1 -

Multiple Removal with Local Plane Waves

Dmitri Lokshtanov

2 -

Content

• Motivation

• WE multiple suppression operator

• Fast 2D/3D WE approach for simple sea-floor

• 2D/3D WE approach for irregular sea-floor

• Conclusions

3 -

Motivation

• Seismic processing and imaging - main challenges:

− Velocity model building for sub-salt and sub-basalt imaging

− Removal of multiples from strong irregular boundaries

Near offset section (no AGC)

Depth migration with water velocity

Input shot gathers (no AGC)

8 -

Multiple suppression

• For multiples from complex boundaries the methods based on periodicity or kinematic discrimination usually don’t work or are not sufficient.

• In such cases the main demultiple tools are based on the Surface Related Multiple Elimination (SRME) or Wave-Equation (WE) techniques.

9 -

SRME (Berkhout, 1982; Verschuur, 1991) – advantages and limitations

• Does not require any structural information. Predicts all free-surface multiples

• As a rule becomes less efficient with increased level of interference of multiples of different orders

• Requires the same dense sampling between sources as between receivers

• Noise in data and poor sampling significantly degrade the prediction quality

• Missing traces required by 3D SRME are reconstructed with least-square Fourier or Radon interpolation; residual NMO correction; DMO/inverse DMO; migration/demigration

10 -

WE approach versus SRME

• SRME is the method of preference for data from areas with deep sea-floor, especially when a thick package of strong reflectors is present below the sea-floor

• WE approach is especially efficient when the main free-surface multiples are just ‘pure’ water-layer multiples and peg-legs. Gives usually better results than SRME when several orders of multiples are involved

• 3D WE approach has less sampling problems than 3D SRME and it gives a flexiblility in methods for wavefield extrapolation depending on complexity of structure

The operator Pg transforms the primary reflection event recorded at receiver 1 into the

multiple event recorded at receiver 2 (Wiggins, 1988; Berryhill & Kim, 1986). The operator Pg transforms the primary reflection event recorded at receiver 1 into the

multiple event recorded at receiver 2 (Wiggins, 1988; Berryhill & Kim, 1986).

12 -

Principles of WE approach

where,)1( :legs-peg side-receiver and multiples pure Remove 1. 1 DPF g

operator.ion extrapolat side-receiver theis data;input are gPD

side-source theis where,)1( :legs-peg side-source Remove 2. 12 PFPF ss

extrapolation operator.

,)1)(1( :multiplelayer -erorder wat-first for theCorrect 3. wsgs DPDPPF

bottom.- water thefrom reflectionprimary the is where wD

13 -

Adaptive subtraction of predicted multiples

)1(,)1)(1( wsgs DPDPPF We apply the ‘scaled version’ of (1) trace by trace to the tau-p transformed CMP or CS gathers. For each p-trace the operator has the form:

)2(),()()()()()()()( tdtrtdtrtdtrtdtf sgsgssgg

results the and data input thefor traces-p are)(),(),(),(where tdtdtdtd sgsg

of extrapolation through the water-layer from the receiver-side, source-side (of muted input data) and source-side after receiver-side respectively.

14 -

Wave-equation approach – main features

• All predicted multiples are split into 3 terms, where each term requires the same amplitude correction

• All source-side and receiver-side multiples of all orders are suppressed simultaneously in one consistent step

• The prediction and the adaptive subtraction of multiples are performed in the same domain

• Fast version (WEREM) for a simple sea-floor. Slower version for irregular sea-floor

15 -

Why in the tau-p domain

• Easier to apply antialiasing protection

• No problems with muting of direct arrival

• Easier to define ‘multiple’ zone of tau-p domain and mute it away

• Estimated reflection coefficients are explicitly angle dependent

2D WEREM – prediction of multiples - 1

T h e i n p u t R a d o n t r a n s f o r m e d C M P g a t h e r s ),( ypD c a n b e r e p r e s e n t e d a s f o l l o w s :

,exp),(2

),( ddd dpypippRypD

( 3 )

w h e r e .2,2 dsdg pppppp F o r ‘ l o c a l l y ’ 1 D w a t e r - b o t t o m a n d a r b i t r a r y

2 D s t r u c t u r e b e l o w i t t h e r e s u l t s o f r e c e i v e r - s i d e e x t r a p o l a t i o n ),( ypD g i s :

dgddg dphqypippRypD )2(exp),(2

),(

( 4 )

,2)(exp),(2

dxdphqxypixpD dgd

w h e r e h i s t h e ‘ l o c a l ’ w a t e r - b o t t o m d e p t h , w h i l e q g i s t h e v e r t i c a l s l o w n e s s .

2D WEREM – prediction of multiples - 2

For each pair x , y the sta tionary po in t stdp corresponds to a sim ple re la tion :

)5(.2andsinwhere stdgggg pppcptghyx

G eom etrica l illustra tion o f the resu lt (5 ). x and y are C M P positions o f the p rim ary and

ghRRyx tg)(5.0 :by related areThey ly.respective events multiple 21 .

w ater b ottom

R 2 R 1 SC M Py x

*

g

Velocity model used to generate synthetic FD data

Constant P sections (angle at the surface is about 3º)

Input After Werem After Remul

Constant P sections (angle at the surface is about 15º)

Input After Werem After Remul

Velocity model 2 used to generate synthetic FD data

Constant P sections (angle at the surface is about 3º)

Input After Werem

m. residual

T. Shetland

T. Draupne

T. Brent

Stack before multiple suppression Stack after Werem

Constant P sections (angle at the surface is about 10º

Input After Werem

Stack before multiple suppression (left) and after Werem multiple suppression (right).The pink line shows the expected position of the first-order water-layer peg-leg from theTop Cretaceous (black line). The multiple period is about 140 msec.

Constant P sections (angle at the surface is about 8º

Input After Werem multiple suppression Difference

raw stack

stack after WEREM

500 m input WEREM

4000 m input WEREM

Improving the results - local prediction / subtraction of multiples

• Within the same prediction term, for the same CMP and the same p we have events reflected at different positions along the water bottom

• Inconsistency between prediction and subtraction in case of rapid variation of sea-floor reflectivity

• The problem is partly solved by applying adaptive subtraction in different time windows

• Or by making prediction dependent on both p and offset (window)

3D WEREM – basic features

• 3D data can be represented as a sum of plane waves with different vertical angles and azimuths from the source-side and receiver-side.

• Current quasi 3D marine acquisition does not allow full 4D decomposition

• Decomposition uniquely defines the direction of propagation from the receiver-side and is an integral over crossline slownesses from the source-side

• The result of decomposition are used for exact prediction of multiples from the receiver-side and approximate prediction from the source-side

• The approximation is that the crossline slowness from the source-side is the same as from the receiver-side (the same azimuth for 1D structures). The approximation allows us to mix data for flip flop shooting

D e c o m p o s i t i o n o f p o i n t s o u r c e d a t a i n t o p l a n e w a v e s ( W e y l I n t e g r a l ) :

,

exp

2exp

12

2

yxz

zyx dpdpiq

zqypxpi

c

Ri

R

w h e r e .0Im,1 2

1

222

zyxz qpp

cq

R e f l e c t i o n f r o m a 3 D E a r t h a s a s u m o f p l a n e w a v e s

)increasing ofd irection the tooppositeare,ofd irection(positive sxrxs xpp

,ex p),,(

2

ex p),(),,,(~

2

),,0,(

3

3

4

4

yrxrxsryrrxrsxsxsyrxr

yrxrysxsryrrxrsxsysxsysxsyrxr

rrss

d pd pd pypxpxpipppR

d pd pd pd pypxpxpippSppppR

yxyxd

.),,,(

~),(

2),,(wh ere ysysxsyrxrysxsxsyrxr d pppppRppSpppR

R a d o n t r a n s f o r m e d C M P g a t h e r s :

.,2

where

,exp2

)2

()2

(exp2

),,(

3

3

3

3

xrxsdxsxr

yrxrxsyrd

yrxrxsyrxrxs

ppppp

p

dpdpdpypxpphiR

dpdpdpyph

xph

xpiRyhxd

.exp),,(and

,exp2

),,(2

2

dydxypxpiDpppR

dpdpypxpiRypxD

yrdyrd

yrdyrd

3 D p r e d ic t io n f r o m t h e r e c e iv e r - s id e :

.sin)~(0,cos)~(0

:conditionphaseStationary

.2,,1

where

,2)~()~(

.2)~()~(exp),,(

2~~exp2

)~,,~(

21

222

1

2

2

2

2

rryr

rrd

dxryrxrrrr

ryrd

yrdryrd

yrdryrdr

tgzyyp

tgzxxp

pppppppc

q

zqyypxxp

dydxdpdpzqyypxxpiyxpD

dpdpzqypxpiRypxD

37 -

water bottom

R2 R1 SCMPy x

*

g

I l l u s t r a t i o n o f t h e s t a t i o n a r y p h a s e r e s u l t : x a n d y a r e C M P p o s i t i o n s o f t h e p r i m a r y a n d m u l t i p l e e v e n t s r e s p e c t i v e l y . T h e y a r e r e l a t e d b y :

gtghRRyx )(5.0 21

Approxim ate 3D prediction from the source-side :

.sin)~(0,cos)~(0

:conditionphaseStationary

.and2,,1

where

,2)~()~(

.2)~()~(exp),,(

2~~exp2

)~,,~(

21

222

1

22

2

2

ssyr

ssd

yrysd

xsysxssss

syrd

yrdsyrd

yrdsyrds

tgzyyp

tgzxxp

pppppppppc

q

zqyypxxp

dydxdpdpzqyypxxpiyxpD

dpdpzqypxpiRypxD

Input constant P section (small angles)

Predicted multiples – R-side (small angles)

Input constant P section (small angles)

Constant P section – after prediction / subtraction (small angles)

Difference (Input – 3D WEREM), small angles

Input constant P section (larger angles)

Constant P section – after 3D WEREM (larger angles)

46 -

Werem - conclusions

• Very efficient when the main assumptions are met: strongest multiples are water-layer multiples and peg-legs and the sea-floor is simple

• Very fast - each predicted p trace is simply obtained as a sum of time-delayed input traces with the same p from the neighbour CMPs

47 -

WE for irregular sea-floor

• Kinematic prediction of multiples (extrapolation through the water layer) takes into account coupling between incident and reflected / scattered plane waves with different slownesses

• Both multiple reflections and diffractions are predicted

• The procedure starts from the Radon transformed CS gathers (no interleaving is required)

• In 3D exact prediction from the receiver side; approximate prediction from the source side

48 -

2D prediction of multiples from the receiver side for irregular sea-floor

1 . E x t r a p o l a t e R a d o n t r a n s f o r m e d C S g a t h e r ),,( sr xpD d o w n t o t h e s e a - f l o o r :

,)()(exp),(2

),(, rrsrsrs dpxzqxxpixpDxxzxW

2 . C a l c u l a t e t h e a m p l i t u d e ),( sscg xpD o f t h e r e f l e c t e d / s c a t t e r e d p l a n e w a v e w i t h

s l o w n e s s scp ( W e n z e l e t a l . , 1 9 9 0 )

.)()(exp1),(,),( dxxzqxxpiq

p

dx

dzxxzxWxpD scssc

sc

scssscg

49 -

2D prediction of multiples from the source side for irregular sea-floor

1 . U s e F F T t o d e c o m p o s e t h e i n p u t d a t a ),,( sr xpD i n t o c o n t r i b u t i o n s w i t h

d i f f e r e n t p r o p a g a t i o n a n g l e s ( w a v e n u m b e r s ) f r o m t h e s o u r c e s i d e :

,exp),,(),,( sssrr dxikxxpDkpR

w h e r e t h e s o u r c e - s i d e w a v e n u m b e r sk i s d e f i n e d a s : kpk rs .

2 . E x t r a p o l a t e t h e r e s u l t s o f d e c o m p o s i t i o n d o w n t o t h e s e a - f l o o r :

.)(exp),(2

1),(, dkxzkxkikpRpxzxW szsrr

3 . C a l c u l a t e r e f l e c t e d / s c a t t e r e d r e s p o n s e s f o r e a c h k a n d t h e n u s e i n v e r s e F F T t o d e f i n e ),,( srs xpD . A l l s t e p s a r e p e r f o r m e d i n a d o u b l e l o o p o v e r p r a n d o v e r .

Velocity model for FD modelling

Input P-section (zero angle) Receiver-side prediction

Input P-section (zero angle) Receiver-side prediction

Input P-section (20 degrees) Receiver-side prediction Source-side prediction

Input CS gather After prediction + subtraction

Input CS gather After prediction + subtraction

Raw stack with final velocity / mute libraries

Stack after WE + VF

Raw CMP (no AGC) CMP after WE + VF (no AGC)

Input Constant P section R-side prediction

Input Constant P section R-side prediction

Input Constant P section After adaptive subtraction

62 -

3D prediction from the receiver-side - 1

Decompose the recorded CS data into plane waves and then extrapolate each plane wave down to the sea-floor:

.1~,

1where,)(~)(

pointstationaryForion.approximatphasestationarythebycalculatedisintegralInner

)(expexp),(2

exp),(2

),,(

21

22

21

222

2

122

2

2

2

2

xzyxzrzzry

rxyzryxrx

yxzyxyx

pc

qppc

qyyzqzqyyp

dydpdpzqyypixpiypD

dpdpzqypxpippRzyxW

63 -

3D prediction from the receiver-side - 2

Extrapolate the wavefield along the sea-floor W(x,y) up to the free-surface and calculate Radon transformed CS gathers after prediction:

ion.approximatphasestationarythebycalculatedisintegralinner above, As

),()(expexp),(2

),(

dydxdpyxzqyypiCxpiyxW

ypD

yzryx

rxg

Model for 3D ray tracing of primaries and sea-floor multiples

Modelled CS gathers

primary

peg-legs

Radon transformed CS gathers

Constant P sections (small angle) for line with crossline offset 250m.

Input 3D prediction 2D prediction

Input 3D prediction 2D prediction

Constant P sections (larger angle) for line with crossline offset 250m.

Constant P sections (small angle) for line with crossline offset 250m.

Input quasi 3D prediction 2D prediction

Constant P sections (larger angle) for line with crossline offset 250m.

Input quasi 3D prediction 2D prediction

71 -

WE for irregular sea-floor

• Both multiple reflections and diffractions are predicted

• Exact 3D prediction of pure water-layer multiples and peg-legs from the receiver-side

• Quasi 3D prediction of peg-legs from the the source side

• 3-5 times slower than WEREM

72 -

Conclusions

• SRME is the method of preference for data from areas with deep sea-floor, especially when a thick package of strong reflectors is present below the sea-floor

• As a rule the method becomes less efficient when several orders of multiples are involved

• For such data we use the wave-equation schemes, especially when the main free-surface multiples are just water-layer multiples and peg-legs

• The 3D WE approach has fewer sampling problems than 3D SRME and it allows us to use different WE extrapolation schemes for different complexities of sea-floor and structure below it

73 -

Thank you