Direct horizontal image gathers without velocity or “ironing” Fang Liu, Arthur B. Weglein, Kristopher A. Innanen, Bogdan G. Nita, Jingfeng Zhang M-OSRP 2006 Annual Meeting, June 7, 2007 M-OSRP report pages: 160-179

Key Points Flattening of the common-image gather without knowing the velocity and waveform distortion Best-effort plane-wave scenarios for the Zoeppritz equation in AVO analysis Totally deterministic procedure and rich structures in the common-image gather

Outline Common image gather Theory The promise of the imaging subseries and our current capture Numerical examples Conclusions and acknowledgements

Outline Common image gather Theory The promise of the imaging subseries and our current capture Numerical examples Conclusions and acknowledgements

What is common-image gather (CIG)? One more degree of freedom in the data than in the migration section. Mapping 3 2 Different ways of mapping D(xg,xs,t) to M(x,z) at the same x location constitute a CIG. Since there is only one earth, migration with different parameters should achieve the same depth, i.e., flat (horizontal) CIG.

CIG : current procedures Velocity driven: If the velocity is correct, the CIG should be flat (horizontal). Flat CIG is a necessary condition for correct velocity. Some may produces NMO stretches and other waveform distortions. If CIG is not flat, “ironing” procedure can damage zero-crossing and other valuable information. Not totally deterministic.

CIG : Inverse series approaches Driven by the promise of the inverse scattering series: they should automatically give the same depth Direct formula: flat CIG is no longer a necessary or sufficient condition to strive for No waveform distortion in the plane-wave world A natural by-product of the imaging subseries, and a totally deterministic procedure

Where does it come from?

Where does it come from? Migrated section Input data Horizontal red lines are drawing to bench-marking the flatness of events

Outline Common image gather Theory The promise of the imaging subseries and our current capture Numerical examples Conclusions and acknowledgements

Theory Solving for the wave equation, with the help of wave propagation in the much simpler reference medium,

Solution: the inverse scattering series

Solution for the linear term Solution in the wave-number domain The triple Fourier transform of the data equals to the double Fourier transform of . We should choose a slice of spectrum in the data to reach an image. We choose different slices of spectrums such that each slice corresponds to a plane-wave incidence experiment.

Essential element 1 : angle θ It is the incidence angle of the plane-wave. Zoeppritz equation is for many plane waves with different incidence angles. We don’t have plane wave from the original data. The plane wave can be synthesized by Radon transform (slant stacking, or tau-p transform). H. Zhang & Weglein 2004

Essential element 2 : CMP gather Image is formed in the CMP gather (i.e., NMO stacking, Clayton & Stolt 1981) Liu et al. 2005

Combining two elements (1) (2) Each slice with a fixed angle θ corresponds to the data from the experiment of a plane-wave with angle θ as the incidence angle.

First part of our imaging formula: the linear term Construct a plane-wave in the CMP gather Receiver location Source location Time

Second part of our imaging formula: Higher-order imaging subseries (HOIS) Is the partial capture of the imaging capability of the imaging subseries More imaging capability than the leading-order subseries to deal with large contrast Amplitude is left untouched for later AVO analysis Lightening speed

Outline Common image gather Theory The promise of the imaging subseries and our current capture Numerical examples Conclusions and acknowledgements

The promise of the imaging subseries Accurate image of all reflectors at depth using water-speed, for any angle θ. Imaging results (locations) from different angle should be the same. Amplitude is left untouched.

The capability captured by current HOIS Imaging reflectors very close to the actual depth using water speed. Imaging results (locations) from different angle are much closer than the linear image. Amplitude is left untouched.

Assumptions Remove direct wave and ghosts Known source wavelet Remove free-surface multiples Remove internal multiples

Outline Common image gather Theory The promise of the imaging subseries and our current capture Numerical examples Conclusions and acknowledgements

Numerical examples x z Big contrast 1500 (m/s) 2328.75 (m/s)

Shot records x Source at = 0 m x Source at = 3000 m t t Conflicting hyperbola

Linear image : θ=0° D C A B

Linear image : θ=9°

Higher-order image : θ=0°

Higher-order image : θ=9°

Case 1: right boundary D

D

D

Case 2: Middle C

C

C

Sum of 11 angles ( )

Higher-order imaging :

Sum of 11 angles ( )

Higher-order imaging :

Outline Common image gather Theory The promise of the imaging subseries and our current capture Numerical examples Conclusions and acknowledgements

Conclusions Flattening of the common-image gather without knowing the velocity and waveform distortion Best-effort plane-wave scenarios for the Zoeppritz equation in AVO analysis Totally deterministic procedure and rich structures in the common-image gather

Acknowledgments M-OSRP members. GX-Technologies for the scholarship. M-OSRP sponsors. NSF-CMG award DMS-0327778. DOE Basic Energy Sciences award DE-FG02-05ER15697.