Reverse-Time Migration Geol 757 Advanced Seismic Imaging and Tomography

References Paul Sava and Stephen J. Hill, Tutorial: Overview and classification of wavefield seismic imaging methods: The Leading Edge, February 2009, v. 28, p. 170-183, doi:10.1190/1.3086052. Edip Baysal, Dan D. Kosloff, and John W. C. Sherwood, Reverse time migration: Geophysics, v. 48, no. 11 (Nov. 1983), p. 1514-1524. Matthew H. Karazincir and Clive M. Gerrard, Explicit high-order reverse time pre-stack depth migration: Expanded Abstracts, Soc. Explor. Geophys. New Orleans 2006 Annual Meeting, p. 2353-2357.

From Sava & Hill, 2009 What defines a WE migration? Classification based on: Assumptions of algorithms Domain of implementation Imaging Principle

WEM Classifications Single Scattering – no multiples in data Born approximation Wave-Equation Solutions – acoustic forward modeling Not Kirchhoff summation The acoustic equation cannot get close to Zoeppritz Not full-wave inversion

WEM Classifications Imaging and Wavefield Reconstruction Shot record migration – sequential, independent Survey-sinking migration - simultaneous

WEM Classifications Implementations in Sava & Hill: Shot record, 2-way in time, time domain Shot record, 1-way in depth, frequency domain Survey-sinking, 1-way in depth, frequency domain

The Wavefield 2D world Constant velocity Impulse source at t=0 at z=0 red dot

The Wavefield Constant-depth slices Hyperbolas Diffractions

The Wavefield Constant-time slices Semicircles Wave propagation

Migration Migration = Wavefield continuation + Imaging condition Continuation of full multi-dimensional wavefields

Migration Two different imaging conditions: Shot record, sequential imaging Survey-sinking, simultaneous imaging

Shot Record, Sequential Imaging Constant velocity Examine: Data Wavefields Image At: Source Receiver

Shot Record, Sequential Imaging (a) Model that generates data: Flat reflector above Dipping reflector below 2D Survey in x: Split spread Look at one shot record

Shot Record, Sequential Imaging (b) Fire impulsive source: t=0 z=0 Shot gather data: Two reflections Impulsive waves

Shot Record, Sequential Imaging (c) Source impulse data: Single red impulse t=0, z=0 Data at source, just like receiver data

Shot Record, Sequential Imaging (d) Exploding reflectors: Blue = horizontal Green = dipping Cones in const.-V From t=0 at recorded depth point

Shot Record, Sequential Imaging (e) Source radiation: Wavefield cone From t=0 From source x

Shot Record, Sequential Imaging (f & g) Imaging condition – Ws-R-Wr model: Scatterer exists at the spatial coordinate (x and z) that contains coincident, nonzero wavefield amplitudes in both the source and the receiver wavefields

Shot Record, Sequential Imaging (f & g) Imaging condition – Ws-R-Wr model: Reflectors exist where incident and reflected wavefields are coincident in time and space

Shot Record, Sequential Imaging (f & g) Imaging condition – Ws-R-Wr model: Ws and Wr coincide (nonzero) at some time t Doesn’t matter what t it was - only the coincidence

Shot Record, Sequential Imaging (h) (g) Ws(t) contains one nonzero value (red) at (x*, z*) (f) Wr(t) has two non-0 values (blue, green) at (x*, z*)

Shot Record, Sequential Imaging (h) This (x*, z*) is on upper reflector Ws(t) • Wr(t) gives non-0 at reflector

Shot Record, Sequential Imaging (h) Post nonzero Ws(t) • Wr(t) at (x*, z*) in (x, z) image Correlate at other (x, z) points and post their nonzero amplitudes Add in migrated sections for other shot gathers

Shot Record, Sequential Imaging Ws-R-Wr model, Berkhout (1982) Need the source and scattered wavefields Source wavefield carries energy to the reflector Scattered wavefield carries energy away from the reflector For 2D data, the wavefields are 3D W(x, z, t) For 3D data, the wavefields are 4D W(x, y, z, t)

Sequential Imaging Needs 1. Wavefield reconstruction that generates the source and scattered wavefields, WS and Wr, at all locations in space x, z and all times t from data recorded at the surface, and 2. An imaging condition that extracts reflectivity information, i.e. the image I, from the reconstructed source and scattered wavefields WS and Wr.

Imaging Principle Single-scattering assumption The incident and scattered wavefields are identical at the scatterer, except for: The reflection coefficient. Kinematically accurate- timing & structure Dynamically inaccurate- poor R, impedance, AVO Scattering cannot change wave phase. If there are multiples, the cross-correlated amplitude will be too high.

Wavefield Reconstruction Velocity Model Must be known a priori. In a smooth-velocity area, uncertainty will not prevent imaging. In the presence of strong lateral velocity contrasts, their complete characterization is essential. Code the velocity model into a procedure for generating wavefields from sources.

Wavefield Reconstruction Generating the Source Wavefield Ws Simulate each shot gather’s source, forward in time from its true position. Generating the Receiver Wavefield Wr Simulate each shot gather trace’s receiver position as a virtual source, at that receiver’s true position. Feed each receiver’s recorded data into each receiver “source,” as a source time function. Produces a “reversed time” wavefield from the data, projecting recorded amplitudes back onto the scatterers.

Wavefield Reconstruction Successful wavefield reconstruction relies on the single-scattering assumption for seismic imaging, i.e., Recorded wavefields have scattered only once in the subsurface (there are no multiples in the data), and No scattering occurs in the process of wavefield reconstruction. Full-wave modeling methods may not work well, since they always implement scattering with propagation.

Wavefield Reconstruction One-way Paraxial wave-propagation modeling will work well, since it cannot create reflections. Paraxial is also faster. Two-way modeling procedures can work so long as they do not introduce scattering – downward continuation, WKBJ ray tracing, deterministic traveltimes, etc. Any modeling method capable of handling lateral variations will introduce scattering. More reasons RTM is kinematic, not dynamic

Wavefield Reconstruction Axis Depth marching Downward continuation Paraxial wavefield extrapolation in the frequency domain Time marching Reverse-time migration with acoustic finite-difference modeling in the time domain

Extended Imaging Conditions Zero-lag, h=0 cross-correlation: Space and time shifts λx, λy, λz, τ:

Extended Imaging Conditions Create a multidimensional image I(x, y, z, λx, λy, λz, τ) Try amplitude-vs.-angle analysis Determine wavefield reconstruction error from very approximate wavefield reconstructions (one-way, low-order) from velocity error from multiples in the data from problems with acquisition coverage from incomplete subsurface illumination

Marmousi Model

Marmousi Model

Marmousi Model