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 (p_yh). As demonstrated in the previous section, for dipping reflectors the dip spectrum is not centered around the zero dip (p_yh=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 p_yh 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:  

$$\widehat{k_{y_h}}= k_{y_m}\frac{\sqrt{ \frac{\omega^2}{v^2({... ...{{\bf s},z})} - \frac{1}{4} \left(k_{x_m}-k_{x_h}\right)^2 }}.$$ (1)

where $\omega$ is the temporal frequency, k_xm and k_ym are the midpoint wavenumbers, k_xh and k_yh are the offset wavenumbers, and $v({{\bf s},z})$ and $v({{\bf s},z})$ are the local velocities. Ideally we would like to define a range of k_yh 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 k_yh by setting a minimum velocity $v_{\rm min}$and a maximum velocity $v_{\rm max}$,and define

$$k_{y_h}^{\rm min} = k_{y_m}\frac{\sqrt{ \frac{\omega^2}{v_{\... ..._{\rm min}^2} - \frac{1}{4} \left(k_{x_m}-k_{x_h}\right)^2 }}.$$ (2)

and

$$k_{y_h}^{\rm max} = k_{y_m}\frac{\sqrt{ \frac{\omega^2}{v_{\... ..._{\rm max}^2} - \frac{1}{4} \left(k_{x_m}-k_{x_h}\right)^2 }}.$$ (3)

The disadvantage of this solution is that the choice of $v_{\rm min}$ and $v_{\rm max}$ is somewhat arbitrary.

The central point of the k_yh range is then defined as a function of $k_{y_h}^{\rm min}$ and $k_{y_h}^{\rm max}$as  

$$\widebar{k_{y_h}}=\frac{k_{y_h}^{\rm max}+k_{y_h}^{\rm min}}{2},$$ (4)

and the range as  

$$\widebar{k_{y_h}} - \left(\frac{N_{y_h}}{2}-1\right)dk_{y_h} \leq k_{y_h}\leq \widebar{k_{y_h}} + \frac{N_{y_h}}{2}dk_{y_h},$$ (5)

where N_yh is the number of crossline offsets and dk_yh is the sampling in k_yh.

The two narrow azimuth schemes that I propose and tested differ in the definition of the crossline-offset wavenumber sampling dk_yh. The first, and simplest, uses a constant value for dk_yh; that is  

$${dk_{y_h}}_{1}=\frac{2\pi}{N_{y_h} \Delta y_h}.$$ (6)

The second one allows dk_yh to vary as  

$${dk_{y_h}}_{2}=\frac{k_{y_h}^{\rm max}-k_{y_h}^{\rm min}}{N_{y_h}}.$$ (7)

Varying dk_yh is equivalent to vary the width of the crossline-offset strip, and, at constant N_yh, is also equivalent to vary the sampling $\Delta y_h$.At lower frequencies dk_yh 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, dk_yh is larger and $\Delta y_h$ 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.