Continuous wavelet transform of function f(t) at time relative to wavelet kernel at frequency scale f:

"Multiscale reconstruction of shallow marine sediments using wavelet correlation"A. Kritski1, A.P. Vincent2, D. A. Yuen3

1 Statoil Research Centre, Postuttak, N-7005 Trondheim, Norway (akr@statoil.com) 2 Departement de Physique Universite de Montreal, Centre-Ville, Canada

3 Department of Geology and Geophysics and Minnesota Supercomputing Institute, U.S.A.

Wavelet Cross-Correlation Function for fx(t) and fy(t):

IntroductionThe subject of studying surface waves is receiving new attention because of the potential in using surface waves for prediction of physical properties of near surface marine sediments. However, processing of surface waves normally relies on algorithms, which are do not allow sufficient discrimination between surface waves modes. This work extends and recast the results of our previous studies on the wavelet cross-correlation analysis of surface waves. We are introducing multiscale cross-correlation in the spatial and time domains. We applied the wavelet transform to seismic traces 'Wa(t[ime],f[requency])' and 'Wb(t[ime],f[requency])', then calculated the cross-correlation function in the time domain {< Wa(t,f,X)*Wb(t-tau,f,X)>} and additionally carried out the cross correlation of wavelet fields in horizontal direction X (distance along the interface) {<WCR(tau,f,X)*WCR(tau,f,X-X')>}. The moduli and phases of multiscale cross-correlation function present the group and phase velocities values at given frequencies and time delay with the spatial resolution defined as a minimum spatial lag X' (minimum distance between receivers). We present a comparison of using 1D and 2D multiscale cross-correlation techniques in terms of resolving the phase and group velocities of surface waves.

direct arrivals

sourcewater column

marine sedimentsinterface waves

127 m/s

0.5

1.0

0 5 10 15 20 25 30 350

5

10

15bottom hydrophone

tim e , s e c

ran

ge

, k

mR

ang

e, k

m

0.5

1.0

1.5

2.0

2.5

1534 m/s

257 m/s 137 m/s77 m/s

1D Wavelet Cross-Correlation for traces: 5 (2.5 km) and 3 (1.5 km)

Synthetic Data Modeling

Data acquisition scheme

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5-0 . 5

0

0 . 5

1

1 . 5

2

2 . 5

3S y n t h e t ic s e is m o g ra m s

t im e , s e c

ran

ge

, k

m

75 m/s

130 m/s

250 m/s

Signal 5

Signal 4

Signal 3

Signal 2

Signal 1

)(t

dbabWfabWfT

WC y

T

T

xxy ),(),(1

lim2/

2/

Experimental data

Synthetic data

2D Wavelet Cross-Correlation for selected frequency range from 1.5 to 2.3 Hz

)]1)(2

1

1

1([)]([_ 1

2211 x

VVxxxphaseWCR

jj

n

jj

n

j

n

k

kkj

W C R _ p h a s e d e s c rib es th e p h as e s h if t d if fere n c e b e tw e e n tw o s ig n a ls a t tw o d if fere n t loc a tio n s f or g ive n tim e d e la y a n d

f req u e n c y j .T h is c h a n g e of p h a s e is c a u s e d b y c h a n g in g of p h as e ve loc ities b e tw e e n loc a tio n p o in ts : )2/11/1( jj VV . V 1

a n d V 2 – p h a s e ve lo c it ies of s u rfac e w a ve s rec o rd e d a t d if fere n t loc a tio n s . W h en V 1 = V 2 - th ere a re n o c h a n g es in p h ys ic a l

p ro p er ties b e tw e e n rec e ive rs .

Conclusion 2D spatial and time domains Cross-correlation of the wavelet transformed seismic traces extracts the information of coherent strength (moduli) and phase in terms of periods (frequencies), time delay and spatial shift and allows to monitor the changes of these parameters in both time and horizontal distance.The phase velocity dispersion can be studied directly from the phase field of 2D wavelet cross-correlation function. The peaks of wavelet correlation moduli perform the relative energy distribution in surface wave (and its modes) showing group velocity dispersion.

We consider harmonic surface waves propagating on the ocean-sediment interface. The surface wave is a combination of propagational motion along the interface, where the phase is moving along horizontal coordinate, and syn-phase oscillations (modes) along z where amplitude is rapidly decaying with depth. In the case of the depth-dependence shear modules, phase and group velocity of the surface waves perform frequency dependence - dispersion.

Layer1

Layer2

Layer3

Depthshort wavelength

Depthlong wavelength

Particle motion Particle motion

1 2 3

frequency

Group velocity shear properties

variability

dtf

ttf

ffWf )()(

1),(

dispersion

2D Wavelet Cross-Correlation function

Cross-correlation of the wavelet transformed seismic data in X direction (distance along the interface) in addition to the time domain:

n

j

n

k j

jj V

phaseDWCR )(_2_

2/

2/

2/

2/00

),(),(1

lim1

limX

X

y

T

T

xTX

xy dtdxftWfftWfTX

WC

Here Vj is a difference in the phase velocity between x=0 and x=Xmax for the given frequency range. WCR-2D_phase field can be inverted

for the phase velocities values at given frequencies with the spatial resolution defined as a minimum distance between seismic receivers: Xmin. For different distances the phase velocity of the surface wave can be found directly from WCR-2D_phase and then inverted into the shear velocities, and then into shear strength in sediment layers as a function of distance and depth (frequency).

Cross-correlation in time domain