Making CMP’s

From chapter 16 “Elements of 3D Seismology” by Chris Liner

Outline•Convolution and Deconvolution•Normal Moveout•Dip Moveout•Stacking

Outline•Convolution and Deconvolution•Normal Moveout•Dip Moveout•Stacking

Convolution means several things:

•IS multiplication of a polynomial series•IS a mathematical process•IS filtering

Convolution means several things:

•IS multiplication of a polynomial series

E.g., A= 0.25 + 0.5 -0.25 0.75]; B = [1 2 -0.5];

0 1 2 30.25 0.5 0.25 0.75A z z z z 0 1 22 0.5B z z z

A * B = C

C = [0.2500 1.0000 0.6250 0 1.6250 -0.3750]

Convolutional Model for the Earthinput

output

Reflections in the earth are viewed as equivalent to a convolution process between the earth and

the input seismic wavelet.

Convolutional Model for the Earthinput

output

SOURCE * Reflection Coefficient = DATA(input) (earth)

(output)where * stands for convolution

Convolutional Model for the Earth

(MORE REALISTIC)

SOURCE * Reflection Coefficient = DATA(input) (earth)

(output)where * stands for convolution

SOURCE * Reflection Coefficient + noise = DATA(input) (earth)

(output)

s(t) * e(t) + n(t) = d(t)

Convolution Convolution in the TIME TIME domain is equivalent to MULTIPLICATIONMULTIPLICATION in in

the FREQUENCYFREQUENCY domain

s(t) * e(t) + n(t) = d(t)

s(f,phase) x e(f,phase) + n(f,phase) = d(f,phase)

FFT FFT FFT

Inverse FFT

d(t)

CONVOLUTION as a mathematical operator

0

j

j j k kkD s e

2-1/2

-1

z

Reflection Coefficients with depth (m)

1/41/2-1/43/4

1/41/2

-1/43/4

Reflection Coefficient

signalsignal has 3 terms (j=3)has 3 terms (j=3)

earthearth has 4 terms (k=4)has 4 terms (k=4)

time

000

1/41/2-1/43/4000

000

-1/2210000

xxxxx

=====

000000

+

000

1/41/2-1/43/4000

000

-1/22-10000

xxxxx

======

0000000

+

000

1/41/2-1/43/4000

000

-1/2210000

xxxxxxx

=======

00000000

+

000

1/41/2-1/43/4000

000

-1/2210000

xxxxxxxx

========

000

1/40000

1/4

+

000

1/41/2-1/43/4000

000

-1/2210000

xxxxxxxxx

=========

000

1/21/200001

+

000

1/41/2-1/43/4000

000

-1/2210000

xxxxxxxxxx

==========

000

-1/81

-1/40000

5/8

+

000

1/41/2-1/43/4000

000

-1/2210000

xxxxxxxxxx

==========

0000

-1/4-1/23/40000

+

000

1/41/2-1/43/4000

000

-1/2210000

xxxxxxxx

========

000

1/81 1/2

000

1 5/8

+

000

1/41/2-1/43/4000

000

-1/2210000

xxxxxxx

=======

000

-3/8000

-3/8

+

000

1/41/2-1/43/4000

000-12

-1/20000

xxxxxx

======

0000000

+

c = 0.2500 1.0000 0.6250 0 1.6250 -0.3750

%convolutiona = [0.25 0.5 -0.25 0.75]; b = [1 2 -0.5];c = conv(a,b)d = deconv(c,a)

2 3 40.25 0.5 0.25 0.75a z z z z

MATLAB

2 32 5b z z z

matlab

Outline•Convolution and Deconvolution•Normal Moveout•Dip Moveout•Stacking

Normal Moveout

22 2

0 2

xT TV

22

0 0 02( ) ( ) xT x T x T T TV

x

T

Hyperbola:

Normal Moveoutx

T

“Overcorrected”

Normal Moveout is too large

Chosen velocity for NMO is too (a) large (b) small

Normal Moveoutx

T

“Overcorrected”

Normal Moveout is too large

Chosen velocity for NMO is too (a) large (b) smallsmall

Normal Moveoutx

T

“Under corrected”Normal Moveout is too small Chosen velocity for NMO is

(a) too large(b) too small

Normal Moveoutx

T

“Under corrected”Normal Moveout is too small Chosen velocity for NMO is

(a) too largetoo large(b) too small

Vinterval from Vrms

122 2

1 1interval

1

n n n n

n n

V t V tVt t

Dix, 1955

2i i

RMSi

V tV

t

Vrms

V1

V2

V3

Vrms < Vinterval

Vinterval from Vrms

Vrms T Vinterval from Vrms ViViT VRMS from V interval1500 0 01500 0.2 1500 450000 15002000 1 2106.537443 4000000 20003000 2 3741.657387 18000000 3000

SUM 3.2 22450000

Primary seismic eventsx

T

x

T

Primary seismic events

x

T

Primary seismic events

x

T

Primary seismic events

Multiples and Primariesx

TM1

M2

Conventional NMO before stackingx

T NMO correctionV=V(depth)

e.g., V=mz + B

M1

M2

“Properly corrected”Normal Moveout is just right

Chosen velocity for NMO is correct

Over-correction (e.g. 80% Vnmo)x

T NMO correctionV=V(depth)

e.g., V=0.8(mz + B)

M1

M2

x

TM1

M2

f-k filtering before stacking (Ryu)x

T NMO correctionV=V(depth)

e.g., V=0.8(mz + B)

M1

M2

x

T

M2

Correct back to 100% NMOx

T NMO correctionV=V(depth)

e.g., V=(mz + B)

M1

M2

x

TM1

M2

Outline•Convolution and Deconvolution•Normal Moveout•Dip Moveout•Stacking

Outline•Convolution and Deconvolution•Normal Moveout•Dip Moveout•Stacking

Dip Moveout (DMO)

How do we move out a dipping reflector in our data set?

z

m Offset (m)

TWTT (s)

(Ch. 19; p.365-375)

Dip MoveoutA dipping reflector:

• appears to be faster•its apex may not be centered

Offset (m)

TWTT (s)For a dipping reflector:Vapparent = V/cos dip

e.g., V=2600 m/s

Dip=45 degrees,Vapparent = 3675m/s

Offset (m)

TWTT (s)

Vrms for dipping reflector too low &

overcorrects

Vrms for dipping reflector is correct but

undercorrects horizontal reflector

3675 m/s

2600 m/s

CONFLICTING DIPS Different dips CAN NOT

be NMO’d correctly at the same time

DMO Theoretical Background (Yilmaz, p.335)

2 22 2

0 2

cos( ) xT x TV

(Levin,1971)

22 2 2

0 2( ) (1 sin )xT x TV

2 2sin cos 1

2 22 2 2

0 2 2( ) sinx xT x TV V

“NMO”

is layer dip

DMO Theoretical Background (Yilmaz, p.335)

2 22 2

0 2

cos( ) xT x TV

(Levin,1971)

22 2 2

0 2( ) (1 sin )xT x TV

2 2sin cos 1

2 22 2 2

0 2 2( ) sinx xT x TV V

“DMO”

2 22 2 2

0 2 2( ) sinx xT x TV V

“DMO”“NMO”

(1) DMO effect at 0 offset = ? (2) As the dip increases DMO (a) increases (B) decreases(3) As velocity increases DMO (a) increases (B) decreases

Three properties of DMO

2 22 2 2

0 2 2( ) sinx xT x TV V

“DMO”“NMO”

(1) DMO effect at 0 offset = 00 (2) As the dip increasesincreases DMO (a) increasesincreases (B) decreases(3) As velocity increasesincreases DMO (a) increases (B) decreasesdecreases

Three properties of DMO

Application of DMOaka “Pre-stack partical migration”

•(1) DMO after NMO (applied to CDP/CMP data)• but before stacking•DMO is applied to Common-Offset Data •Is equivalent to migration of stacked data•Works best if velocity is constant

DMO Implementation before stack -I

2 22 2 2

0 2 2( ) sinx xT x TV V

(1) NMO using

background Vrms

Offset (m)

TWTT

(s)

22 2 2

0 2( ) sinxT x TV

Reorder as COS data -II

2 22 2 2

0 2 2( ) sinx xT x TV V

Offset (m)

TWTT

(s)

2 22 2 2

02 2( ) sinx xT x TV V

NMO

(s)

DMO Implementation before stack -II

f-k COS data -II

NMO

(s)

X is fixed

f

k

NMO

(s)

DMO Implementation before stack -III

f-k COS data -II NM

O (s

)

X is fixed

f

k

NMO

(s)

f-k COS data -II NM

O (s

)

X is fixed

f

k

NMO

(s)

Outline•Convolution and Deconvolution•Normal Moveout•Dip Moveout•Stacking

NMO stretching

V1

V2

T0

“NMO Stretching”

NMO stretching

V1

V2

T0

“NMO Stretching”

V1<V2

NMO stretching

V1

V2V1<V2

0 0T T0T 1T

1 1T TNMO “stretch” = “linear strain”

Linear strain (%) = final length-original length original length

X 100 (%)

NMO stretching

V1

V2V1<V2

0 0T T0T 1T

1 1T T

X 100 (%)

original length = 1T final length = 0T

NMO “stretch” = 0 1

1

T TT

X 100 (%)0

11T

T

0T

NMO stretching

X 100 (%)0

11T

T

220 2

0 0 0

( )xd TdT TVdT dT T

12 22

0 0 2122

xT TV

12 22

0 0 2xT TV

12 2

2 20

1 1xT V

X 100 (%)

Where,

“function of function rule”

Assuming, V1=V2:

NMO stretching1

2 220 2

0

0

xTVdT

dT T

12 2

2 20

1 xT V

So that…

X 100 (%)0

11T

T

stretching for T=2s,V1=V2=1500 m/s

Green line assumes V1=V2

Blue line is for general case,where V1, V2 can be different and delT0=0.1s (this case: V1=V2)

Matlab code

Stacking

+ + =

+ + =

Stacking improves S/N ratio

+ =

Semblance Analysis

221 1 2

221 1 2

221 1 2

“Semblance”

+

223 33

2 2 2

X

Twtt

( s)

+ =

Semblance Analysis

+

X

Twtt

( s)

V3

V1

V2

V

Peak energy