Multi-Offset Data Volume Example
This page illustrates the use of a special type of stacking chart to slice
common-midpoint (CMP) gathers out of a volume assembled from multiple shot gathers.
Thus the traditional CMP sorting operation becomes a simple oblique slicing
in 3-d data space.
The slices also demonstrate the geometries of flat and dipping reflections in
multi-offset data space.
By clicking on an image here you can get a high-resolution version in
Adobe Acrobat PDF format.
Consider an off-end survey across the synthetic, constant-velocity section
at left.
We have 50 shots at 2 m intervals pushing to the right a spread of ng=50
receiver groups also spaced at 2 m, for a minimum offset of 0 m and
a maximum offset of 98 m.
We set up the data volume as having 50 planes, one for each shot, with
each plane as a constant-shot gather.
However, unlike the usual stacking chart, we do not stagger each shot gather
according to the receiver roll-along, but instead make the horizontal axis
in the chart at left a receiver offset axis.
We want to find a line of constant midpoint so we will be able to slice the
data volume in that direction to obtain CMP gathers.
The shot gather at Xsource=50 m and its trace at Xoffset=100 m has its midpoint
at X=100 m. The trace at Xoffset=0 m and Xsource=100 m has the same midpoint,
Xmidpoint=100 m.
The line on the stacking chart tilted 30 degrees up from the Xoffset axis
is thus a line of constant midpoint.
Any line parallel to it will intersect traces that all have the same midpoint.
The width of those traces will define the fold, or the number of traces
at that midpoint, which has a maximum of 25 for this survey.
The line perpendicular to the fold axis is the Xmidpoint axis.
The 3-d view at left above shows the synthetic acoustic data volume, rendered so
the higher positive amplitudes appear as solid objects in warmer colors.
The stacking chart labels the top of the volume, at time t=0.
The front right plane of the volume is the shot gather for Xsource=100 m,
and the rear plane, out of sight, is the shot gather at Xsource=0 m.
The left front plane is the zero-offset section, and the right rear plane,
out of sight, is the far-offset section.
The image at above at right shows a plot of the zero-offset section.
The way to tell the origin of a reflection in the volume is to look at it
in the zero-offset section, which bears similarities to the model section
at the top of the page that we are used to looking at from our experience
with the migration of zero-offset sections.
Working down in the zero-offset section from zero time, we see the direct
wave, a reflection from the upper flat reflector, a multiple of that reflection,
the dipping reflection, a multiple of the dipping reflection, and at the
bottom the lower flat reflection.
Note that in the shot gather the dipping reflection has its hyperbolic apex
at a non-zero offset.
On the left side of the zero-offset section are also hyperbolic diffractions
that are computational artifacts due to the truncation of reflectors at the
edge of the model finite-difference grid.
You can experiment with slicing this volume yourself.
It is provided here in Sun's .vff format, which you can
examine using the SunVision SunCube tools on SunOS, or using
MacCubeView on
a Macintosh.
This file contains the substance attribute and
transparency definitions used in SunCube for the renderings here.
Now, to show a CMP gather instead of just a common-shot or common-offset gather,
we make an oblique but still vertical slice of the volume parallel to the line
of constant midpoint.
On the left is a CMP gather that has high fold, or many traces, and on the
right is a lower-fold gather from a midpoint near the right end of the survey.
Note that in the CMP gathers every reflection has its hyperbolic apex at zero
offset.
These are similar views of a data volume that has been corrected for simple
normal moveout using the correct model velocity.
The NMO correction flattens the asymptotes of reflection hyperbolas to put the
entire reflection at the times the original hyperbolas had a zero-offset.
It reduces multi-offset reflection times to zero-offset reflection times.
Note two effects of the simple geometric NMO correction: the far offsets of the
reflections have experienced ``NMO stretch'', which artificially lowers their
frequency; and the dipping reflection has been over-corrected and now bends
slightly upwards.
The over-correction is due to the fact that the asymptotic moveout velocity of
a dipping reflection is increased by dividing the real velocity by the cosine
of the dip. So the NMO correction, even given perfect velocity information,
is only appropriate for zero-dip reflections.
The final part of the traditional stacking process is to simulate a zero-offset
section by summing the NMO-corrected CMP gathers along lines of constant time.
Thus the 3-d multi-offset data volume is reduced to an approximation of the
time section on the zero-offset plane, with random noise reduced by the summation.
At left above is a view of the NMO-corrected data volume right in the direction
that stacking proceeds.
This view matches the completed stack shown at right above.
Note that the stack includes information at midpoints toward the right side of the
model, where there were no zero-offset traces.
The stack at those midpoints incorporates the reflections recorded on the
farther-offset traces.
J. Louie, Oct.30 1996