Theory of Interferometric Turning Wave Migration
Gerard Schuster
migrate the double-turning ray reflection by conventional
2-way migration, but this might encounter severe defocusing problems induced by migration velocity errors and the
extrapolation of wavefields over the entire double-turning
ray shown in Figure 2. The longer the double-turning ray
the more severe the defocusing errors in migrating data.
Can the extra defocusing error due to the double-turning
ray migration be reduced to that of a single turning ray
migration? The answer is yes, by interferometric P mP
migration.
ABSTRACT
A theory is developed for interferometric migration of
turning waves. Instead of backprojecting reflections
over the doubly-turned reflection raypaths, interferometric migration extrapolates the data over just a
single turning ray. This property is similar to the
SSP→SSP correlation transform that transforms 1storder free-surface multiples into primary reflections.
The advantages of interferometric turning wave migration (ITWM) compared to standard turning wave migration (TWM) are that 1). the defocusing errors induced by migration velocity errors will be reduced by
one half because the raypath length of a single turning
ray is half that of a double-turning ray, 2). the salt
flank with gentle dips is blind to standard forwardbackward rays that coincide with one another (e.g.,
underside of a salt flank) but can be, theoretically, imaged by ITWM, 3). source statics are eliminated, and
4). there is no need to know the source location, the
source wavelet, and the source excitation time. In theory, interferometric turning wave migration promises
to more fully illuminate the undersides of salt flanks
compared to TWM alone.
INTERFEROMETRIC IMAGING OF PMP
DATA
Assume the acoustic medium in Figure 2 characterized
by a velocity profile that increases in depth such that a
single-turning P wave and a double-turning P mP wave
are recorded by a dense distribution of geophones at the
free surface B0 . Underlying the free surface is the saltbottom interface B1 , and the goal is to image this reflector from the recorded P mP and P events. These events
are generated by a surface source at the unknown location x with an unknown excitation time. The smooth
background P-velocity model is assumed to be known.
The P and P mP pressure fields in the frequency domain
are denoted, respectively, by DP (y|x) and DP mP (y|x),
where y denotes the observation position and x denotes
the location of the source. The key imaging condition
for delineating the B1 boundary is that the ratio of the
P mP wavefield DP mP (y|x) and the P wavefield DP (y|x)
at the yB1 boundary has the same phase1 for all frequencies, and so can be coherently summed over all frequencies
and source positions. Mathematically, this ratio at a planar reflection interface is an estimate of the plane-wave
reflection coefficient RP mP (y|x) at the B1 boundary:
INTRODUCTION
Turning waves were proposed as events that could be used
to image steeply dipping salt flanks (Hale, 1991, 1992; Foley et al., 1991; Hussein et al., 1997) by migrating salt reflections associated with the forward-backward refracted
rays depicted in Figure 1. This technology has proven
to be quite useful for wide-offset surveys over salt bodies with steeply dipping flanks. However, conventional
migration of turning waves requires that the flank reflection return along almost the same path as the incident
ray illustrated by the forward-backward rays in Figure 1.
This means that conventional TWM can be blind to gently
dipping flanks as illustrated in Figure 2. It is possible to
RP mP (y|x)
1 This
=
D P mP (y|x)
D P (y|x)
≈
assumes that P mP events are a pure composition of either
pre-critical or post-critical reflections, but not both.
31
32
Schuster
DP mP (y|x)DP (y|x)∗
,
|DP (y|x)|2 + (1)
where is a small positive value that is known as the
damping parameter; it protects against instabilities in
the reflection coefficient estimate when the magnitude of
DP (y|x) is small. The important property of the, e.g.
post-critical, reflection coefficient RP mP (y|x) is that its
phase is the same for any frequency and any post-critical
angle of incidence. Hence, at any specified reflection point
yB1 the summation over all frequencies should be coherent for all post-critical reflections, and tends to be incoherent when y is more than one-half of a wavelength from
B1 . Thus, the image r(y) of the reflection boundary is
estimated as
XX
rP mP (y) ≈
RP mP (y|x),
ω
=
x
|DP (y|x)|2 + ,
(2)
where the first summation is over the useable data frequencies with bandwidth between −ωmax and ωmax . The
summation in x is over the different sources of a surface
seismic profile.
The problem with equation 2 is that the data are measured near the earth’s free surface gB0 , not along the
salt reflector boundary at B1 . Therefore, the P mP data
DP mP (g|x) and P data DP (g|x) recorded just below the
free surface gB0 should be downward extrapolated in
depth using the asymptotic Green’s functions G0 (y|g):
G0 (y|g)
= A(y, g)eiωτyg ,
(3)
where A(y, g) accounts for geometrical spreading effects
and is a solution to the transport equation. The traveltime field τgy represents the traveltime for a P wave to
propagate from y to g and is a solution of the eikonal
equation for the smooth P-wave velocity distribution. In
practice, this traveltime function is computed by some
type of ray tracing procedure and is valid for a smooth
velocity distribution.
Formally, the extrapolated waves can be estimated by
a far-field approximation to the reciprocity equation of
correlation type (Wapenaar, 2004);
P
y, xV ; DP (y|x)
= 2ik g G0 (y|g)∗
y, xV ; DP mP (y|x)
e−iωτgy DP mP (g|x),
DP (g|x) + DP (y|x)∗ ;
P
= 2ik g G0 (y|g)∗
DP mP (g|x) + DP mP (y|x)∗ (4)
where k is the wavenumber for a P-wave measured at the
geophone. Replacing the Green’s functions in the above
equations by the asymptotic approximation in equation 3
yields
P
y, xV ; DP (y|x)
≈ 2ik g A(y, g)
(5)
where the acausal component Green’s function on the
right-hand side is harmlessly neglected. The migration image only uses the causal part of the extrapolated field and
so these neglected terms will not stack coherently along
the reflector when summing the migration images for all
frequencies, source positions, and geophone positions.
The extrapolated field values DP (y|x) and DP mP (y|x)
in equation 5 can be inserted into equation 2 to give the
migration image rP mP (y) of the salt reflector:
P P P P
yV ; rP mP (y)
= −4k 2 ω x g g0
A(y, g0 )A(y, g)e−iω(τgy −τg0 y )
DP mP (g|x)DP (g0 |x)∗ ,
x
X X DP mP (y|x)DP (y|x)∗
ω
y, xV ; DP mP (y|x)
e−iωτgy DP (g|x);
P
≈ 2ik g A(y, g)
(6)
where the denominator in equation 2 is ignored. The
above equation is that for ITWM without the deconvolution term.
Ray Diagram Interpretation. Estimating the reflectivity by backward extrapolating P events and P P events
and dividing these fields by one another (see equation 1)
can also be interpreted as interferometric migration (Schuster, 2009). The corresponding ray diagrams for correlating two traces to produce a virtual trace are shown in
Figure 3(a).
Relationship to Interbed Multiple Prediction. The
double-turning ray can be considered as a special case of
an interbed multiple. Earlier work by Keydar et al. (1997)
interpreted interbed multiples, which was used by ? to
eliminate interbed multiples in seismic data. He states
”...the combination of two primaries (SR’ and S’R in Figure 4) minus a third primary (S’R’ in Figure 4). Thus the
primary reflection from the interbed multiple generator
can be used to remove the effect of S’R’ from the combination of SR’ and S’R through correlation.”. Thus, the earlier interbed multiple prediction work is related to P mP
interferometry in that they both use cross-correlation of
traces to remove the effects of propagation along certain
portions of the ray path.
Benefits. The benefits of ITWM can be ascertained by
noting that equation 2 divides the P mP event by the P
event, which means that the source statics and wavelet
are divided out. This also means that the source excitation time and source location are not needed for migration! Moreover, the backward extrapolation of the direct
P wave back from the geophone2 and the imaging at the
reflector is equivalent to extrapolating waves over a single
turning ray (see righmost rays in Figure 3(a)). This means
2 Conventional
migration forward extrapolates the source wavefield from the source to the image point.
Interferometric Turning Wave Migration
that the defocusing induced by migration velocity errors
should be half that of standard TWM which extrapolates
waves over a double-turning raypath.
33
hoff formula is given by
rP mP (y)
XXX
≈
A
Extension to Shear Waves. The benefits of this procedure can also be applied to P and P mS reflection data,
where P mS represents the converted P -to-S reflection
recorded on the earth’s surface and P represents the recorded
P wave. Other combinations of multiply reflected waves
can be used to image subsurface reflectors as long as these
events are somewhat visible and can be isolated in the
records. A special case is when the reflector is the free
surface so that P mP becomes P P ; in this case the P P
reflections can be transformed into P waves.
Liabilities. A potential liability of interferometric TWM
is that one of the extrapolation paths is through the salt,
as illustrated in the rightmost diagrams in Figure 3(a).
The strong velocity contrast across the salt-sediment interface makes it difficult to accurately extrapolate waves
through salt and so might render ITWM ineffective. Another potential problem is that the double-turning ray
that bounces beneath the salt, as shown in Figure 2, requires a very large source-receiver offset. Such long offsets promote poor signal-to-noise ratios in the far-offset
traces and demand expensive (and possibly impractical)
wide-offset surveys3 . Finally, Xiao et al. (2006) noted the
difficulty in migrating traces with a subtraction imaging
condition, similar to that in equation 6. However, it appears that Xiao’s (2008) more recent results suggest that
the use of reverse time migration largely overcomes these
difficulties.
Workflow. The workflow for implementing ITWM might
be the following.
• Fourier transform the shot gathers in time to get the
frequency-domain data D(g|x).
• Identify the wide-offset P arrival for receivers around
the salt body and window them in the shot gathers to get DP (g|x). Using modeling, estimate the
time window for P mP events and window the shot
gathers to get DP mP (g|x). These windows are not
restricted to just contain the desired P mP events,
they can contain many other spurious events because
they will not coherently contribute when the correlated data are migrated.
• Correlate DP mP (A|x) with DP (B|x) and sum over
all relevant sources to get the virtual trace φ(A, B):
X
φ(A, B) =
DP mP (A|x)DP (B|x)∗ . (7)
x
• Migrate the virtual traces φ(A, B) by either Kirchhoff migration or reverse time migration. The Kirch3 Extremely
OBS surveys
wide-offset surveys might not be too impractical with
−iω(τAy −τBy )
e
.
B
k 2 φ(A, B)
ω
(8)
To implement the deconvolution described by equation 2, a more precise window about the DP (A|x)
events should be used. The estimate for the reflectivity function assumes a local plane-wave approximation, so perhaps deconvolution migration should
be performed in the wavenumber-frequency domain.
SUMMARY
A theory is developed for interferometric migration of
double-turning waves. Instead of extrapolating waves over
both parts of the double-turned waves, ITWM extrapolates over just one part of the double-turned ray. The advantage of ITWM compared to standard turning wave migration is that 1). the defocusing errors induced by migration velocity errors will be reduced by approximately half
because the raypath length of a single turning ray is half
that of a double-turning ray for a sufficiently wide sourcereceiver offset, 2). the salt flank with gentle dips is blind
to standard forward-backward rays that coincide with one
another but can be, theoretically, imaged by ITWM, 3).
source statics are eliminated, and 4). there is no need
to know the source location, the source wavelet, and the
source excitation time. Interferometric turning wave migration seems to have the possibility of more fully illuminating the undersides of salt flanks compared to TWM
alone.
Interferometric TWM is similar to the local VSP imaging method described in the PhD dissertation of Xiao
(2008) who used VSP P and P P data to image the reflectivity distribution around a VSP well without knowing
the source location or source excitation time. To complete
the analogy, the VSP receiver well is analogous to the horizontal line of receivers on the free surface, the VSP direct
P wave plays the role of the surface seismic profile (SSP)
P arrival, and the VSP P P reflections act as the SSP
P mP reflections. A key issue will be if there is enough
source-receiver coverage to allow this procedure to work
for the P mP case. This theory now waits for numerical
validation with both synthetic and field data.
REFERENCES
Foley, W., W. Abriel, and R. Wright, 1991, Application of
turning wave migration to a 3-d seismic survey in main
pass block 299, offshore louisiana, and its impact on
field development: SEG Expanded Abstracts 10, 1175–
1178.
Hale, D., 1991, Migration of seismic turning waves: European Patent, EP0513448.
——–, 1992, Migration of seismic turning waves: US
Patent, 51348584.
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Schuster
a). Conventional Reflection Migration
b). Standard Turning Wave Migration
Figure 1: a). Conventional migration of reflections and b). turning wave migration. Illustration is modified from a
Western-Geco advertising brochure.
PmP
P
B0
illuminated area
x
B1
unilluminated area for forward rays
coincident with backward rays
Figure 2: Double-turning P mP and single-turning P events, where m denotes the depth of the reflecting salt interface.
Because of the gentle dip of the salt interface the double-turning P mP ray does not return along the incident ray as
shown by the rays in Figure 1.
Interferometric Turning Wave Migration
PmP
A
P
B
x
B
A
P
P
B
35
x
B
1
A
B
x
B
1
C
1
C
Virtual source
advanced by TCB
(a)
PmP
P
P
A
B
A
B
P
A
C
B
C
Virtual source
advanced by TCB
(b)
Figure 3: (Top) P mP event at A correlated with P event recorded at B yields the virtual trace at A shown on the far
right ray diagram. Similar to the interferometric migration of PS events in Xiao et al. (2006), the event on the far right
diagram can be interpreted as a P wave generated by a virtual source at the reflection point C advanced in time by
TBC . (Bottom) Same as top except the return ray almost coincides with the forward ray, which is the case for imaging
steep flanks. In both the top and bottom diagrams, the interferometric rays on the right are about half the length of the
double-turned rays on the far left for a wide-offset survey.
Figure 4: Interbed multiple rays (solid) and related primary S’R’ denoted by the dashed rays (diagram extracted from ?.
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Schuster
Hussein, S. J., Desler, and G. Miller, 1997, Imaging salt
substructures in the gulf of mexico using 3d turning
wave migration: The Leading Edge, 1487–1495.
Keydar, S. E., B. Landa, Gurevich, and B. Gelchinsky,
1997, Multiple prediction using wavefront characteristics of primary reflections: Extended Abstracts, 59th
Conference EAGE, Paper A016.
Schuster, G. T., 2009, Seismic interferometry: Cambridge
Press, under press.
Wapenaar, K., 2004, Retrieving the elastodynamic green’s
function of an arbitrary inhomogeneous medium by
cross correlations: Phys. Rev. Lett., 93, 254301–1–
254301–4.
Xiao, X., 2008, Local reverse-time migration with vsp
green’s functions: PhD dissertation, U of Utah.
Xiao, X., M. Zhou, and G. T. Schuster, 2006, Salt-flank
delineation by interferometric imaging of transmitted
p-tos waves: Geophysics, 71, SI197–SI207.