# 3-D poststack 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 Figure 4.1-15  Principles of migration based on diffraction summation. (a) Zero-offset section (trace interval, 25 m; constant velocity, 2500 m/s), (b) migration. The amplitude at input trace location B along the flank of the traveltime hyperbola is mapped onto output trace location A at the apex of the hyperbola by equation (4).

To understand 3-D migration, consider a point scatterer that is buried in a constant-velocity medium. The zero-offset traveltime curve in two dimensions is a hyperbola given by equation (4-4). The curvature of the hyperbolic trajectory for amplitude summation is governed by the velocity function. The equation for this trajectory derived from the geometry of Figure 4.1-15 is rewritten here as

 $t^{2}=\tau ^{2}+{\frac {4x^{2}}{v_{rms}^{2}}}.$ (12)

Refer to Figure 4.1-15 to review the 2-D migration based on diffraction summation. Given the output time τ, which coincides with the the apex of the hyperbola, compute the input time t at sample location B, and deposit the amplitude at that input location on the output section at location A, corresponding to the output time τ at the apex of the hyperbola. Assuming a horizontally layered earth model, the velocity function used to compute the traveltime trajectory described by equation (12) is the rms velocity at the apex of the hyperbola at time τ (normal moveout).

We can imagine that the zero-offset response of a point scatterer in three dimensions is a hyperboloid of revolution described by the equation

 $t^{2}=\tau ^{2}+{\frac {4(x^{2}+y^{2})}{v_{rms}^{2}}},$ (13)

where x and y are the inline and crossline coordinates of the input sample at two-way zero-offset time t. Migration in three dimensions amounts to summing amplitudes at times t over the surface of the hyperboloid and placing the result at time τ that coincides with the apex of the hyperboloid.

In migration principles, we learned that the simple diffraction summation technique for migration can be improved by making the appropriate amplitude and phase corrections based on the far-field term of the Kirchhoff integral solution to the scalar wave equation before summation. As for the 2-D case (equation 4-5), the output 3-D image Pout(x0, y0, z = /2, t = 0) from Kirchhoff summation at a subsurface location (x0, y0, z) is computed from the 3-D zero-offset wavefield Pin(x, y, z = 0, t), which is measured at the surface (z = 0), by a summation over an areal aperture in the inline and crossline directions given by

 $P_{out}={\frac {\Delta x\Delta y}{4\pi }}\sum _{y}\sum _{x}\left[{\frac {\cos \theta }{v_{rms}r}}\rho (t)\ast P_{in}\right],$ (14)

where vrms is the rms velocity at the output point (x0, y0, z), and $r={\sqrt {(x-x_{0})^{2}+(y-y_{0})^{2}+z^{2}}},$ which is the distance between the input (x, y, z = 0) and the output (x0, y0, z) points. The output image Pout is computed at (x0, y0, z = /2, t = 0) using the input wavefield Pin at (x, y, z = 0, t − r/v). For a formal mathematical treatise of the Kirchhoff integral solution to the 3-D scalar wave equation, see Section H.1.

Equation (14) can be used to compute the wavefield at any depth z. To obtain the migrated section at an output time τ, equation (14) must be evaluated at z = /2 and the imaging principle must be invoked by mapping amplitudes of the resulting wavefield at t = 0 onto the migrated section at output time τ. The complete migrated section is obtained by performing the summation in equation (14) over an areal aperture and setting t = 0 for each output location. For 2-D migration, as many as 300 traces may be included in the summation. In three dimensions, this implies that as many as 70,000 traces may need to be included in the summation. Among the early works in 3-D migration are French  who provides an excellent and physically intuitive description of the process based on wavefield modeling, and Schneider  who provides a detailed theoretical discussion on 3-D Kirchhoff migration with some field data examples.

The rho filter ρ(t) in equation (14) corresponds to the time derivative of the measured wavefield, which yields the 90-degree phase shift and adjustment of the amplitude spectrum by the ramp function ω of frequency (Table A-1, Section A.1). Since the rho filter is independent of the spatial variables, it actually can be applied to the output of the summation in equation (14). Finally, the far-field term in equation (14) is proportional to the cosine of the angle of propagation (the directivity term or the obliquity factor) and is inversely proportional to vr (the spherical spreading term) in three dimensions.