Reflector steepening

Consider a vertical wall, a limiting case of a dipping bed. Its reflections, the asymptotes of a hyperbola, have a nonvertical steepness. This establishes that migration increases the apparent steepness of dipping beds. I use the words apparent steepness because it is the slope as seen in the (x,t)-plane that has steepened. Migration really produces its output in z, but z/v is often overlain on t to create a migrated time section . When we say a hyperbola migrates to its apex, we are of course thinking of the migrated time section. Let us determine the steepening as a function of angle.

Consider a point $(x_{0+} , t_{0+} ) \ =$$(x_0 \,+\,\Delta$, $t_0 \,+\,p \Delta )$ neighboring the original point (x₀ , t₀ ). By equations (4) and (5), this neighbor migrates to

$$\begin{eqnarray} t_{m+} \ \ \ &=&\ \ \ { (t_0 \,+\,p\,\Delta ) } \ { \sqrt { 1 ... ...\ \ \ &=&\ \ \ x_0 \ +\ \Delta \ -\ (t_0 \,+\,p\,\Delta ) \,p\,v^2\end{eqnarray}$$ (6) (7)

Now we compute the stepout p_m of the migrated event

$$\begin{eqnarray} p_m \ \ \ &=&\ \ \ {dt_{m+} \over d x_{m+}} \ \ =\ \ {dt_{m+}... ... \sqrt { 1 \,-\,p^2 \, v^2 } } \ \ =\ \ { \tan \, \theta \over v }\end{eqnarray}$$ (8)

So slopes on migrated time sections, like slopes in Cartesian space, imply tangents of angles while slopes on unmigrated time sections imply sines.

It may seem paradoxical that dipping beds change slope on migration whereas flanks of hyperbolas do not change slope during downward continuation. One reason is that migration is downward continuation plus imaging (selecting t=0). Another reason is that a hyperbola is a special event that comes from a single source at a single depth whereas a dipping bed is a superposition of point sources from different depths.

 

dip
Figure 7 Left is a superposition of many hyperbolas. The top of each hyperbola lies along a straight line. That line is like a reflector, but instead of using a continuous line, it is a sequence of points. Constructive interference gives an apparent reflection off to the side. Right shows a superposition of semicircles. The bottom of each semicircle lies along a line that could be the line of an observed plane wave. Instead the plane wave is broken into point arrivals, each being interpreted as coming from a semicircular mirror. Adding the mirrors yields a more steeply dipping reflector.

Figure 7 shows how points making up a line reflector diffract to a line reflection, and how points making up a line reflection migrate to a line reflector.