Reverse-Time Migration

Geol 757

Advanced Seismic Imaging

and Tomography

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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.

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From Sava & Hill, 2009 What defines a WE migration? Classification based on:

Assumptions of algorithms Domain of implementation Imaging Principle

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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

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WEM Classifications Imaging and Wavefield Reconstruction

Shot record migration – sequential, independent

Survey-sinking migration - simultaneous

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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

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The Wavefield 2D world Constant velocity Impulse source

at t=0 at z=0 red dot

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The Wavefield Constant-depth

slices Hyperbolas Diffractions

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The Wavefield Constant-time

slices Semicircles Wave

propagation

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Migration Migration =

Wavefield continuation + Imaging condition

Continuation of full multi-dimensional wavefields

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Migration Two different

imaging conditions:

1. Shot record, sequential imaging

2. Survey-sinking, simultaneous imaging

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Shot Record, Sequential Imaging Constant velocity Examine:

Data Wavefields Image

At: Source Receiver

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Shot Record, Sequential Imaging (a) Model that generates

data: Flat reflector above Dipping reflector below

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2D Survey in x: Split spread Look at one shot

record

Shot Record, Sequential Imaging (b) Fire impulsive source:

t=0 z=0

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Shot gather data: Two reflections Impulsive waves

Shot Record, Sequential Imaging (c) Source impulse data:

Single red impulse t=0, z=0

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Data at source, just like receiver data

Shot Record, Sequential Imaging (d) Exploding reflectors:

Blue = horizontal Green = dipping

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Cones in const.-V From t=0 at

recorded depth point

Shot Record, Sequential Imaging (e) Source radiation:

Wavefield cone

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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

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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

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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

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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*)

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

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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

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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)

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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.

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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.26

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.

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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.

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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.

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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

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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

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Extended Imaging Conditions

Zero-lag, h=0 cross-correlation:

Space and time shifts λx, λy, λz, τ:

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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

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Marmousi Model

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Marmousi Model

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Marmousi Model

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