# Phase-shift migration

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Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

## Contents

Theory of the frequency-wavenumber (f − k) migration techniques is left to Section D.7. For now, we briefly review the f − k migration algorithms as follows:

1. Just as any other migration algorithm, start with the two-way scalar wave equation (12).
2. Assume constant velocity and perform 3-D Fourier transform and obtain the dispersion relation between the transform variables (equation 13a).
3. Then, adapt the dispersion relation to the exploding reflectors model by halving the velocity for the upcoming waves (equation 13b).
4. Operate on the pressure wavefield P and inverse transform in z to obtain the differential equation (20).
5. Obtain the solution given by equation (21).

The discrete form of this solution given by equation (22) is the basis for phase-shift migration in which velocity can be varied at each depth step in the vertical direction.

The phase-shift method involves the following steps:

1. Start with the stacked section — an approximation to the zero-offset section P(x, z = 0, t), and perform 2-D Fourier transform to get P(kx, z = 0, ω).
2. By using equation (22), for each frequency ω, extrapolate the transformed wavefield P(kx, z, ω) at depth z with a phase-shift operator exp(−ikzΔz) to get the wavefield P(kx, z + Δz, ω) at depth z + Δz. At each z step, a new extrapolation operator with the velocity defined for that z value is computed.
3. As for any other migration, invoke the imaging principle t = 0 at each extrapolation step to obtain the migrated section P(kx, z, t = 0) in the transform domain. The imaging condition t = 0 is met by summing over all frequency components of the extrapolated wavefield at each depth step. This is easily shown from the integral representing the inverse Fourier transform of the extrapolated wavefield (equation D-84).
4. Repeat steps (b) and (c) for downward continuation and imaging, respectively, for all depth steps to get the migrated section in the transform domain P(kx, z, t = 0).
5. Final step involves inverse transforming in the x direction to get the migrated section P(x, z, t = 0). Figure 4.1-29  Flowchart for Gazdag’s phase-shift method of migration.

Figure 4.1-29 shows a flowchart of the phase-shift method.

The phase-shift method  can only handle vertically varying velocities. A way to accommodate lateral velocity variations judged to be acceptable for time migration is to first stretch the CMP-stacked section in the time direction so as to make it correspond to a velocity field ${\bar {v}}(z)$ that only varies vertically. This velocity field is obtained by averaging the original velocity field associated with the unstretched CMP-stacked section in the x direction. Following the stretching operation, the stacked section is migrated using the velocity function ${\bar {v}}(z)$ in the standard phase-shift migration scheme. Finally, the migrated section is unstretched.

Gazdag and Squazzero  extended the phase-shift method to handle lateral velocity variations. To achieve this, first the input wavefield is extrapolated by the phase-shift method using a multiple number of laterally constant velocity functions and a series of reference wavefields are created. The imaged wavefield then is computed by interpolation from the reference wave-fields. This migration method is known as phase-shift-plus-interpolation. An alternative extension of phase-shift migration to handle lateral velocity variations is presented by Kosloff and Kessler (1987).

## Equations

 ${\frac {\partial ^{2}P}{\partial x^{2}}}+{\frac {\partial ^{2}P}{\partial z^{2}}}-{\frac {1}{v^{2}(x,z)}}{\frac {\partial ^{2}P}{\partial t^{2}}}=0,$ (12 )

 $k_{z}=\mp {\sqrt {{\frac {\omega ^{2}}{v^{2}}}-k_{x}^{2}}},$ (13a )

 $k_{z}={\frac {2\omega }{v}}{\sqrt {1-\left({\frac {vk_{x}}{2\omega }}\right)^{2}}},$ (13b)

 $P(k_{x},z,\omega )=P(k_{x},0,\omega )\exp(-ik_{z}z).$ (21 )

 $P(z+\Delta z)=P(z)\exp(-ik_{z}\Delta z),$ (22a )

 $P(z+\Delta z)=P(z)(1-ik_{z}\Delta z),$ (22b )

 $P(z+\Delta z)=P(z)\left[{\frac {1-ik_{z}\Delta z/2}{1+ik_{z}\Delta z/2}}\right].$ (22c )