M-OSRP 2006 Annual Meeting, June 6, 2007 Implementation of first term in the inverse scattering series: practical strategies and issues. Fang Liu, Arthur B. Weglein, Bogdan G. Nita , Kristopher A. Innanen, Jingfeng Zhang M-OSRP 2006 Annual Meeting, June 6, 2007 M-OSRP report pages: 180-203

Key points Physics guided us towards the best implementation of . is the input to all of our leading-order and higher-order imaging subseries. Its accurate calculation is a critical step. Computational issues: fast and accurate linear inversion without interpolation, better approximation of the singular spectrum.

Objective For traditional migration, is the objective. For our purposes, is the beginning of our calculation. Hence there is a very high bar on being . Striving towards that high-bar in the face of imperfect acquisition, and algorithmic issues. To obtain a best-effort linear term for inverse scattering series (ISS) applications.

What is our ? The first term in ISS (Weglein et al. 2003). Closest to: Migration & inversion (Clayton & Stolt 1981). Similar to current seismic migration methods. Similar to the term used in the internal multiple attenuation algorithm (Weglein et al. 1997).

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

Solution: the inverse scattering series

Definitions and notations Ignoring constants (zg, zs)

Solution for the linear term Solution in the wave-number domain (km , kz) There are 3 degrees of freedom in the data. There are 2 degrees of freedom in the object. Select a 2D slice from a 3D spectrum space. I’m finding for a fixed angle in the data. all depend on the fixed angle. Their sum, , is independent of the fixed angle.

A common implementation 3D FFT (km , kz) Choose (km, kz) Divide factor Interpolation to make sampling rate in kz uniform 2D FFT

Our implementation (fixed θ) 1D Radon 1D FFT 1D integral with 1D FFT

Choosing a portion of the spectrum Our subset of the data is for variable ω, variable km, and fixed angle. Remarks: This represents a plane-wave in the CMP gather. Can be reduced to the previous result of S. Shaw and H. Zhang. Best effort to fit the plane-wave logic of the Zoeppritz equation. Closest in form to the physics underlying AVO.

Important relationship Dispersion relation Invert Uniform sampling in kz implies non-uniform sampling in ω , this is not available for Fast Fourier transform. One approach to solve this issue is to interpolate. The other approach is to grab the spectrum on need without the benefit of Fast Fourier transform.

This has not proven to be an issue for the imaging subseries. Missing spectrum The dispersion relation precludes filling certain portions of the spectrum of This has not proven to be an issue for the imaging subseries.

formula We compute (ignoring the constant θ) : Issue, the integral is no longer a Fast Fourier transform since the sampling interval in ω is no longer uniform

Speeding up the transform with a pre-calculated kernel

Speeding up the transform with a pre-calculated kernel

First: crunch the dkz integral

Second: crunch the dxm integral

Third : crunch the dτ integral

Computing kernel . ωis computed via Sampling in z and kz is kept constant to allow FFT. Calculate once and used over and over again. Fourier transform over time, and interpolation are skipped. Saving in computation time.

Ideal and adequate data A simple Fourier transform requires infinitely small sampling interval and infinitely large aperture. Do we really need this? NO!

On sampling interval Sampling rate in time is fixed : Deepest target : We have looked at synthetic experiments with dx=5, dx=10, dx=20, dx=40, for instance dx of 40 is sufficient for our purposes.

On aperture Tmax=4.2(s) For fixed recording time, there exists a maximal useful/necessary horizontal aperture. Beyond the blue lines, there is no additional information.

Data reconstruction Data reconstruction & regularization techniques:

Pre-processing Wavelet estimation. Data reconstruction & regularization. De-ghosting (including direct removal). Free-surface multiple removal. Internal multiple removal.

Pre-processing literatures wavelet estimation and de-ghosting

Pre-processing literatures free-surface and internal multiple

Wavelet physics What is our wavelet? Source term in the wave equation. That is the wavelet we want to deal with. What is it not? It is not any event affected by propagation in the earth.

Best wavelet in theory Lippmann-Schwinger equation is written as if . is not required for the inverse scattering task-specific subseries, i.e., free-surface multiple removal, internal multiple removal, and imaging. The imaging subseries responds well to a Ricker wavelet, that is twice integrated.

Conclusions Physics guided us towards the best implementation of . is the input to all of our leading-order and higher-order imaging subseries. Its accurate calculation is a critical step. Computational issues: an approach to fast and very accurate linear inversion.

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.