Post on 14-Apr-2018
transcript
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.