Seismic imaging in the curvelet domain: achievements and

perspectives

Hervé Chauris(1) & Jianwei Ma(1,2)

EAGE 2009 - Amsterdam

(1) Centre de Géosciences, Mines ParisTech, France(2) Institute of Seismic Exploration, School of Aerospace,

Tsinghua University, Beijing, China

Locally coherent events

Time common-offset

sectionSame section after

depth migration

Locally coherent events

Context : seismic modeling / migration

Diffraction point Globally coherent

eventLocally coherent

event

Curvelets and other …lets

EPW

Ridgelet

Surfacelet Seislet

Contourlet ShearletWavelet

Bandlet, …

Taking into account band-limited data:

Curvelet

Potential of curvelets

Single curveletSeismic data

Introduction – Curvelets

Curvelets

for different

coefficients

cμ(x,z)shift

stretchrotation

Content

Introduction – what are curvelets

Curvelets and seismic processing tasks• pre-/post-processing• seismic migration• seismic demigration/migration – velocity estimation

Conclusions

Curvelets and seismic applications

Review:• Ma and Plonka, 2009• Workshop EAGE 2007 (London), Chauris and Douma

Data denoising, interpolation and compression• Hennenfent and Herrmann, 2006 Droujinine et al., 2007• Herrmann et al., 2008a Sacchi et al., 2007• Herrmann et al., 2008b Fomel, 2007• Lin and Herrmann, 2007• Neelamani et al, 2008

Seismic modeling and migration• Douma and de Hoop, 2007• Chauris, 2006

Velocity model estimation• Chauris and Nguyen, 2007• Chauris and Nguyen, 2008

redundant

transform

Seismic propagation/migration

Theoretical results• Candes and Donoho, 2003

Practical results• Douma and de Hoop, 2006• Douma and de Hoop, 2007• Chauris, 2006

Curvelet processingMigrated sectionin initial model

Perturbed section

Fastcurvelettransform

Fast inversecurvelettransform

Use of digital curvelets

Migration in heterogeneous models

Kirchhoff migration First-order curvelet migration

Smooth heterogeneous 2-D model

Migration in heterogeneous modelsK

irchh

off

mig

ratio

n

Firs

t-or

der

curv

elet

mig

ratio

n

Migration in heterogeneous models

Kirchhoff migration of a few curvelets

First-order approximation

not good enough

Demigration/migration

Sensitivity of the migrated result with respect to the velocity

model

Initial velocity model Velocity perturbation

(up to 200 m/s)

Triplicated

ray field

Demigration/migration

Given velocity

model

Local velocity

perturbation

Sensitivity of a migrated image with respect to the background velocity model

Migration

Migration ?

Input data

Ray+Born 2-D synthetic data set (offsets from 0 to 2 km)

Initial imageOffset 600 m

Demigration/migration

Sensitivity of the migrated result with respect to the velocity

model

Initial image Exact imagePredicted image

Demigration/migration

Initial / exact Predicted / exact

Depth difference

reduced from 60 m

to less than 2 m

Demigration/migration

Sensitivity of the migrated result with respect to the velocity

model

Unperturbed part Modified part

Common Image Gathers

Reference

image

Prediction

with curvelets

Exact

The prediction takes into account the lateral velocity variations

Demigration/migration

Given velocity

model

Local velocity

perturbation

Sensitivity of a migrated image with respect to the background velocity model

Migration

Migration ?

Velocity estimation

Given velocity

model

Optimal velocity

perturbation?

Sensitivity of a migrated image with respect to the background velocity model

Migration

Migration?Cost function

Improved

seismic

section

DSO in curvelet domainFeasibility study

2-D synthetic ray+Born data set

Initial After 1 iteration Exact

DSO in curvelet domain

Initial

After 1

iteration

Exact

Stack of offsets

between 100 and

800 m

Conclusions & perspectivesSeismic imaging operators:• Curvelets more suited for demigration/migration than

for migration or demigration (modeling) alone• Applications limited to smooth background velocity

models

A similar analysis should be conducted for (non-smooth) general background velocity models (without the use of geometrical optics)

Perspectives: new transform (e.g. with explicit curvature)?

Acknowledgements

We would like to thank

F. ten Kroode (Shell E&P)

for fruitful discussions and support

M. Noble and P. Podvin

(Mines ParisTech)

Shell E&P for partly funding the project