Next: Migration tests Up: Biondi: Narrow-azimuth migration Previous: Analysis of common-azimuth migration

# Two schemes for narrow-azimuth migration

The kinematic analysis presented in the previous section suggests a generalization of common-azimuth migration based on the downward continuation of a narrow strip around the zero crossline offset. The computational cost of such a generalization is obviously proportional to the number of crossline offsets used to represent this narrow strip. The width of the strip is dependent on the reflector geometry and on the velocity model, but the sampling depends on the crossline-offset dip spectrum. To minimize the cost it is crucial to define an optimal criterion to define the range of crossline-offset dips (pyh). As demonstrated in the previous section, for dipping reflectors the dip spectrum is not centered around the zero dip (pyh=0), and thus a symmetric range would be wasteful. I exploit the information provided by the common-azimuth equation to define a range of crossline-offset dips. For this reason I named my generalization narrow-azimuth migration, even if narrow crossline-offset would be a more accurate name.

The common-azimuth equation provides the crossline-offset dip pyh as a function of the other dips in the data when the data are propagated along a constant azimuth Biondi and Palacharla (1996). In the frequency-wavenumber domain the common-azimuth relationship is:
 (1)
where is the temporal frequency, kxm and kym are the midpoint wavenumbers, kxh and kyh are the offset wavenumbers, and and are the local velocities. Ideally we would like to define a range of kyh that is varying with depth, as a function of the local velocities. However, that may lead to complex implementation. For the moment I chose a simpler solution. I define a range of kyh by setting a minimum velocity and a maximum velocity ,and define
 (2)
and
 (3)
The disadvantage of this solution is that the choice of and is somewhat arbitrary.

The central point of the kyh range is then defined as a function of and as
 (4)
and the range as
 (5)
where Nyh is the number of crossline offsets and dkyh is the sampling in kyh.

The two narrow azimuth schemes that I propose and tested differ in the definition of the crossline-offset wavenumber sampling dkyh. The first, and simplest, uses a constant value for dkyh; that is
 (6)
The second one allows dkyh to vary as
 (7)
Varying dkyh is equivalent to vary the width of the crossline-offset strip, and, at constant Nyh, is also equivalent to vary the sampling .At lower frequencies dkyh is smaller, and thus the maximum crossline offset is larger. The migrated results benefit because the lower frequencies are the most affected by boundary artifacts and a wider strip reduces the artifacts caused by the boundaries. On the contrary, as the frequency increases, dkyh is larger and smaller, and thus spatial aliasing is avoided.

The main disadvantage of this second scheme is that the transformation between space and wavenumber domains becomes more cumbersome. Mixed space/wavenumber domain downward-continuation methods Biondi (1999) become more expensive, and thus the scheme becomes less attractive when the velocity is laterally varying.

Next: Migration tests Up: Biondi: Narrow-azimuth migration Previous: Analysis of common-azimuth migration
Stanford Exploration Project
4/29/2001