Prestack depth migration in angledomain using beamlet decomposition

Prestack depth migration in angle-domain using beamlet decomposition: Local image matrix and local AVA Ru-Shan Wu and Ling Chen Modeling and Imaging Laboratory/IGPP University of California, Santa Cruz ---------------------------†Presently at Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, China

v Beamlet decomposition: Wave field in angle-domain v Local image matrix and local scattering matrix v Effect of acquisition aperture v Local AVA: Preliminary tests v Conclusion

True-reflection imaging in angle-domain Preserving the relative amplitudes of scattered waves w. r. t. incident waves. Benefits: • • • Improve image (total strength image) quality, especially for steep reflectors. Reduce artifacts (angle-domain filtering). Provide basis for local AVA and local inversion

True-amplitude imaging in angle-domain Amplitude corrections (for ray theory see Hubral et al. , Bleistein et al. , Xu et al. , Audebert et al. , ……): • Transmission loss (boundary reflection and scattering) • • • Geometric spreading (for ray method) Nonuniform information distribution: Jacobian (Beylkin determinant) Acquisition aperture effects (hit-count for ray method) • Intrinsic attenuation (Anelasticity)

True-reflection imaging in angle-domain for wave-equation based methods Preserving the relative amplitudes of scattered waves w. r. t. incident waves: • • • Nonuniform information distribution: Jacobian Acquisition aperture effects (in angle-domain) (including the geometric spreading and hit-count for ray method) Transmission and anelastic losses are less important, especially for small-angle reflections

Ø Windowed plane waves G-D frame atoms is a Windowed Plane wave (each beamlet is a windowed plane wave)

Ø Local plane waves Local plane wave: a superposition of windowed plane waves of the same local wavenumber from all neighboring windows: Partial reconstruction of wavefield (mixed domain wavefield: local phase–space): The corresponding propagating angle:

Source ** Receiver High-velocity body Target area Local Scattering Matrix Local Image Matrix (includes aperture and propagation effects)

Point scatterer Planar reflector shallow deep Local image matrix in homogeneous medium (total 201 shots with 176 left-hand receivers )

Ø Local image matrix: matrix image condition in beamlet domain and mixed domain ä Forward-propagated source field: ä Backward-propagated scattered field:

Local image matrix: Where Serves as the Jacobian Ws and WRs are the wave fields in angle-domain by beamlet decomposition

Stacking over frequency to get the final image In the local angle-domain: The final image in space domain:

Local Reflection Analysis (AVA): For planar reflectors: the local image matrix can be represented as: with

CDAI (common dip-angle image) gathers Sum up all reflections for a common dipangle: CDAI gather Obtain the dip-angle of the local reflectors from CDAI.

CRAI (common reflection-angle image) gathers Sum up reflected energy for a common reflection-angle for all possible dip-angles: CRAI. Performing local AVA from CRAI gathers. The calculation of local reflection coefficients:

Local AVA for an oblique interface in homogeneous background

Local image matrices at a point on the middle of dipping interface 14° obtained from 80 shots with a two-side receiver array (513 receivers). The dotted line corresponds to theoretical prediction without aperture effect.

CDAI gathers for a local reflector at its central location Obtain the dip-angle of local reflectors from the CDAI gathers

Calculated reflection coefficients from CRAI gathers . Angle-dependent reflection coefficients at the interface using 256 shots with 513 two-side detectors for the horizontal layered model with different velocity contrasts: (a) 10%; (b) 25%; (c) 50%; (d)150%

Dotted: synthetic; Red: 513 points two-sides Blue: 257 points one-side; Green: 129 points two-sides Angle-dependent reflection coefficients at the interface obtained from LIM in case of 10% velocity contrast for the horizontal layered model

³Local image matrix and the local scattering matrix The local image matrix has the acquisition-aperture and propagation effects included. The purpose of the imaging/inversion is to recover the real local scattering matrix and obtain the local reflection coefficients. To achieve the true-reflection imaging, we need to estimate the acquisition-aperture effect and apply the correction.

Acquisition-Aperture Efficacy (Effect of the source-receiver configuration) • • • Acquisition-Aperture Efficacy (AAE) Matrix Acquisition-Aperture Dip-response function Aperture corrections

Source ** Receiver Overburden structures Assume scattering Coefficients as 1 Target area Acquisition-Aperture Efficacy: (includes propagation effects)

Ø Acquisition-aperture efficacy matrix With unit impulses at both source and receivers, the local acquisition-aperture efficacy matrix is obtained as: Where G’s are the Green’s functions in beamlet domain

Ø Acquisition-aperture dip-response function Acquisition-aperture dip-response as a function of dip-angle of local interface (reflector), which reduce the AAE matrix into a vector: with

Acquisition-Aperture Dip-Response (Acquisition Configuration Response) S 2 S 3 * * * S 1

image by common-shot prestack G-D migration Acquisition-Dip-Response (horizontal reflector) from all the 325 shots Acquisition-Dip-Response (45 down from horizontal) from all the 325

G-D beamlet prestack migration image Acquisition-Dip Response for 45 o reflectors Improved image after Directional illumination correction

Conclusion • • Local image matrix can be obtained from the local incident and scattered plane waves based on beamlet decomposition The goal of true-reflection imaging in angle-domain is to remove the acquisition aperture effect and propagation effect through directional illumination analysis and the corresponding corrections

Conclusion (continued) • • • CDAI and CRAI gathers can be deduced from local image matrices (after corrections) CDAI gathers can be used to determine the dip-angles of local reflectors CRAI gathers can be used for local AVA analysis (and further for local inversion)

Acknowlegement § We thank the support from WTOPI Research Consortium at UCSC § We thank the support from DOE Project at UCSC ______________________ Welcome to visit our Consortium booth #2745