An Unsophisticated Look at Curvelets and How to use them for Seismic Data Processing

Mostafa NaghizadehUniversity of Alberta(Currently at the University of Calgary)

CSEG Lunchbox Calgary

20th April 2010

Outlines:

Introduction

Curvelet transform

Curvelet interpolation

Synthetic and real data examples

Curvelet ground-roll elimination

Synthetic example

Conclusion

Introduction I:

Curvelet transform:

F-K filteringintegrated with

frame and operator theory

*** This talk focuses on f-k filtering concept and leaves out the frame theory part. This facilitates a physical understanding of curvelets rather than getting stuck in technical details of computing them.

Introduction II:

Curvelet transformCandes and Donoho (2003) (http://www.curvelet.org/)

Curvelets in seismic data processing Interpolation

Hennenfent and Herrmann (2008)De-noising (random and coherent)

Yarham and Herrmann (2006)Multiple removal

Herrmann and Verschuur (2004)Imaging

Douma and de Hoop (2007)Chauris and Nguyen (2008)

…..

Curvelet Transform

Problem definition

Seismic data in T-X domain

Scale Angle Time Distance

Curvelet functions

Curvelet coefficients

Inner product of data and curvelet functions

Forward curvelet transform

Adjoint curvelet transform

F-K domain tiling of curvelet transform

0 0.25 0.5-0.25-0.50.0

0.25

0.5

Norm

alized frequency

Normalized wavenumber

scale 1

scale 3

scale 5

Plotting Curvelet coefficientsa)

0 0.25 0.5-0.25-0.50.0

0.25

0.5

Nor

mal

ized

freq

uenc

y

Normalized wavenumber

1

23 4 5 6

7

8

b)

12345678

Curvelet panels have different sizes but for illustration purposes they can be scaled into a constant panel size (50x50 for plots in this presentation).

Synthetic seismic section

Curvelet windows in F-K domain

Curvelet panels

Synthetic example

T-X F-K Curvelets

Only the 4th scale of curvelet domain

T-X F-KCurvelets

Data with only one angle of the 4th scale of curvelets

T-X F-KCurvelets

One can built a super-redundant resemblances of curvelets by just applying F-K filtering for each curvelet tile in the F-K domain.

A single curvelet coefficient at scale 4

T-X F-KCurvelets

Curvelet interpolation*

*Accepted for publication in GEOPHYSICS. The article is accessible online at:http://www.phys.ualberta.ca/~mnaghi/Files/Research/Papers/curvelet_interpolation.pdf

Introduction I:

Herrmann and Hennenfent (2008) used curvelets for interpolation of irregularly sampled seismic data.

They reported failure of curvelet interpolation for regularly sampled aliased data. They recommended using jitter sampling strategy in the acquisition stage in order to avoid having regularly sampled data.

Introduction II:

In this presentation curvelets are used for interpolation of regularly and irregularly sampled aliased seismic data. The method can be considered as a combination of:

1.The F-X (Spitz,1991) or F-K (Gulunay,2003) interpolation methods which utilize the low frequency information for beyond-alias interpolation of high frequencies.

2.Minimum Weighted Norm Interpolation (MWNI) method with the exception that here we will use curvelettransform instead of Fourier Transform.

Problem definition

Sampling matrix

Interpolated data

Available data

Inverse Curvelet

Mask function

Curvelet coefficients

Least-squares curvelet interpolation

Cost function

Mask function

Maximum alias-free scale

Synthetic example 1(Regularly sampled aliased data)

Original synthetic data

T-X F-K

Decimated data by factor of 4

T-X F-K

Zero-interlaced data

F-K panel of zero-interlaced data

F-K panel of zero-interlaced data

F-K panel of zero-interlaced data

Curvelet panels of zero-interlaced data

The mask (weight) function

Curvelet panels of interpolated data

Interpolated data using curvelets

T-X F-K

The difference section

Interpolated DifferenceOriginal

Synthetic example 2(Irregularly sampled data)

Curvelet interpolation of irregularly sampled data

original interpolated

missing difference

F-K panels of data

original interpolatedmissing

Curvelet panels

missing

interpolated

Synthetic example 3(Data with conflicting dips)

Synthetic example with conflicting dips

original interpolateddecimated

Zero-interlaced data

F-K

T-X

Curvelet panels of zero-interlaced data

The mask (weight) function

Curvelet panels of interpolated data

Real data example 1(Shot record)

Original shot gather from the Gulf of Mexico

Interpolated shot gather

F-K panel of data

original interpolated

Curvelet panels of zero-interlaced data

The mask (weight) function

Curvelet panels of interpolated data

Real data example 2(Near-offset section)

Original near-offset section

Interpolated near-offset section

F-K panel of data

original interpolated

Curvelet panels of zero-interlaced data

The mask (weight) function

Curvelet panels of interpolated data

Ground-roll elimination

Synthetic data contaminated by ground-roll

Synthetic data contaminated by ground-roll

Curvelet panels of data

Mask function (from high to low frequency)

Projecting maskfunction fromhigher scales

to lower scales

Filtered curvelet panels using mask function

Ground-roll eliminated section using curvelets

F-K dip filtered data

Conclusions:Curvelet transform is a local decomposition of data based on

some predefined scales and directions. It can be conceived as an F-K filtering combined with frame theory principles to obtain an optimal and efficient redundant representation of data.

For Interpolation of data in curvelet domain:Extract a mask function from alias-free scales (low frequencies) and

project it to alias-contaminated scales (high frequencies).Form a least-squares fitting algorithm using the sampling operator

and mask function. In the case of irregularly sampled data, iterative thresholding of

curvelet coefficients (IRLS) suffices for interpolation purposes.For Ground-roll elimination in Curvelet domain:

Extract mask function from non-contaminated scales (high frequencies) and use it to eliminate ground-roll in the contaminated area (low frequencies).

Acknowledgments:

Dr. Mauricio Sacchi for his insightful supervision during my PhD program and after.

Authors of CurveLab [http://www.curvelet.org/], Emmanuel Candes, Laurent Demanet, David Donoho, and Lexing Ying for providing access to their curvelettransform codes.

Sponsors of SAIG for their financial support.