Prestack depth migration and illumination maps - OnePetro

PreStack Depth Migration and illumination maps
Renaud Laurain*, and Vetle Vinje, NORSAR.
Summary
Designing a survey is often based on fold and offset distribution
under the assumption of a flat subsurface geometry. In real
situations, this is hardly the case. Recently, ray tracing has frequently
been used to predict the illumination at selected
reflectors corresponding to given seismic surveys. Illumination
maps are used as an aid to choose the best possible survey
when planning a seismic acquisition. In this 2-D synthetic
study we compare hit density and illumination amplitude maps
from ray tracing with the result from prestack depth migration
(PSDM). We show that (i) holes (i.e. non-illuminated zones) in
the ray-tracing based illumination maps correspond well with
holes in the PSDM images and (ii) illumination amplitude
maps only crudely approximate the amplitudes in the PSDM
images.
Introduction
When designing a seismic acquisition, it has become common
practice to model a hypothetical reflection data set using ray
tracing in order to plan the survey geometry (i.e. source and
receiver positions) to obtain an optimum image of the subsurface
(Bear et. al., 1999, Sassolas et. al., 1999, Pereyra et. al.,
1999, Rosland and Drivenes, 2000). For a given survey in a
given model, ray tracing may be used to estimate the subsurface
illumination at selected horizons in terms of e.g. hit density
and amplitude distribution.
On a model designed to combine the effects of both a syncline
with a steep flank and a lens (e.g. a salt body), a full finite difference
data set has been generated and migrated using a
PSDM algorithm. Then, the correspondence between PSDM
Root Mean Square amplitude (RMS amplitude) profiles along
the target horizon and hit density and amplitude illumination
maps are studied.
Model
Figure 1 displays the model we used to generate a synthetic
marine data set with classic 2-D finite-difference (FD) code.
The upper layer represents water (velocity 1.5 km/s). The sediment
layer, delimited by the sea bottom and the target horizon,
is affected by a vertical gradient function (from 2.0 to 3.0
km/s at the deepest point). The target geometry is chosen so
that the steepest part of the syncline can not be illuminated
properly except by using a recording time longer than 5.0 s. In
the sediment layer, a salt lens with constant velocity of 4.0 km/
s has been introduced to study the effects of shadow zones
induced on the target horizon.
Depth (km)
Distance (km)
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0
P-velocity (km/s)
1.5 2.0 2.5 3.0 3.5 4.0
Fig.1: Model used for generating synthetic data. Shots are
plotted as red triangles at the surface. Target horizon is plotted
in red.
The data acquisition scheme is designed as a marine survey
with a nominal fold of 10 in bin cells of 5 m size. Shots are
located on the surface, every 50 m between 3.0 and 14.5 km.
For each shot, 100 receivers with offsets from 10 m to 1000 m
and a 10.0 m interval are used (figure 1). An example of constant-offset
FD section is given in figure 2.
sea bottom
top salt
bottom salt
Target Horizon
Fig.2: Constant-offset section for offset 10.0 m.
target horizon
SEG Int'l Exposition and Annual Meeting * San Antonio, Texas * September 9-14, 2001
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2.0
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4.0
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PreStack Depth Migration
The finite difference data is migrated on two local targets
using a un-weighted PreStack Depth Migration algorithm. For
each point of the defined target, the migration process performs
a sum of amplitudes along diffraction-traveltime curves.
The stacking is done with wide aperture and unit weights
along the diffraction-traveltime curves. The traveltime maps
required for this process are generated with a first arrival
Eikonal solver in a background velocity model where the target
interface has been removed (figure 3). The first target area
(denoted as target 1) covers the syncline. The second target
area (denoted as target 2) has been chosen beneath the salt lens
in order to study the induced focusing/defocusing effects.
Migrated sections using the source-receiver pairs as input
(every shot, all offsets, traveltimes up to 8.5 s) are displayed in
figures 4 and 5.
Depth Depth (km) (km)
0.0
0.0
2.0
2.0
4.0
4.0
6.0
6.0
8.0
8.0
10.0
10.0
( )
12.0 Distance 14.0 (km)
12.0 14.0 0.0
0.0
1.0
Target 1
1.0
2.0
2.0
3.0
3.0
4.0
Target 2
4.0
5.0
5.0
P-velocity (km/s)
1.5 2.0 2.5 3.0 3.5 4.0
Fig.3: Velocity model used for performing the depth migration.
Target areas are limited by the red boxes. Red squares represent
the shots used as input data.
Fig.4: Migrated section of target 1. Target horizon is superimposed
in black.
Fig.5: Migrated section of target 2. Target horizon is superimposed
in black. The red line superimposed on the horizontal
axis represents the salt lens extension.
PreStack Depth Migration and illumination maps
Illumination maps
In order to generate illumination maps of the target horizon, a
full data set is modeled by wavefront construction (Vinje et.
al., 1996). Reflection points on the target horizon and amplitudes
of the corresponding rays from each shot to each
receiver in the survey are computed. To be comparable to the
finite difference data set, ray tracing is done with cylindrical
sources. The wavefront construction is performed in a 2.5-D
model extrapolated from the 2-D model (see figure 6) using
the same survey as used for the finite difference data.
Receiver
Fig.6: 2.5-D model used to generate illumination maps. Shots
are shown in black. Reflection points along the target horizon
are plotted in red.
To perform the calculation of the illumination maps, the surface
of the reflector is divided in bin cells of given constant
area. In each cell, the following quantities are computed:
•hit density (i.e. number of reflection points per unit area)
•illumination amplitude
∑
A
i
A = -------------------------------i
Bin Cell Area
where A
i
is the complex amplitude coefficient in the
receiver for ray number i reflecting in the bin cell.
Amplitude illumination and migration
Shot
Even using a fairly small aperture (1 km), the survey illuminates
the whole target horizon as may be seen by the reflection
points on the target horizon in figure 6. The steepest part of the
syncline presents a low illuminated zone (x=3.4 km). This is
obvious on both hit density and amplitude illumination curves
(see figure 7). RMS amplitude along the migrated horizon (figure
8) is smoother than the illumination amplitude curve, even
if the general variations show some similarities.
SEG Int'l Exposition and Annual Meeting * San Antonio, Texas * September 9-14, 2001

Illumination Amplitude amplitude density
Fig.7: Hit density and amplitude illumination for the syncline
(target 1). Sampling interval is 25.0 m.
Fig.8: RMS amplitude along the migrated horizon for the syncline
(target 1).
Illumination Amplitude amplitude density
Hit density
x 10
4
4
3
2
1
15
10
0.00
Hit density
5
3
2
1
15
10
2.5 3 3.5
Horizontal coordinate
4 4.5
2.5 3 3.5
Horizontal coordinate
4 4.5
x 10
4
4
5
9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5
Horizontal coordinate
9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5
Horizontal coordinate
Fig.9: Hit density and amplitude illumination for the area
under the salt lens (target 2). Sampling interval is 25.0 m. The
red line on the horizontal axis represents the salt lens extension.
Low-amplitude zone
focusing
Low-amplitude
Fig.10: RMS amplitude along the migrated horizon under the
salt lens (target 2). The red line superimposed on the horizontal
axis represents the salt lens extension.
PreStack Depth Migration and illumination maps
Under the salt lens (red line on figures 9 and 10), the hit density
returns to normal (compared to the hit density before
x=10.0 km) whereas the amplitude illumination remains lower
due to the energy losses occurring when the wavefield is transmitted
through the salt. This effect is also noticeable on the
RMS amplitude extracted from the migrated section (figure
10).
Near the left edge of the salt lens (x=10.7 km in figure 9), there
is a low-amplitude zone where very little energy is reflected
even if the illumination density is high. This low-amplitude
zone is also visible at x=10.7 km in the RMS section in figure
10, but in this case the low-amplitude zone is larger. This is not
an intrinsic limitation of the PSDM process, but rather a side
effect introduced when using first arrival traveltime maps. The
Eikonal solver computes first arrival traveltimes, which generally
does not represent the most energetic part of the wavefield
(Geoltrain and Brac, 1993). In this case, the first arrival is
associated to a low energy event (figure 11) passing through
the salt. The consequence is that the imaging will take place
through the salt giving lower amplitudes in the PSDM image
from about 9.5 to 11 km. This effect could be corrected by
using only the traveltimes for the most energetic events
instead, or all arrivals.
Distance (km)
8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0
most energetic
arrival
Fig.11: Multi ray path example on the edge of the salt lens.
Generally speaking, RMS amplitude profiles along both
migrated horizons (figures 8 and 10) are smoother than the
corresponding illumination and amplitude profiles. The PSDM
process is performed on a finite difference data set on which
Fresnel-zone effects introduce a smoothing (Bear et. al., 1999,
Thore and Juliard, 1999). Moreover, the PSDM algorithm performs
another smoothing when stacking amplitudes along the
diffraction-traveltime curves. Therefore, RMS amplitude variations
are not as drastic as they appear on the illumination
maps. Nevertheless, several general aspects remain coherent.
Illumination holes and migration
first arrival
Analyzing normal incidence ray tracing data, it is evident that,
in this special case, the steep flank of the syncline is only illuminated
by rays with a traveltime above 5.0 s. On a zero-offset
section extracted from the finite difference data, a PSDM has
SEG Int'l Exposition and Annual Meeting * San Antonio, Texas * September 9-14, 2001
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een run twice, once using the full time and once with the
traces truncated at 5.0 s (figures 12 and 13). The correspondence
between the area of no image in figure 12 b) and the
shadow zone in the illumination amplitude in figure 14 b) is
good.
a) b)
Fig.12: Migrated section of zero-offset data using the full
traces up to 8.5 s (a) and truncating the traces at 5.0 s (b). Vertical
dotted lines in b) delimit the non-illuminated zone
according to the illumination amplitude map in figure 14.
Fig.13: RMS amplitude measured along the migrated reflector
using the full traces up to 8.5 s (solid line) and truncating the
traces at 5.0 s (dashed line).
Illumination Amplitude amplitude density
b)
Illumination Amplitude amplitude density
a)
15
10
5
1.5
1
0.5
0
x 10 −4
2.5 3 3.5 4 4.5
x 10 −3
Illumination Amplitude amplitude density full time full time
Illumination Amplitude amplitude density cut cut at 5s at 5.0 s
Non-illuminated zone
2.5 3 3.5
Horizontal coordinate
4 4.5
Fig.14: Amplitude density profile using the full traces up to
8.5 s (a) and truncating the traces at 5.0 s (b).
PreStack Depth Migration and illumination maps
Conclusions
PSDM introduces a smearing of the recorded traces along isochrones
tangential to the reflectors. This is not taken into
account in the illumination maps, as they are made without
any knowledge of the source pulse, using a simple process of
adding amplitudes in bin cells.
In spite of this, the synthetic example studied here shows that
the variation of the illumination amplitude is comparable to
the PSDM amplitude. Important effects like focusing/defocusing
and the reduction of amplitude due to energy losses in the
overburden is seen on the illumination amplitude profiles. This
synthetic case also shows that there is a good correspondence
between non-illuminated zones predicted by the illumination
maps and shadow zones on the PSDM seismic sections.
Acknowledgments
The authors would like to thanks Isabelle Lecomte (NORSAR)
for implementing the PSDM algorithm.
References
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The construction of subsurface illumination and amplitude
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SEG Int'l Exposition and Annual Meeting * San Antonio, Texas * September 9-14, 2001