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 A * B = C E.g., A= 0.25 + 0.5 -0.25 0.75]; B = [1 2 -0.5]; C = [0.2500 1.0000 0.6250 0 1.6250 -0.3750]

Convolutional Model for the Earth output input Reflections in the earth are viewed as equivalent to a convolution process between the earth and the input seismic wavelet.

Convolutional Model for the Earth output input SOURCE * Reflection Coefficient = DATA (input) (earth) (output) where * stands for convolution

Convolutional Model for the Earth SOURCE * Reflection Coefficient = DATA (input) (earth) (output) where * stands for convolution (MORE REALISTIC) SOURCE * Reflection Coefficient + noise = DATA (input) (earth) (output) s(t) * e(t) + n(t) = d(t)

s(f,phase) x e(f,phase) + n(f,phase) = d(f,phase) Convolution in the TIME domain is equivalent to MULTIPLICATION in the FREQUENCY domain s(t) * e(t) + n(t) = d(t) FFT FFT FFT s(f,phase) x e(f,phase) + n(f,phase) = d(f,phase) Inverse FFT d(t)

CONVOLUTION as a mathematical operator signal has 3 terms (j=3) -1 2 -1/2 earth Reflection Coefficient has 4 terms (k=4) 1/4 1/4 1/2 time z 1/2 -1/4 3/4 -1/4 3/4 Reflection Coefficients with depth (m)

-1/2 2 1 1/4 1/2 -1/4 3/4 x = +

-1/2 2 -1 1/4 1/2 -1/4 3/4 x = +

-1/2 2 1 1/4 1/2 -1/4 3/4 x = +

-1/2 2 1 1/4 1/4 1/2 -1/4 3/4 x = +

-1/2 2 1 1/2 1 1/4 1/2 -1/4 3/4 x = +

-1/8 1 -1/4 5/8 x = 1/4 1/2 -1/4 3/4 -1/2 2 1 +

-1/4 -1/2 3/4 x = 1/4 1/2 -1/4 3/4 -1/2 2 1 +

1/8 1 1/2 1 5/8 x = 1/4 1/2 -1/4 3/4 -1/2 2 1 +

-3/8 x = 1/4 1/2 -1/4 3/4 + -1/2 2 1

x = 1/4 1/2 -1/4 3/4 + -1 2 -1/2

MATLAB %convolution a = [0.25 0.5 -0.25 0.75]; b = [1 2 -0.5]; c = conv(a,b) d = deconv(c,a) c = 0.2500 1.0000 0.6250 0 1.6250 -0.3750 matlab

Outline Convolution and Deconvolution Normal Moveout Dip Moveout Stacking

Normal Moveout Hyperbola: x T

Normal Moveout “Overcorrected” Normal Moveout is too large x T “Overcorrected” Normal Moveout is too large Chosen velocity for NMO is too (a) large (b) small

Normal Moveout “Overcorrected” Normal Moveout is too large x T “Overcorrected” Normal Moveout is too large Chosen velocity for NMO is too (a) large (b) small

Normal Moveout “Under corrected” Normal Moveout is too small x T “Under corrected” Normal Moveout is too small Chosen velocity for NMO is (a) too large (b) too small

Normal Moveout “Under corrected” Normal Moveout is too small x T “Under corrected” Normal Moveout is too small Chosen velocity for NMO is (a) too large (b) too small

Vinterval from Vrms Dix, 1955

Vrms V1 V2 Vrms < Vinterval V3

Vinterval from Vrms

Primary seismic events x T

Primary seismic events x T

Primary seismic events x T

Primary seismic events x T

Multiples and Primaries x M1 T M2

Conventional NMO before stacking x M1 NMO correction V=V(depth) e.g., V=mz + B T M2 “Properly corrected” Normal Moveout is just right Chosen velocity for NMO is correct

Over-correction (e.g. 80% Vnmo) x x M1 M1 NMO correction V=V(depth) e.g., V=0.8(mz + B) T T M2 M2

f-k filtering before stacking (Ryu) x x M1 NMO correction V=V(depth) e.g., V=0.8(mz + B) T T M2 M2

Correct back to 100% NMO x x M1 M1 NMO correction V=V(depth) e.g., V=(mz + B) T T M2 M2

Outline Convolution and Deconvolution Normal Moveout Dip Moveout Stacking

Outline Convolution and Deconvolution Normal Moveout Dip Moveout Stacking

How do we move out a dipping reflector in our data set? Dip Moveout (DMO) (Ch. 19; p.365-375) How do we move out a dipping reflector in our data set? m Offset (m) TWTT (s) z

For a dipping reflector: Dip Moveout A dipping reflector: appears to be faster its apex may not be centered Offset (m) For a dipping reflector: Vapparent = V/cos dip TWTT (s) e.g., V=2600 m/s Dip=45 degrees, Vapparent = 3675m/s

CONFLICTING DIPS Different dips CAN NOT be NMO’d correctly at the same time Offset (m) TWTT (s) 3675 m/s 2600 m/s Vrms for dipping reflector too low & overcorrects Vrms for dipping reflector is correct but undercorrects horizontal reflector

DMO Theoretical Background (Yilmaz, p.335) (Levin,1971) is layer dip “NMO”

DMO Theoretical Background (Yilmaz, p.335) (Levin,1971) “DMO”

Three properties of DMO “NMO” “DMO” (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 “NMO” “DMO” (1) DMO effect at 0 offset = 0 (2) As the dip increases DMO (a) increases (B) decreases (3) As velocity increases DMO (a) increases (B) decreases

aka “Pre-stack partical migration” Application of DMO aka “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 Offset (m) (1) NMO using background Vrms TWTT (s)

DMO Implementation before stack -II Reorder as COS data -II Offset (m) TWTT (s) NMO (s)

DMO Implementation before stack -III f-k COS data -II X is fixed k NMO (s) f NMO (s)

f-k COS data -II X is fixed k NMO (s) f NMO (s)

f-k COS data -II X is fixed k NMO (s) f NMO (s)

Outline Convolution and Deconvolution Normal Moveout Dip Moveout Stacking

NMO stretching T0 V1 V2 “NMO Stretching”

NMO stretching V1 T0 V2 “NMO Stretching” V1<V2

NMO stretching V1 V1<V2 NMO “stretch” = “linear strain” V2 Linear strain (%) = final length-original length original length X 100 (%)

NMO stretching original length = final length = V1 V1<V2 V2 X 100 (%) X 100 (%)

“function of function rule” NMO stretching X 100 (%) Assuming, V1=V2: X 100 (%) Where, “function of function rule”

NMO stretching So that…

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 X 100 (%)

Stacking + + =

Stacking improves S/N ratio + + =

Semblance Analysis X + + = Twtt (s) “Semblance”

Semblance Analysis X V + + = V1 V2 Twtt (s) V3 Peak energy