Green’s function retrieval by iterative substitution or inversion (?) of the Marchenko equation

Joost van der Neut(Delft University of Technology)

Acknowledgements

Kees Wapenaar, Evert SlobJan Thorbecke

Satyan Singh, Jyoti Behura, Roel Snieder

Ivan Vasconcelos

Filippo Broggini, Dirk-Jan van Manen

Bowen Guo, Jerry Schuster

Andrey Bakulin

Introduction

Input for Marchenko Redatuming

Data Multiple-freeWavelet-freeGhost-freeData

SourceFunction

SurfaceReflectivity

Data

1^^^^ ^

(Van Groenestijn & Verschuur, 2009;Lin and Herrmann, 2013)

EPSI output

Input for Marchenko Redatuming

Data Multiple-freeWavelet-freeGhost-freeData

SourceFunction

SurfaceReflectivity

Data

1

2

Focal point

^^^^ ^

(Van Groenestijn & Verschuur, 2009;Lin and Herrmann, 2013)

EPSI output

BackgroundGreen’s function

Marchenkoredatuming

X

gup

gdown

g0

The aim of Marchenko redatuming

gup gdown

= *

Retrieve e.g. by inversion

Xdatum

Redatuming below a complex overburden

Examples

R T0

Conventional image Marchenko image

Example 1

Example 2

Example 2 – Conventional image

Example 2 – Marchenko image

Model

Target area Image

Example 3 - Conventional

Model

Target area Image

Example 3 – Marchenko(with adaptive subtraction)

Example 4 - ConventionalVelocity Density Image

Red=

without multiples

Yellow=

with multiples

Yellow Red

7000m/s

0 4000kg / m3

0

Example 4 - MarchenkoVelocity Density Image

Red=

without multiples

Yellow=

Marchenko result

Yellow Red

7000m/s

0 4000kg / m3

0

Example 4 with erroneous (constant) velocity - ConventionalVelocity Density Image

Red=

without multiples

Yellow=

with multiples

Yellow Red

7000m/s

0 4000kg / m3

0

Example 4 with erroneous (constant) velocity – MarchenkoVelocity Density Image

Red=

without multiples

Yellow=

Marchenko result

Yellow Red

7000m/s

0 4000kg / m3

0

Example 5 - Sigsbee

Behura et al. (2014)

ConventionalMarchenko

Theory

The focusing function (Wapenaar et al., 2014)

Focal point

Focusing functionResponse Earth

Heterogeneous

The focusing function (Wapenaar et al., 2014)

Focal point

Focusing functionResponse Earth

The focusing function (Wapenaar et al., 2014)

Response

Heterogeneous

Focal point

Focusing functionEarth

Data acting on the focusing function

Reflection response

Heterogeneous

Focal point

Focusing function

Heterogeneous

Time-reversedFocusing function& Green’s function

Representation in matrix-vector notation

Time-reversal Focusingfunction

Green’sfunction

Convolutionwith data

Focusingfunction

Representation in matrix-vector notation

Time-reversal Focusingfunction

Green’sfunction

Convolutionwith data

Focusingfunction

Representation in matrix-vector notation

Time-reversal Focusingfunction

Green’sfunction

Convolutionwith data

Focusingfunction

Unknown Unknown

Exploiting causality

Time-reversedFocusing function

Green’s function

Exploiting causalityWindowfunction

10

Time-reversedFocusing function

Green’s function

Exploiting causality

Windowfunction

Focusingfunction

10

Timereversal

Exploiting causality

10

Windowfunction

Green’sfunction

Exploiting causality

10

01

Windowfunction

BackgroundGreen’s function

Green’sfunction

Green’s function representation

Apply

Intial Green’sfunction (model)

Window Convolutionwith data

Focusingfunction

TimeReversal

Green’s function representation

Apply

Intial Green’sfunction (model)

Window Convolutionwith data

Focusingfunction

TimeReversal

Velocity Density

7000m/s

0 4000kg / m3

0

Retrieved Green’s functions - 1D results from spgl1

Iterative solution(50 iterations)

spgl1 inversion(100 iterations)

Applications for Least-Squares Migration?

Least-Squares Migration

perturbation

Data =

Born approximation

perturbation

Data =

Using Marchenko-based Green’s functions?

perturbation

Data =

Marchenko mapping:

Discussion

Revisiting the problem

1. EPSI (inversion):

2. Focusing functionRetrieval (inversion):

3. Green’s functionRetrieval (forward):

4. Least-SquaresMigration(inversion): δδ