Wei Wang Sanya

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Abrupt Feature Extraction via theCombination of Sparse Representations

Wei Wang, Wenchao Chen, Jinghuai Gao, Jin XuInstitute of Wave & Information,

Xi’an Jiaotong University

International Symposium on Geophysical Imaging with Localized WavesSanya, Hainan, 24-28 July, 2011

Institute of Wave & Information, Xi'an Jiaotong University, Xi'an, P.R. China



Co-Authors

Wenchao Chen Jinghuai Gao Jin Xu



• Introduction• Morphological Component Analysis

• Abrupt Feature Extraction Methodology• Synthetic and field data examples• Conclusions

Outline



• Problems for stratigraphic interpretation – Complexity of channels in 3D seismic data always

makes detailed interpretation challenging; – Definition of sand bars and beaches also becomes

complicated as they are only partially visible whenseismic amplitude is examined.

• How to improve imaging these features

– Current interpretation workflows adopt advancedcolor and opacity based co-rendering techniques tomerge multiple stratigraphic attributes.

Introduction



• Current multi-attributes trap – Instantaneous attributes, e.g., the peak frequency

depicts horizontal distribution of channels and theenvelope highlight bright spots and dim spots;

– Coherence attributes are taken to delimit the edgesof stratigraphic units;

– Spectral decomposition generates narrow-bandinstantaneous amplitude volumes and allows better imaging channel details.

Time-consuming to interpret many seismic attributes;Multi-attribute analysis techniques, e.g., multi-attribute3D visualization, PCA, cluster analysis, sometimeslack explicit physical meaning.



• Motivation

– Raw seismic time samples contain all informationrepresented by various seismic attributes; – The amplitude and phase spectra and the spatial

architecture within an analysis window form specific

waveform patterns.• Our approach

Adopt the morphological component analysis (MCA)theory (Starck, et al., 2005), to extract waveforms of meaningful stratigraphic targets from seismic data;Different waveform dictionaries are chosen torepresent specific waveform patterns by exploringtheir sparsity.



• Assumptions of the MCAFirstly, the signal is assumed as a linear combination of different morphological featureswhich are sparsely represented by n differentdictionaries:

Morphological Component Analysis

1 1

(1)n n

k k k k k

s s

where each signal component can be described aswith an overcomplete dictionary and a

sparse representation .k k k s

k

k



• Assumptions of the MCA

Secondly, the MCA assumes that for any givencomponent s k the sparsest decomposition over theproper dictionary yields a highly sparse description,while its decomposition over the other dictionaries,Φ

j≠k, is highly non sparse, that is

• Sparsity-promoted signal separation

In the MCA framework, the decompositioncoefficients of corresponding signal componentsare the solutions of

0 0

, 1, , ; . (2)T T k k k jk j n j k s s

k



• Sparsity-promoted signal separation

By substituting the norm by an norm, andrelaxing the equality constraint, the MCA algorithmfinally seeks a solution by solving the convexminimization problem (Starck, et al., 2005),

NP-Hard Optimization Problem

10

, , 1 1Argmin .

n

n n

k k k k k

subject to s

0l

1l

1

2

1, , 1 1 2

1Arg min (3)

2n

n nT

k k k s s k k

s s s



• Signal decomposition example

“Bumps+Cosine” Separation

DCT dictionary

Wavelet dictionary



• Signal model assumption According to morphologic appearance in verticalsections, volumetric flattened seismic data arecomposed of coherent events and abrupt features ;

We thus model these two kinds of seismic featuresas linear structures and punctate structuresrespectively. A vertical section s along the inlinedirection is formulated as

Abrupt Feature Extraction Methodology

1 2 n (4) s s s

: abrupt features; : coherent events;n: a zero-mean Gaussian noise matrix with astandard deviation .

1 s

2 s



• Nonlinear optimization model

1 2

21 1 1 2 2 2 1 2 21 1,

1Arg min (5)2

T T

s s s s s s s

& : representing dictionaries ( s 1 & s 2);& : controlling how much the sparseness

priors are emphasized on the model.

1 2

1

2

The success of MCA relies on the incoherencebetween sub-dictionaries and each sub-dictionaryshould lead to sparse representations of thecorresponding signal component.



• Choice of representations

The 2D-UWT presents only three directionalelements independent of scales, and there are nohighly anisotropic elements. We expect the 2D-UWTto be non-optimal for detection of highly anisotropicfeatures and adopt it to detect punctate features.

Fig.1 2D-UWT atomsalong the horizontal,vertical, and diagonaldirection, with threedifferent scale indexes.



• Choice of representations

The curvelet transform is a redundant dictionary andthe curvelet elements are anisotropic and obey theparabolic scaling, which makes it the best choice for the detection of anisotropic structures such ascoherent wavefronts.

Fig.2 Discrete curveletsindexed by different scale,orientation, and location.



• Implementation

To deal with 3D seismic data, our algorithm is performedline-by-line, each vertical slice separated into abruptfeatures and coherent events via the iterative-shrinkagealgorithm (Bruce et al., 1998; Daubechies et al., 2005):

1) Initialize N , ,11 12 1 N 21 22 2 N 2) Iterative procedure, n=1,2, …,N ,

a) Update x 1, assume x 2 fixed,

1,1, 1 1, 1, 1 2, 2 1n

T n n n n s x T x x x

b) Update x 2, assume x 1 fixed,

2,2, 1 2, 1, 1 1 2, 2 2n

T n n n n s x T x x x

3) 1 1, 1 , N s x2 2, 2 N s x



Synthetic data examples

(a) The earth model with a thin channelwith the size of 100-m wide and 10-mthick embedded. Both the horizontal andvertical sampled at 2.5-m.

(b) The migrated seismic section isgenerated by the PSDM algorithmusing a Ricker wavelet with dominantfrequency 60 Hz.

The reflection waveforms of the channel appear as a bright spot and overlap

the coherent reflection events.



Separated synthetic sections by the

proposed method: (a) The migratedseismic section of the earth model;(b) The extracted abrupt featurespart and (c) the coherent events part.



Field data examples

(a) An inline section from volumetric

flattened 3D seismic data

(b) The separated coherent events

part using the proposed method

R e

l a

t i

v e

G e o

l o g

i c a

l T i

m e

(

s )

CMP No. CMP No.



(a) An inline section from volumetric

flattened 3D seismic data

(c) The extracted abrupt features


R e l

a t i

v e

G e o

l o g

i c a

l T i

m e

(

s )

CMP No. CMP No.



(a) Zoomed part of theinline section

(b) Zoomed part of thecoherent events component

(c) Zoomed part of theabrupt features component

R e

l a

t i

v e

G e o

l o g i c a

l

T i

m e

(

s )

CMP No. CMP No. CMP No.

Top of Channel

Bottom of Channel



(b) The extracted abrupt features


(a) An horizontal slice from

volumetric flattened 3D seismic data



(a) An horizontal slice from

volumetric flattened 3D seismic data

(b) The extracted abrupt features




The MCA technique is utilized to extract sedimentaryfeatures from 3D volumetric flattened seismic data;

Since sedimentary features modeled as punctate

structures, the 2D-UWT is chosen to represent thesedimentary features while the curvelet transform, ischosen to sparsify coherent events representation;

Both synthetic & field data examples show the efficiencyof our method for interpreting sedimentary features. Theextracted abrupt feature waveforms can be used for subsequent quantitive analysis and reservoir modeling.

Conclusions



• This work has been partially supported by the NSFC(No. 40730424 & No. 40674064), the NHTRDPC (No.2006AA09A102-11), and the INSTSP (No. 2008ZX05023-005-005 & No. 2008ZX05025-001-009).

• We would like to thank Exploration and DevelopmentResearch Institute of Daqing Oilfield Company Ltd. fortheir supporting us seismic data.

Acknowledgements

Wei Wang Sanya

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