Tieman's conversion of common-midpoint slant stacks to common-source
Hans Tieman spoke to the Stanford Exploration Project on Jan 23, 1996
about depth imaging with slant slacks. Among his techniques
is a clever method of converting slant stacks of midpoint gathers
into equivalent slant stacks of source gathers.
A source gather best represents a physical experiment
that can be modeled easily by wave-equation methods.
Midpoint gathers, however, better include the coherence of steep
reflections and better avoid aliasing.
A conversion takes advantage of the best of both domains.
Seismic amplitudes are
recorded over time as a function of surface horizontal
positions for source
and receiver . Although these positions are one-dimensional we must also
be prepared to think of them as vectors indicating a surface
position. The time coordinate is sampled evenly and densely enough that
we can think of it as continuous.
For a given source, we have a limited range of receivers (perhaps
3–5 kilometers), and vice versa.
Receiver positions are often sampled
two or four times as densely as source positions. In marine
data, both are relatively evenly sampled, but a spatial Fourier transform
must pay attention to aliasing or
edge effects from the short span. Land data will be much more
arbitrarily sampled.
Define the coordinates of full offset
and midpoint
. Resorted data can be written as
. The well-sampled midpoint coordinate covers
the entire span of the survey.
Slant stacks are commonly applied to unsorted data, one
shot at a time. This form is well suited to deconvolution
of multiple reflections from flat reflectors. Such multiple
reflections are periodic at zero-offset, but not at a single
finite offset .
A slant stack attempts to describe our recorded data as
a sum of dipping lines.
A dip will measure the slope of
time with offset holding a source position constant.
(1)
With ideal sampling and infinite offsets,
this equation would describe a plane-wave
source on the surface. A plane wave reflecting from
flat reflectors would produce periodic multiples at any .
Predictive deconvolutions can detect this periodicity and
remove multiple reflections.
The simplest slant-stack sums data over all lines within
a feasible range of dips.
Let be the intersection at zero offset of our imaginary
plane wave in the shot gather.
(2)
In practice the integral over offset must be a discrete
sum with a limited range of offsets.
The inverse of this transform looks much like another slant
stack, with some adjustments of the spectrum. Papers
are readily available to explain this inverse. I will concentrate
instead on the conversion of one type of slant stack to another.
Because the time axis is well sampled and unaliased, we can safely
Fourier transform the data between time and frequency :
(3)
Tildes will indicate Fourier transforms.
Transform the slant stack from to its frequency :
(4)
The slant stack simplifies numerically.
Substitute the transform (3) into the slant stack (2),
then take the transform (4) of both sides of the equation:
(5)
Rearranging terms, we reduce integrals
The second step uses the Fourier transform of a delta function
.
The third uses
the behavior of a delta function in an integral
.
We should prefer a transform of a single source gather
because these gathers correspond to a physical experiment
that can be modeled easily by wave-equation methods.
Unfortunately,
reflections from dipping layers and point scatters may have
a very complicated expression in a source gather.
We may be obliged to use many dips to capture their coherence.
Worse, many reflections will have minimum times at finite offset,
and a slant stack will alias some of their energy.
If the data are first sorted by midpoint and half-offset ,
then reflections from dipping lines and from points will
still remain symmetric about zero offset. A slant
stack of a midpoint gather will better capture the
coherence of the reflections:
(7)
where
(8)
The Fourier version of a common-midpoint slant stack
can be derived exactly as before.
Let be the Fourier frequency of :
(9)
Unfortunately, this slant stack does not correspond to any
single seismic experiment, and wave-equation modeling is much
more awkward.
Fortunately, we can convert this common-midpoint transform (9) into
an equivalent common-source transform (6).
Let us make two additional Fourier transforms over spatial dimensions
of and for the spatial frequencies and :
and
To place the second integral (11) in the form of the
first (10), we should change the variables of integration
from and to and . (The Jacobian of this transformation
is
.) Substituting we get
Thus, a two-dimensional stretch of the midpoint-gather transform
becomes equivalent to the source-gather transform.
For a given dip over offset in a midpoint gather
, we can identify a dip over midpoint
(13)
The adjustment of
subtracts
half of this midpoint dip from the offset dip.
With a careful application of the chain rule, and carefully
distinguishing partial derivatives, we could arrive at the
same result
(14)