# Prestack Stolt migration

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

The first method for migration velocity analysis that we shall review is based on migrating prestack data using a range of constant velocities and creating constant-velocity migration (CVM) panels [1]. Since migrations are performed using constant velocities, an appropriate choice for the algorithm would be prestack frequency-wavenumber migration (Section E.6). A flowchart for the CVM approach for migration velocity analysis is shown in Figure 5.4-3.

Figure 5.4-3  A flowchart of an algorithm for prestack Stolt migration.

If the medium velocity is constant, then migration can be expressed as a direct mapping [2] from temporal frequency ω to vertical wavenumber kz (Section E.6). The equation for Stolt mapping is

 ${\displaystyle P(k_{y},k_{h},k_{z},t=0)=\left[{\frac {v}{2}}{\frac {k_{z}^{2}-k_{y}^{2}k_{h}^{2}}{\sqrt {(k_{z}^{2}+k_{y}^{2})(k_{z}^{2}+k_{y}^{2})}}}\right]\times P\left[k_{y},k_{h},0,{\frac {v}{2k_{z}}}{\sqrt {(k_{z}^{2}+k_{y}^{2})(k_{z}^{2}+k_{h}^{2})}}\right],}$ (40)

where y, h, and t are the variables for midpoint, offset and event time in the unmigrated position, and ky, kh, and ω are the associated Fourier transform variables.

Note that Stolt migration involves, first, mapping from ω to kz for a specific ky and kh by using the dispersion relation for prestack wave extrapolation (Section E.6):

 ${\displaystyle \omega ={\frac {v}{2k_{z}}}{\sqrt {(k_{z}^{2}+k_{y}^{2})(k_{z}^{2}+k_{h}^{2})}}.}$ (41)

The output of mapping is then scaled by the quantity

 ${\displaystyle S={\frac {v}{2}}{\frac {k_{z}^{2}-k_{y}^{2}k_{h}^{2}}{\sqrt {(k_{z}^{2}+k_{y}^{2})(k_{z}^{2}+k_{h}^{2})}}}.}$ (42)

Stolt migration output normally is displayed in two-way vertical zero-offset time τ = 2z/v. In practice, mapping in the f − k domain really is from ω to ωτ rather than from ω to kz, where ωτ is the Fourier dual of τ, and is simply kz scaled by v/2. Accordingly, equations (40), (41), and (42) are recast in terms of ωτ = (v/2)kz when implemented in practice.

Migration velocity analysis based on Stolt’s prestack algorithm for constant velocity thus involves the following steps:

1. Starting with prestack data P(y, h, t) in coordinates of midpoint y, offset h and two-way event time t in the unmigrated position, perform 3-D Fourier transform to obtain the transformed volume of data P(ky, kh, w), where ky, kh, and ω are the Fourier transform duals of the variables y, h, and t, respectively.
2. For each trial constant velocity v, use equation (41) to map the transform variable ω — the temporal frequency associated with the input data P(ky, kh, ω), to ωτ — the temporal frequency associated with the migrated data P(ky, kh, ωτ; v). This mapping of complex numbers is the basis for constant-velocity prestack Stolt migration (Section E.6).
3. Apply the scaling factor of equation (42).
4. Invoke the imaging principle by setting t = 0 and obtain P(ky, kh, ωτ, t = 0).
5. Sum over the offset wavenumber kh to obtain the image at zero offset, yet in the transform domain, P(ky, h = 0, ωτ; v).
6. Perform 2-D inverse Fourier transform to obtain the constant-velocity migrated zero-offset section, P(y, τ; v).
7. Repeat steps (b) through (f) for a range of constant velocities to obtain the migration velocity volume P(y, τ; v). By viewing this volume, it can be incised to obtain the surface of optimum migration velocity field with an accompanying image derived from prestack time migration.

Practical issues related to prestack Stolt migration include spatial aliasing along the offset axis and cost of Stolt mapping in steps (b) and (c). The spatial sampling along the offset axis often is too coarse for shallow events with low velocity; this gives rise to large moveout on CMP gathers. A linear moveout may be applied to CMP gathers to circumvent spatial aliasing. Equation (40) for Stolt mapping is then modified accordingly [3].

The Stolt mapping of amplitudes for prestack data involves interpolation of complex numbers in the transform domain. This involves the three input variables ky, kh, and ω, and the output variable ωτ, and thus is quite costly when one has to consider as many as 100 or more constant velocities. A way to reduce the computational cost is to perform prestack migration using a set of constant velocities at coarse interval, followed by poststack residual constant-velocity migrations of the zero-offset sections from prestack migration to fill in between the coarsely sampled migration velocity panels [3].

## References

1. Shurtleff, 1984, Shurtleff, R., 1984, An F − K procedure for prestack migration and velocity analysis: Presented at the 46th Ann. Mtg. European Asn. Expl. Geophys.
2. Stolt, 1978, Stolt, R.H., 1978, Migration by Fourier transform: Geophysics, 43, 23–48.
3. Li et al., 1991, Li, Z., Lynn, W., Chambers, R., Larner, K. and Abma, R., 1991, Enhancements to prestack frequency-wavenumber (f − k) migration: Geophysics, 56, 27–40.