relations between reflection and transmission responses of 3-d inhomogeneous media kees wapenaar...
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Relations between reflection and transmission responses of 3-D inhomogeneous media
Kees Wapenaar
Department of Applied Earth Sciences Centre for Technical Geoscience Delft University of Technology
Dallas SEG 2003October 29
Claerbout, 1968 SEG, 2002‘Acoustic daylight imaging’
T R Berkhout, 1982‘Multiple elimination’
0R R
Imaging ‘from below’
0 0R R
R R Schuster, 2001‘Seismic interferometry’
0 0R T Herman, 1992Wapenaar, 1993‘True amplitude imaging’
Claerbout, 1968 SEG, 2002‘Acoustic daylight imaging’
T R Berkhout, 1982‘Multiple elimination’
0R R
Imaging ‘from below’
0 0R R
R R Schuster, 2001‘Seismic interferometry’
0 0R T Herman, 1992Wapenaar, 1993‘True amplitude imaging’
One-way reciprocity theorems
‘State A’
‘State B’
A
A
B
B
),(),( tptp ABBA xx
Reciprocity interrelates states A and B
‘State A’
‘State B’
A
A
B
B
),(),( ABBA PP xx
Reciprocity interrelates states A and B
‘State A’ ‘State B’
AP
AP
BP
BP
Downgoing waves
AP
AP
BP
BP
and upgoing waves
One-way reciprocity theorems
‘State A’ ‘State B’
Convolution-type propagation invariant:
AP
AP
BP
BP
D D
2{ }dA B A B
D
P P P P
x constant
One-way reciprocity theorems
‘State A’ ‘State B’
0D 0D
mDmD
AP
AP
BP
BP
AP
AP
BP
BP
Convolution-type reciprocity theorem:
One-way reciprocity theorems
0
2 2{ }d { }dm
A B A B A B A B
D D
P P P P P P P P
x x
‘State A’ ‘State B’
Correlation-type propagation invariant:
*AP *
AP
BP
BP
D D
* * 2{ }dA B A B
D
P P P P
x constant
One-way reciprocity theorems
‘State A’ ‘State B’
0D 0D
mDmD
BP
BP
*AP *
AP
BP
BP
0
* * 2 * * 2{ }d { }dm
A B A B A B A B
D D
P P P P P P P P
x x
Correlation-type reciprocity theorem:
*AP *
AP
One-way reciprocity theorems
‘State A’ ‘State B’
Derivation of relations between reflection and transmission responses
States A and B with or without free surface
‘State A’ ‘State B’
Derivation of relations between reflection and transmission responses
States A and B with or without free surface Convolution- or correlation-type reciprocity
‘State A’ ‘State B’
Derivation of relations between reflection and transmission responses
States A and B with or without free surface Convolution- or correlation-type reciprocity Sources at upper or lower boundary
Ax Bx
‘State A’
Ax Bx
‘State B’
States A and B without free surface Convolution-type reciprocity Sources at upper boundary
One-way source-receiver reciprocity
Ax Bx
‘State A’
Ax Bx
‘State B’
0 ( , )BR x x0 ( , )AR x x
0
2 2{ }d { }dm
A B A B A B A B
D D
P P P P P P P P
x x
( )A x - x ( )B x - x
0 0
Ax Bx
‘State A’
Ax Bx
‘State B’
0 ( , )A BR x x0 ( , )B AR x x
0 0( , ) ( , )B A A BR R x x x x
States A and B without free surface Convolution-type reciprocity Sources at lower boundary
Ax Bx Ax Bx
Ax Bx Ax Bx
0 ( , )A BR x x0 ( , )B AR x x
0 0( , ) ( , )B A A BR R x x x x
States A and B without free surface Convolution-type reciprocity Source A at lower and source B at upper boundary
Ax
Bx
Ax
Bx
Ax
Bx
Ax
Bx
0 ( , )A BT x x
0 ( , )B AT x x
0 0( , ) ( , )B A A BT T x x x x
Ax Bx
‘State A’
Ax Bx
‘State B’
States A and B with free surface Convolution-type reciprocity Sources at upper boundary
Ax Bx
‘State A’
Ax Bx
‘State B’
( , )A BR x x( , )B AR x x
( , ) ( , )B A A BR R x x x x
States A and B with free surface Convolution-type reciprocity Sources at lower boundary
Ax Bx Ax Bx
Ax Bx Ax Bx
( , )A BR x x( , )B AR x x
( , ) ( , )B A A BR R x x x x
States A and B with free surface Convolution-type reciprocity Source A at lower and source B at upper boundary
Ax
Bx
Ax
Bx
Ax
Bx
Ax
Bx
( , )A BT x x
( , )B AT x x
( , ) ( , )B A A BT T x x x x
Summary of one-way source-receiver reciprocity
Without free surface:
0 0( , ) ( , )B A A BR R x x x x
0 0( , ) ( , )B A A BR R x x x x
0 0( , ) ( , )B A A BT T x x x x
With free surface:
( , ) ( , )B A A BR R x x x x( , ) ( , )B A A BR R x x x x
( , ) ( , )B A A BT T x x x x
Ax Bx
‘State A’
Ax Bx
‘State B’
States A and B without free surface Correlation-type reciprocity Sources at upper boundary
From reflection to transmission
x x
Ax Bx Ax Bx
0 ( , )AT x x
x x
0 ( , )ABR x x
0 ( , )BT x x
0 ( , )BAR x x
* 20 0{ ( , )} ( , )d ( )
m
A B B A
D
T T
x x x x x x x
0
* 20 0{ ( , )} ( , )dA B
D
R R
x x x x x
0D 0D
mDmD
Ax Bx Ax Bx
0 ( , )AT x x
x x
0 ( , )ABR x x
0 ( , )BT x x
0 ( , )BAR x x
0D 0D
mDmD
Herman, 1992 Wapenaar and Herrmann, 1993 Application in inverse wave field extrapolation
Ax Bx
‘State A’
Ax Bx
‘State B’
States A and B with free surface Correlation-type reciprocity Sources at lower boundary
From transmission to reflection
x x
Ax Bx Ax Bx
x x
( ),AT x x ( ),BT x x
* 2{ ( , )} ( , )dm
A B
D
T T
x x x x x
( , ) ( )2 ReA B A B
R x x x - x
0D 0D
mDmD
1500 m/s
2000 m/s
Acoustic daylight imaging
Syncline model
Acoustic daylight imaging
* 2{ ( , )} ( , )dm
A B
D
T T
x x x x x( ),AT x x
Acoustic daylight imaging
*{ ( )} ( )obs A obs BT T x x( )AobsT x
Acoustic daylight imaging
Poster presentation Draganov et al.
Ax Bx
‘State A’
Ax Bx
‘State B’
State A without and state B with free surface Convolution-type reciprocity Sources at upper boundary
Multiple elimination
Ax Bx Ax Bx
( , )A BR x x0 ( , )B AR x x
0 ( , ) ( , )B A A BR R x x x x
0
20{ ( , ) ( , )}dA B
D
R R
x x x x x
Berkhout, 1982; Verschuur, Berkhout and Wapenaar, 1992 Fokkema and van den Berg, 1993, Van Borselen et al, 1996
Multiple elimination (van Borselen et al., 1996)
( , )A BR x x 0 ( , )B AR x x
Ax Bx
‘State A’
Ax Bx
‘State B’
State A without and state B with free surface Correlation-type reciprocity Sources at upper boundary
Seismic interferometry
Ax Bx Ax Bx
( , )A BR x x0 ( , )B AR x x
0
* 2
0( , ) ( ) { ( , )} ( , )dA B A B B
D
AR R R
x x x - x x x x x x
* 20{ ( , )} ( , )d
m
A B
D
T T
x x x x x
Schuster, 2001, EAGE
( , )A BR x x0
* 2
0{ ( , )} ( , )d
B
D
AR R
x x x x x
Seismic interferometry
Ax
Bx
‘State A’
Ax
Bx
‘State B’
States A and B without free surface Correlation-type reciprocity Source A at upper and source B at lower boundary
Reflection ‘from below’
x
x
Ax
Bx
Ax
Bxx
x0 ( , )AR x x
0 ( , )B AT x x
0 ( , )A BT x x
0 ( , )BR x x
* 20 0{ ( , )} ( , )d
m
A B
D
T R
x x x x x
0
* 20 0{ ( , )} ( , )dA B
D
R T
x x x x x
‘State A’ ‘State B’
Derivation of relations between reflection and transmission responses
States A and B with or without free surface
‘State A’ ‘State B’
Derivation of relations between reflection and transmission responses
States A and B with or without free surface Convolution- or correlation-type reciprocity
‘State A’ ‘State B’
Derivation of relations between reflection and transmission responses
States A and B with or without free surface Convolution- or correlation-type reciprocity Sources at upper or lower boundary
Claerbout, 1968 SEG, 2002‘Acoustic daylight imaging’
T R Berkhout, 1982‘Multiple elimination’
0R R
Imaging ‘from below’
0 0R R
R R Schuster, 2001‘Seismic interferometry’
0 0R T Herman, 1992Wapenaar, 1993‘True amplitude imaging’
One-way reciprocity theorems
Monitoring the universe by an array of telescopes
Monitoring the subsurface by an array of geophones
Combined monitoring the universe and the subsurface:Different applications; shared infrastructure
25% of antennas in central core
High speed data transport
Antenna cluster (Station)
Central processor
X
Shared infrastructure (1)
Shared infrastructure (2)
1 3 0 0 0 D a ta F a b ric s
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