GEOPHYSICS, VOL. 58, NO. 7 (JULY 1993), P. 997-1001, 4 FIGS., 2 TABLES.
A two-layer stacking procedure to enhance converted waves
B. L. N. Kennett*
situations where conversions between P- and S-waves have
occurred. Such conversions between wave types can be
significant for marine profiles where the bottom is quite hard
as, for example, where carbonate layers occur at the sea
floor in tropical waters (Tatham and Stoffa, 1976; Tatham
and Goolsbee, 1984). P-waves generated in the water layer
by an air-gun source are converted to S-waves beneath the
seafloor, and can be reflected back from structure at some
depth before reconverting back to P-waves in the water to be
recorded by a seismic streamer. Such PSSP reflections have
a symmetric raypath in a horizontally layered medium. The
process of conversion is most effective at larger angles of
incidence and the resultant converted waves have their
largest amplitudes on the outer traces of long (4 km) cables.
The PSSP arrivals contain information on the S-wave
structure beneath the seafloor, and it is desirable to exploit
this information in the form of an S-wave stack, but this
cannot be readily achieved with conventional stacking procedures because the dominant energy occurs at large offsets.
As noted by Tatham and Stoffa (1976) the S-wave velocity is
an important attribute for interpretation. The use of both Pand S-wavespeed information can place useful constraints
on lithology .
The moveout of a PSSP converted wave can be accurately
predicted at short offsets using the hyperbolic trajectory
predicted by the single-layer model. However, at the large
offsets where the converted reflections are most significant
the moveout can be more accurately fit by adopting a
two-layer model in which the upper water layer is separated
from the rock beneath. No analytic formula is available for a
reflector trajectory in the offset-time domain in the two-layer
case, but it is possible to devise efficient numerical schemes
to calculate the required stacking paths. The new procedure
gives clear stacks of the converted PSSP phases when
applied to theoretical pressure records calculated for marine
models including full elastic effects. Interval velocities can
be recovered from the effective velocities in the second layer
by using the Dix relations with an offset in time for the
two-way traveltime for P-waves in the water layer. For the
PSSP reflections the sign of the reflection coefficient can
change with offset, but even so stacking represents an
ABSTRACT
For marine seismic sources quite efficient conversion of P-waves to S-waves can occur at hard seafloors, e.g., carbonate horizons in tropical waters. The
S-waves are reflected back from structures at depth
and are reconverted to P-waves in the water before
detection by the receiver array. Such PSSP reflections
can carry useful information on the structure beneath
the sea bed but are most significant at large offsets and
so are not easily stacked with a conventional normal
moveout (NMO) procedure based on a hyperbolic time
trajectory.
A two-layer stacking procedure that separates the
water layer from the region below the seafloor provides a very effective means of extracting the PSSP
arrivals, but also works well for P-waves. There is no
direct analytic form for the stacking trajectories but
they can be calculated quite efficiently numerically. A
further advantage is that the stacking velocity for
S-waves in the lower layer can be interpreted directly
in terms of S-wave propagation, so that S-wave interval velocities can be found. Stacking procedures based
on such simple physical models are likely to be useful
in other cases where attention needs to be focused on
a particular aspect of the wavefield.
INTRODUCTION
The standard procedures for the stacking of seismic waves
and the estimation of seismic velocities are based on the
NMO relations for a model of a single uniform layer overlying the reflector of interest. For a multilayered model, the
interval velocity for the individual layers can be extracted
from the Dix equation (see e.g., Sheriff and Geldart, 1982).
The conventional approach is well suited to situations
where the energy being stacked has remained as a P-wave
throughout the propagation process. However, the assumptions of the standard procedure are not very suitable for
Manuscript received by the Editor August 25, 1992; revised manuscript received January 4, 1993.
*Research School of Earth Sciences, Australian National University, Canberra ACT 0200, Australia.
© 1993 Society of Exploration Geophysicists. All rights reserved.
997
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998
Kennett
important velocity analysis and signal-to-noise enhancement
tool.
The two-layer algorithm gives a more accurate method of
estimating traveltime variation with offset than the usual
hyperbolic assumption used in normal moveout analysis.
The procedure can be used for P-wave processing but is
more important for the PSSP reflections because the largest
amplitudes for such arrivals occur at large offsets and times
(as can been seen in Figure 2). It is just under these
conditions that the hyperbolic assumption of NMO analysis
is at its weakest.
TWO-LAYER STACKING
For a model with an upper layer of thickness
and
overlying a second layer with thickness
and
velocity
velocity
the traveltime and offset for a reflection from
the underlying boundary are given parametrically as
=
t=
+
2 2)
(1)
2
+
will be significantly more accurate than using a single layer
down to time
Also, since the time and offset effects of the first layer are
included explicitly the wave type for propagation in layer 2
does not have to correspond to that in layer 1. The time
behavior is dictated by the velocities encountered along the
raypaths. For both reflections and the PSSP conversions
the shape of the reflector time curves differ significantly from
simple hyperbolas at large offset; errors can readily exceed
500 ms at 4 km offset.
is being estimated from a sequence
When the velocity
of stacks, the appropriate time has to be found for each offset
for a range of different estimates of the velocity. The root
finding technique we have introduced above works well once
but the convergence is
is a little larger than = 2h
slow when the layer thickness is small. We have therefore
found it advantageous to exclude two-way traveltimes within
25 ms of when calculating the stacks.
For a multiple-layer model, the effective velocity in the
second layer can be found using the Dix formula
(2)
in terms of the horizontal slowness p. Thus, to define the
stacking trajectory starting at a vertical two-way time
=
+
(3)
we have to be able to find the slowness
corresponding to
An effective approach is
a wave that arrives at offset
based on a shooting scheme to bracket the requisite slowThe numerical root finding procedure can be
ness
simplified by using the reciprocal of the NMO velocity as an
estimator for the required slowness. For a sequence of
offsets, we use a slightly higher slowness value than determined for the previous offset as one slowness estimate
and a new value
that for
v is taken as
P2 =
(4)
The offsets
and
corresponding to the slownesses p
and
are then calculated using equation (l), and a new
derived by linear interpolation
estimate
Pe = P2 +
(5)
The estimated offset
calculated from equation (1) for the
is then compared with the required offset and
slowness
accepted if within a specified tolerance, e.g., 1 m. Otherwise, the procedure is iterated with
replacing the least
satisfactory of the earlier slownesses. This procedure generally converges quickly, and while it is slower than the use of
the analytic results for conventional NMO analysis, it is still
fast enough to be useful for stacking. When the second layer
has lower velocity than the first, a slightly more complex
expression than equation (4) is needed to get a satisfactory
since the aim is to
initial estimate for the slowness
bracket the slowness for the given offset.
Note that the velocity
in the second medium plays a
role that is comparable to the rms velocity in the standard
single layer treatment. For a multilayered model, the effecwould be determined by the Dix formula
tive value of
applied to the layers below the first. However, since the full
effect of the first layer is included, the approximation of
using a composite velocity to describe the time trajectories
(6)
i=l
where the summation occurs over the relevant layers below
is determined,
the seabed. Once the effective velocity
the corresponding interval velocities can be found in the
usual way.
EXAMPLE OF TWO-LAYER STACKING
As an illustration of the two-layer stacking approach we
consider the application of both the conventional single layer
stack and the new method to theoretical seismograms calculated for a simple marine model. The velocity model ks3
(Figure 1) is representative of medium hard sediments with a
good contrast at the sea bed and the material properties are
specified in Table 1. Theoretical seismograms calculated for
this model using the reflectivity method (see e.g., Kennett
1983) show significant converted waves especially at the
larger offsets (Figure 2): the traveltime curves for the PSSP
reflections are indicated by the broken lines. The calculation
is for a point source at 10 m depth and a sequence of point
receivers at 15 m depth with offsets from 200 m to 3700 m at
100 m intervals. Near-source and near-receiver ghosts were
included in all theoretical seismogram calculations, full
conversion between P- and S-waves is allowed at each
interface, and up to nine multiples were included in the water
Table 1. Model ks3.
Thickness
m
P velocity
m/s
S velocity
m/s
200
100
400
400
600
1400
1500
2150
2425
2850
3375
4000
4500
0
1121
1288
1544
1868
2264
2600
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Two-layer Stacking
layer. The time signature of the source is a band-passed delta
function filtered with a sine ramp from 5 to 7 Hz, a flat
spectrum from 7 to 50 Hz, and a sine ramp down to zero at
70 Hz. The model includes an allowance for attenuation with
a
of 500 and a
of 250 in each of the solid layers.
In Table 2 we display the times and corresponding stacking velocities expected for model ks3 for single-layer (NMO)
and two-layer analyses for both P- and PSSP-waves. Because the S-wave velocities in the upper part of the medium
are lower than the P-wave velocity in water, the rms
velocities for PSSP reflections in the single layer case
initially drop below 1.5 km/s before rising again for the
deepest reflectors. In the two-layer case we are only concerned with the velocities below the seabed for both P- and
S-waves. As a result the effective velocities for the second
layer
increase monotonically for each wave type.
The results of the single layer NMO analysis for the ks3
model are displayed in Figure 3. The stacking procedure has
been applied with no spatial gain and a linear gain in time. A
mute was applied to exclude refracted arrivals traveling
faster than 1.5 km/s. The traces shown are the stack traces
for each constant rms velocity modulated by the semblance
function for that same velocity. The semblance was calculated over a smooth gate seven samples long centered on the
target time at each offset. Such a semblance-modulated trace
enhances the coherent features in the stacking process. The
locations of the expected rms velocities for P are indicated
by solid symbols, and the equivalent locations for the PSSP
reflections (with conversion at the seafloor) are marked by
open symbols. The simple stacking procedure has given a
good picture of the P reflections. The presence of the PSSP
reflections can be distinguished, although amplitude variations as a function of offset including changes of sign tend to
reduce the stacked amplitude. Because all the PSSP rms
velocities lie close to 1.5 km/s, it is rather difficult to get
constraints on S-wave interval velocities.
The corresponding two-layer stacking analysis is displayed in Figure 4; this took about four times as much
computing as the conventional stack in Figure 3. The display
takes the same form as in Figure 3, with stack traces
S-wave velocity
and density
FIG. 1. P-wave velocity
(p) model ks3 used to generate the theoretical seismograms
shown in Figure 2.
999
modulated by the semblance function. Exactly the same
range of effective velocities are used as in Figure 2 and the
same gain relations were applied in each case. The effective
velocities for each P reflection are again indicated with solid
symbols, and for each PSSP reflection open symbols are
used. The stack displays start 0.1 s after the two-way time
through the water (i.e., 0.367 s). The P reflectors can be
clearly followed as a set of amplitude maxima that are better
defined because the reflection trajectories are more accurately modeled. After the effect of the water layer has been
stripped away, it is easier to interpret the water-layer multiples that occur 0.267 s after the equivalent primaries. For
these multiples the stacking velocities are reduced because
the P-wave velocity in the water (1.5 km/s) is lower than the
velocities beneath the seabed.
The PSSP reflections have stacked well but, as noted
above, the amplitude variation with offset reduces the effectiveness of the stack compared with the P reflections. One of
the major advantages of the two-layer stack is that a definite
sequence of associated stacking maxima can be traced in
time, so that it would be possible to determine S-wave
interval velocities. The stacking peak at 1.4 s is not from a
primary PPSP reflection, but arises from a water layer
multiple of the strong PSSP reflection from the third reflector at 1.13 s, and can be recognized by the 0.267 s displacement in time from the primary reflection. The stacking peak
has also been displaced to slightly higher effective velocity
because the P-wave speed in the water exceeds the S-wave
velocities down to this reflector. Each of the PSSP maxima
for the deeper reflectors is also accompanied by a train of
water multiples.
Because the two-layer stack has a more direct physical
correspondence to the marine case than conventional stacking, it is relatively easy to predict the effect of multiples. A
major advantage is that some degree of velocity resolution
FIG. 2. Theoretical pressure records for the model ks3 for a
delta function source and a frequency band from 5-70 Hz.
Up to nine surface multiples and all conversions between Pand S-waves are included. The traveltime curves for primary
reflections are superimposed on the seismograms. The P
reflection times are shown as a solid line and the PSSP
reflections as a broken line.
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1000
Kennett
and stack enhancement can be offered for PSSP conversions because the appropriate time trajectories are being
used. The effective velocity for the two-layer case is equivalent to that which would be extracted with conventional
analysis from a marine record after downward continuation
to the seafloor, but can be determined with substantially
lower computational effort.
DISCUSSION AND CONCLUSIONS
I have presented the two-layer stacking procedure in the
context of marine seismic studies, but such an analysis can
be applied to any other situation where there is a natural
physical boundary that separates the propagation zone into
two parts.
For the marine case, conversion is most likely at the
seafloor and the two-layer procedure has allowed the converted waves to be tracked in a way that would allow the
extraction of interval velocities for S-waves beneath the
seafloor.
In land surveys, it may be appropriate in some circumstances to separate zones of deep weathering from the
regions beneath, especially if conversion between wave
types could arise at the base of the weathering. A further
Table 2. Effective velocities for single-layer and two-layer stacks.
P reflections
PSSP reflections
time
S
single layer
m/s
two layer
m/s
time
S
single layer
m/s
two layer
m/s
0.360
0.690
0.970
1.326
2.026
1692
2075
2326
2648
3181
2150
2367
2571
2866
3363
0.445
1.066
1.584
2.227
3.463
1361
1319
1397
1547
1836
1121
1253
1375
1554
1861
FIG. 3. Display of ks3 NMO stack traces for the data of
Figure 2, with semblance modulation for a range of rms
velocities. The stacking peaks associated with each of the
primary P (solid symbols) and PSSP reflections (open
symbols) are annotated with the reflector number.
FIG. 4. Display of ks3 2-layer stack traces for the data of
Figure 2, with semblance modulation, using the two-layer
stacking procedure with a range of effective velocities for the
second layer. The stacking peaks associated with each of the
primary P (solid symbols) and PSSP reflections (open
symbols) are annotated with the reflector number.
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Two-layer Stacking
case in which the two-layer procedure could be useful is in
the case of permafrost to reduce the influence of nonhyperbolic moveout on P reflections.
Once the simple analytic formula for the single-layer case
is replaced by a numerical determination of time trajectories
in the offset-time domain, it is possible to introduce other
styles of stacking relation than the simple two-layer function. Such stacking operations could be advantageous when
attention is focused on a particular aspect of the wavefield.
1001
REFERENCES
Kennett, B. L. N., 1983, Seismic wave propagation in stratified
media: Cambridge Univ. Press.
Sheriff, R. E., and Geldart, L. P., 1982, Exploration seismology,
Vol. 1: History, theory and acquisition: Cambridge Univ. Press.
Tatham, R. H., and Goolsbee, D. V., 1984, Separation of S-wave
and P-wave reflections offshore western Florida: Geophysics, 49,
493-508.
Tatham, R. H., and Stoffa, P. L., 1976, Vp/Vs-A potential
hydrocarbon indicator: Geophysics, 41, 837-849.
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