Multi-dimensional depth imaging without an adequate velocity model Fang Liu Arthur B. Weglein Kristopher A. Innanen Bogdan G. Nita University of Houston Houston, Texas April 21th, 2005

Key Points Migration results True Earth Data communicate with itself, no need for adequate velocity model Purposeful perturbation Physical interpretation: Taylor expansion Migration results True Earth

Main issue Success in imaging reflectors in the Earth is closely tied to our ability to find the velocity model. In complex geological environments, we are often unable to find an adequate velocity model.

Objectives Our approach To improve our ability to accurately locate reflectors, point scatters, especially in areas where the velocity model is difficult to estimate. Our approach Inverse scattering series

Background of main issue Current migration and inversion: FK, Phase-shift, Kirchhoff

Background of main issue Current migration and inversion: FK, Phase-shift, Kirchhoff

Inverse scattering approach Linear 2nd Order 3rd Order Free-surface multiple removal Internal multiple attenuation Imaging Inversion

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

Equations to be solved:

Task separation: Second term

Numerical example Geological model: 300 m 400 m 200 m

A sample shot-gather Modeling algorithm : finite difference

X 1000(m) α1 Z Stolt migration 50(m)

X Z 1000(m) Nothing to move the first reflector. 50(m) Move the second reflector.

X 1000(m) Z 50(m)

Task separation: Third term More significant term: Less significant term:

Physical interpretation : Taylor series expansion

Possible patterns and subseries Let’s define: Can be extended by both: As:

Closed forms

Numerical example versus

X 1000(m) Z 50(m)

X 1000(m) Z 50(m)

Model with three interfaces 1500 m/s 300 m 500 m 1600 m/s 200 m 200 m 1500 m/s 1600 m/s

Large contrast model 1500 m/s 2000 m/s 3000 m/s 300 m 500 m 200 m

Tentative model

Preprocessing requirements Objective of the research: 2D acoustic medium with 1 parameter (velocity only) Preprocessing needed in this example: Wavelet estimation Deghosting Free surface multiple removal Internal multiple attenuation

Conclusions All existing tests so far are VERY encouraging. Depth accurate and velocity independent imaging algorithms. Closed forms with demonstrated value.

Acknowledgments M-OSRP members M-OSRP sponsors