Multi-dimensional depth imaging without an adequate velocity model: towards more challenging geology Fang Liu, Arthur B. Weglein, Kristopher A. Innanen, Bogdan G. Nita Annual report: page 245-265 Houston May 11th, 2006

Why ? The difficulty to get a clear image beneath salt or other complicated overburdens. All the current migration procedure require : to get the correct image. Accurate velocity model Adequate propagator

Current Procedures (first approach) Velocity analysis Adequate velocity Seismic migration Correct Depth Image

Our Procedures (second approach) Velocity Independent Imaging Correct Depth Image

How? Inverse scattering series. The only procedure with the capability of directly achieving all the processing objectives without knowing or determining the actual wave propagation. Removed multiples without assuming any subsurface information.

Key points New direct imaging method: (a) accommodate inadequate velocity, (b) non-linear in the data. More imaging terms had being identified and included in subseries with closed-forms. Encouraging numerical results. New understanding & more general framework.

Background: Perturbation Theory Actual Medium Reference Medium Perturbation L – L0 = V Reference Velocity Perturbation Actual Velocity Actual Field - Reference Field = Scattered Field G - G0 = D

Background : Lippmann-Schwinger

Background: Born Approximation FK, Phase-shift, Kirchhoff

Background: inverse series True earth (in perturbation form) Current pre-stack migration-inversion (constant c0) Reference velocity is never updated.

Background: Imaging subseries Inversion Terms with different purposes identified Subseries identified to capture certain part of the imaging capability. Closed-form found

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

Physical interpretation : Cascaded Taylor series in multi-D Express arbitrary function in Taylor expansion : True Earth Current migration-inversion Why cascaded ? The simplest imaging problem is cascaded.

Closed forms Explicitly defined Why should it be cascaded ? Series within another Implicitly defined (1) Shift in the z-direction, (2) amplitude is not modified, (3) Shift is defined in terms of α1 Final Result map Horizontal moving :

Large contrast model 1500 m/s 2000 m/s 3000 m/s Contrast is greatly increased to break the leading imaging subseries

Perfect water-bottom Uneven pull-up What will do ?

The leading order imaging subseries

Shifted leading order imaging subseries An example of recursive modification

The higher order imaging subseries Water-bottom untouched Applying the moving term of simultaneous imaging - inversion

New understanding Why the moving term implicitly defined? Innanen K. A , “Reflector location using high-order inverse scattering terms” M-OSRP Annual report, (2005)

From equation (4) in Kris’s last annual report

The intuitive leap can be derived naturally as a result of proper integration of the δ–function. And ignoring the amplitude term (Jacobian term).

Salt model x z Lateral variation no longer restricted to water bottom Big contrast 1500 (m/s) 2328.75 (m/s) 4600 (m/s) 2463.75 (m/s) 3570 (m/s) 3855 (m/s) 4170 (m/s)

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

Linear imaging result Strong residual diffractions because we used water to migrate Bowtie still visible

Basic features stay the same as simple models

Higher order imaging subseries

Linear imaging

Higher order imaging subseries

Terms dealing with diffractions First term in without 1D analogy The linear image

Second term ( ) Generalize 1D terms Terms with no 1D analog

Tip of an iceberg It belongs to at least the following 2 subseries

More general framework Generalize Issues: missing low frequency

Corresponding closed forms generalize

Linear imaging :

Linear imaging :

Higher-order imaging :

Higher-order imaging :

Sum of 11 angles ( )

Sum of 11 angles ( )

Conclusions New direct imaging method: (a) accommodate inadequate velocity, (b) non-linear in the data. More imaging terms had being identified and included in subseries with closed-forms. Encouraging numerical results. New understanding & more general framework.

Acknowledgments M-OSRP members ExxonMobil Upstream Research Company M-OSRP sponsors NSF-CMG award DMS-0327778 DOE Basic Sciences award DE-FG02-05ER15697