Seismic Trace Interpolation Using Adaptive Prediction Filters

Mostafa Naghizadeh and Mauricio Sacchi

Signal Analysis and Imaging Group (SAIG)Department of PhysicsUniversity of Alberta

CSEG 2008Calgary

Outline:Introduction

Theory PF interpolation for stationary data (Linear events)

Adaptive PF interpolation for non-stationary data

(Hyperbolic and Parabolic events)

ExamplesSynthetic data

Real data

Conclusions

Main Goal:Interpolating curved

(hyperbolic or parabolic) events and dispersive events using prediction

filters

Review of reconstruction methods:

Signal Processing methods (Spectrum)FX (Prediction Filters) methods (Spitz, 1991 and Porsani, 1999)FK (Masking operators) methods (Gulunay, 2003)Minimum Weighted Norm Interpolation (Liu and Sacchi, 2004)Fourier Reconstruction with Sparse Inversion (Zwartjes, 2005)MSAR Reconstruction (Naghizadeh and Sacchi, 2007)…

Wave-equation methods (Physics)NMO and DMO operators (Ronen, 1987)Offset and shot continuation (Bagaini and Spagnolini ,1999)Seismic data mapping and reconstruction (Stolt, 2002)Migration Operators (Malcolm et al., 2005 and Trad, 2003)

….

Theory and Implementation

FX interpolation method (Spitz,1991):

Linear events : Stationary signal in the FX domain

Stationary signal : Parameterized by prediction filters

Prediction filters of the originaldata at low frequencies

=Prediction filters of the interpolated

data at high frequencies

Prediction filters can estimate the missing samples of the signal

Time Shift vs Phase Shift :

τ

X

ttime shift

To F-X

N

n

f

f

fff

2

1

0

X

f

, ....) , (e) , (e , e: f τπf-iτπf-iτπf-in

nnn 322221

Auto-Regressive equivalence of f/2 and f

,....e ,e ,e ,e ,e ,e , :2f nnnnnn fi2-fi2-fi2-fi2-fi2-fi2-

n 625242322221 )()()()()(τπτπτπτπτπτπ

.... ,e ,e ,e ,e ,e ,e , :f nnnnnn f-i2f-i2f-i2f-i2f-i2f-i2n

654321 )()()()()( τπτπτπτπτπτπ

Ordinary FX Interpolation

Estimation of Prediction Filter (PF)

Interpolation using PFs (part 1)

Interpolation using PFs (part 2)

Interpolation using PFs (part 3)

Adaptive FX Interpolation

Non-linear events: Non-stationary signal in the FX domain

Non-stationary signal : Adaptive prediction filters

Currently used methodology:Dividing data into small windows.

Very time consumingThe length of windows.

Our Proposed method in this talk:Computing adaptive local PFs.

Exponentially down-weighting the effects of nearby windows.Utilizing Recursive Least Square (RLS) principles to avoid computational loads.

Adaptive PFs (as local as possible)

Adaptive PFs (using past data with less weight)

forgetting factor

Least Squares solution of adaptive PFs (Forward)

Recursive Least Squares solution of adaptive PFs

Initialization of Recursive Least Square solution(backward)

The only inversionof the algorithm

Interpolation using adaptive PFs (part 1)

Interpolation using adaptive PFs (part 2)

Interpolation using adaptive PFs (part 3)

Interpolation using adaptive PFs (part 4)

Synthetic Examples

Original synthetic section with linear events

Interpolated data using adaptive PFs (λ=1)

Interpolated data using adaptive PFs (λ=.25)

Twice interpolated data using adaptive PFs (λ=1)

Three times interpolated data using adaptive PFs (λ=1)

Original synthetic section with hyperbolic events

Decimated section with hyperbolic events

Original synthetic section with hyperbolic events

Decimated section with hyperbolic events

Interpolated data using Spitz FX method

Interpolated data using adaptive PFs (λ=1)

Interpolated data using windowed Spitz method

Interpolated data using adaptive PFs (λ=0.15)

Residual section (Spitz FX interpolation)

Residual section (Adaptive PFs λ=1)

Residual section (Windowed Spitz method)

Residual section (Adaptive PFs λ=.15)

Phase indicator in the FX domain (decimated data)

Phase indicator in the FX domain (original data)

Phase indicator in the FX domain (interpolated data λ=0.3)

Phase indicator in the FX domain (twice interpolated data)λ=0.6

Synthetic data with conflicting dips

Original Interpolated (λ=0.25)

Original Interpolated (λ=0.25)

FK domain of data with conflicting dips

Original Interpolated (λ=0.25)

Synthetic data with conflicting dips (with high frequencies)

Original Interpolated (λ=0.25)

FK domain of data with conflicting dips (higher frequencies)

Real Data Example

Shot gather from Gulf of Mexico

Decimated shot gather

Interpolated using Adaptive PFs (λ=0.55)

Original shot gather

Residual section

Near offset section from Gulf of Mexico

Interpolated using adaptive PFs (λ=0.2)

Conclusions:Exponentially Weighted Recursive Least Square

Algorithm is used to estimate the adaptive prediction filters.

Adaptive prediction filters computed at low frequencies are used locally to interpolate the high frequencies.

Adaptive F-X interpolation depends on two parameters: operator length and the forgetting factor.

The method eliminates the need of selecting window parameters (window size and amount of overlapping between adjacent windows).

The proposed adaptive F-X interpolation algorithm is robust under strong changes of curvature and continuity.

It can be fully automated and extended to the multi-dimensional data.

AcknowledgmentsSAIG Sponsors:

– BP– Encana– ENI SPA AGIP Division – ConocoPhillips– Divesco– Fugro– Hydro– Statoil– CGGVeritas– GEDCO– Landmark Graphics

• Faculty of Science, University of Alberta• Natural Sciences and Engineering Research Council of Canada

(NSERC)