-Stack Migrationnfor=21, ninv=31, nwin=51

FOCI Post-Stack Migrationnfor=21, ninv=31, nwin=21

FOCI Post-Stack Migrationnfor=7, ninv=15, nwin=0

FOCI Post-Stack Migrationnfor=7, ninv=15, nwin=7

Marmousi Velocity Model

FOCI Pre-Stack Migrationnfor=21, ninv=31, nwin=0, deconvolution imaging condition

FOCI Pre-Stack Migrationnfor=21, ninv=31, nwin=0, deconvolution imaging condition

FOCI Pre-Stack MigrationShot 30

FOCI Pre-Stack MigrationShot 30

FOCI Pre-Stack MigrationStack +50*Shot 30

Detail of Pre-Stack Migration

Marmousi Reflectivity Detail

Marmousi Velocity Model

FOCI Pre-Stack MigrationVelocity model convolved with 200m smoother

FOCI Pre-Stack MigrationVelocity model convolved with 400m smoother

FOCI Pre-Stack MigrationExact Velocities

FOCI Pre-Stack MigrationExact Velocities

FOCI Pre-Stack MigrationVelocities smoothed over 200m

FOCI Pre-Stack MigrationVelocities smoothed over 400m

Detail of Pre-Stack MigrationExact Velocities

Detail of Pre-Stack MigrationVelocities smoothed over 200m

Detail of Pre-Stack MigrationVelocities smoothed over 400m

Run times

Full prestack depth migrations of Marmousi on a single 2.5GHz PC using Matlab code.

20 hours for the best result

1 hour for a usable result

ConclusionsExplicit wavefield extrapolators can be made local and

stable using Wiener filter theory.

The FOCI method designs an unstable forward operator that captures the phase accuracy and

stabilizes this with a band-limited inverse operator.

Reducing evanescent filtering increases stability.

Spatial resampling increases stability, improves operator accuracy, and reduces runtime.

The final method appears to be ~O(NlogN).

Very good images of Marmousi have been obtained.

Research DirectionsBetter phase accuracy.

Extension to 3D.

Extension to more accurate wavefield extrapolation schemes.

Migration velocity analysis.

Development of C/Fortran code on imaging engine.

Acknowledgements

Sponsors of CREWES

Sponsors of POTSI

NSERC

MITACS

PIMSA US patent application has been made for the FOCI process.

We thank:

Detail of Pre-Stack Migration

FOCI Pre-Stack Migrationnfor=21, ninv=31, nwin=0, deconvolution imaging condition

FOCI Pre-Stack Migrationnfor=21, ninv=31, nwin=0, deconvolution imaging condition