Interferometric correlogram-space analysis
Oleg V. Poliannikov, Mark Willis and Bongani Mashele
May 31, 2009
Abstract
Seismic interferometry is a method of obtaining a virtual shot
gather from a collection of actual shot gathers. The set of traces
corresponding to multiple actual shots recorded at two receivers is
used to synthesize a virtual shot located at one of the receivers and
a virtual receiver at the other. An estimate of a Green’s function between these two receivers is obtained by first cross-correlating pairs
of traces from each of the common shots and then stacking the resulting cross-correlograms. In this paper, we study the structure of
cross-correlograms obtained from a VSP acquisition geometry using a
surface source reflected by flat or dipping layers and/or diffracted by
point inclusions. The model is purely acoustic. The shape of events
in the cross-correlogram space can be used to infer the location and
geometry of a subsurface structure. A pilot wavelet created by a curvilinear stacking process is used as a detector of predicted events in the
cross-correlogram. Results of a semblance-based velocity scan of the
cross-correlograms using curvilinear stacks can be used to improve the
quality of the virtual gather.
1
1
Introduction
In this paper, we consider a standard acoustic 2D VSP geometry. Physical
sources are placed on the surface, and several receivers are located in a borehole. It is typically advantageous in imaging applications to have physical
sources as close as possible to an area of interest. Achieving this in practice
is extremely difficult. An interferometric cross-correlation aims to redatum
physical sources on the surface to the receivers in the borehole. If a virtual
gather is an accurate estimate of the physical Green’s function, imaging can
be done as if physical sources were down in the borehole. A virtual trace
is classically obtained by taking two common receiver gathers and stacking
their cross-correlogram over the source locations. This process is intended
to automatically pick stationary phase points and discard everything else.
Provided adequate illumination, the method works and yields a bandlimited
Green’s function between the two receivers.
Realistic source illumination in seismic applications never totally meets
the desired assumptions. Gaps in source coverage can result in an absent
stationary phase point and/or a noisy estimate. Compensating for problems
associated with variable illumination is an important problem.
Conventional cross-correlogram stacking ignores complete geometry of
events in the cross-correlogram space. When the subsurface consists of plane
reflectors and point diffractors, this geometry is a function of their location
and orientation. By making a prediction of the shape of an event based on
an assumed structure, one would hope to detect this structure or confirm its
absence.
We derive moveout equations of reflection and diffraction events on a
correlogram gather. A curvilinear stack is proposed as an extension of the
2
conventional stack to allow for arbitrary moveout curves representing events
in the cross-correlogram space. The sum of cross-correlogram values along a
given curve is a measure of coherence. A high value indicates the presence
of the predicted event. A low value means the event is absent.
We can make this idea more precise by considering a semblance functional, which assigns a high value to a setup confirmed by observed crosscorrelograms. If a reflector with an assumed depth and dip is present in the
medium with an assumed average velocity between itself and the receivers,
then the corresponding semblance is close to one. A semblance close to zero
indicates the absence of such a reflector. This semblance analysis methodology is based on a constant velocity assumption and is therefore prone to
biases associated with vertical and horizontal velocity variations. It does,
however, produce a good approximation for an average velocity for the geometry of the subsurface as long as the velocity above the receivers is taken
into consideration.
The obtained geometry and kinematics can be used to improve the crosscorrelogram. When sources are widely spaced (undersampled), we may be
able to fill in events in the cross-correlogram and thus solve the problem of
spatial aliasing. When a stationary phase point is missing in a given event,
we may be able to extrapolate the latter and predict the location of the
former. Our technique also allows us to filter out undesirable events and
emphasize those we want.
2
Classical interferometry
We begin by assuming a standard two-dimensional acoustic VSP setup for our
problem. A collection of physical sources is placed on the surface {z = 0} at
3
(xs , 0)
r
r
r
r
(0, zr )
r
r
r
r
r
r
r -
x
(xd , zd )
(xℓ , zℓ )r6θℓ
?z
Figure 1: Setup and notation
Figure 2: Simple model with a single flat reflector
4
locations (xs , 0), where x denotes the offset, and z denotes the depth. Several
physical receivers (0, zr ) are located in the borehole {x = 0}. Each shot is a
fixed wavelet of characteristic frequency f or period T = 1/f .
It is typically desirable in applications to place sources so as to maximize
the illumination of an area of interest. This is often technologically impossible
or prohibitively expensive. Interferometry is a seismic data processing technique, which seeks to move or redatum physical sources to a receiver location
to produce a virtual source (Rickett and Claerbout, 1996; Derode et al., 2003;
Bakulin and Calvert, 2004; Schuster et al., 2004; Wapenaar et al., 2005). If
properly constructed, this virtual source can be used for imaging as if we had
physical sources in the borehole.
Assume we fix a pair of receivers and we would like to estimate a seismic
trace that would be recorded at the location of the second receiver had a
physical source been positioned at the place of the first one. The construction
of this estimate is accomplished by computing cross-correlations (Fig 4) of
received signals (Fig. 3) for all available shots (Fig. 2), and by subsequently
stacking them over the sources (Korneev and Bakulin, 2006; Lu et al., 2008).
The estimate is called a virtual trace, the first receiver is a virtual source
(VS), and the second receiver becomes a virtual receiver (VR).
5
CRG at receiver 1
CRG at receiver 2
0
0
0.4
0.3
0.2
0.2
0.3
0.4
0.2
0.6
0.1
0.8
1
0
Direct arrival
1.2
−0.1
1.4 Reflected wave
−0.2
1.6
Recording time (sec)
Recording time (sec)
0.4
Direct arrival
0.2
0.6
0.1
0.8
1
0
1.2
−0.1
1.4
−0.2
1.6
−0.3
1.8
1.8
6
2
−0.4
−2000
−1000
0
1000
Source offset (m)
2000
2
Reflected wave
−2000
−1000
0
1000
Source offset (m)
Figure 3: Common receiver gathers for the simple model
−0.3
2000
Mathematically, this process relies on the following theorem (Wapenaar,
2004; Wapenaar et al., 2005; Schuster and Zhou, 2006).
Theorem 2.1.
VS VR
I
z}|{ z}|{ 1
2
1
2
G xr , xr , −t + G xr , xr , t ∝
G xs , x1r , −t ⋆ G xs , x2r , t dxs ,
|
|
{z
} |
{z
}
{z
}
S
anticausal GF
causal GF
cross-correlation
|{z}
stacking
where
1. x1r , x2r are receiver locations;
2. G is the Green’s function of the medium;
3. xs denotes a source location on a continuous closed curve S surrounding our medium; and
4. ⋆ denotes convolution in time.
The assumption of continuity of the source coverage and its complete
encompassing of the medium are never satisfied in geophysical applications.
However, the method has been shown to work successfully in many practical
settings even when these theoretical constraints are relaxed. Intuitively, the
process of stacking correlograms removes from them all correlated events
except for stationary phase contributions (Snieder, 2004). The latter are
produced by physical sources, whose rays pass through the virtual source
and are received at the virtual receiver (Lu et al., 2008). Other parts of the
cross-correlogram are conventionally regarded as noise, and as a result they
are intended to be filtered out by the stacking operator. This is justified in
ideal circumstances (Fig. 5, left column) because the stack so obtained can
7
Crosscorrelogram of the two CRGs
−2
0.25
−1.5
Reflected x Direct
Reflected x Reflected
0.2
0.15
−1
0.1
Lag (sec)
−0.5
0.05
0
0
−0.05
0.5
−0.1
1
−0.15
Direct x Direct
−0.2
1.5
Direct x Reflected
2
−0.25
−2000
−1000
0
1000
Source offset (m)
2000
Figure 4: Cross-correlogram of the two CRGs for the simple model
be shown to contain a bandlimited Green’s function from the virtual source
to the virtual receiver exactly. In less ideal situations, stacking leads to errors
and undesirable artifacts in the estimate of the Green’s function.
Consider a model with flat reflectors when all sources are located to one
side of the borehole (Fig. 5, middle column). The physical source that would
provide a stationary phase point in the cross-correlogram is absent, and the
correct arrival in the resulting stack is replaced with an artifact caused by
source truncation (edge effect). Alternatively, if the angular coverage is adequate in range but sparse (Fig. 5, right column) then the cross-correlogram
becomes spatially undersampled, which leads to correlated events which do
not align and stack out. The stack will contain a “ringing” noise, which will
bury the correct event in the virtual trace (Mehta et al., 2008a,b)..
8
normal
no stationary point
Interferometric stack under favorable conditions
spatial aliasing
Spatialy aliased stack
Interferometric Stack
12
4
2
10
9
1.5
8
3
1
Ringing due to aliasing
Edge effect at incorrect time
2
6
0.5
Spike at correct time
4
1
0
2
−0.5
0
−1
−2
−1
−4
−1.5
−6
−2
−8
−2
0
−1.5
−1
−0.5
0
0.5
Lag (sec)
1
1.5
2
−2.5
−2
−2
−1.5
−1
−0.5
0
0.5
Lag (sec)
1
1.5
2
−3
−2
−1.5
−1
Figure 5: Common receiver gathers for the simple model
−0.5
0
0.5
Lag (sec)
1
1.5
2
As evidenced by these simple examples, compensating for problems associated with variable illumination can be an important problem. Conventional interferometry regards the complete geometry of events in the crosscorrelogram space inconsequential. When the standard approach fails due
to practical complications, a possible solution lies in taking the geometry of
the correlogram space into account with the hope of extracting additional
information about the subsurface, and hence improving the interferometric
method.
3
Curvilinear stack
Throughout this paper, we will assume that the subsurface contains plane
reflectors and point diffractors (Fig. 1). The former are specified by their
location (xℓ , zℓ ) and dip θℓ , and the latter are fully described by their position
(xd , zd ). The geometry of events in the correlogram space are then functions
of those parameters.
Suppose, a flat reflector is embedded in a medium with a constant velocity
v. The cross-correlogram space will then consist of four events (Fig. 4), each
being a cross-correlation of a direct or reflected wave from the first gather
with same from the second one (Fig. 3). The equations for delays of those
events are as follows:
p
p
x2s + zr,2 2
x2s + zr,12
−
;
τDD =
v
p
p v
x2s + zr,1 2
x2s + 4zℓ2 − 4zℓ zr,2 + zr,2 2
−
;
τDR =
vp
v
p
x2s + 4zℓ2 − 4zℓ zr,1 + zr,1 2
x2s + z22
−
;
τRD =
v p
p v
x2s + 4zℓ2 − 4zℓ zr,2 + zr,2 2
x2s + 4zℓ2 − 4zℓ zr,1 + zr,1 2
τRR =
−
,
v
v
10
(1)
where DD is the correlation of the direct with the direct wave, DR correlation
of direct with reflection, RD correlation of reflection with direct, RR correlation of reflection with reflection. Equations corresponding to a reflector with
an arbitrary dip are more complicated but still easily derived. Similarly, for a
point diffractor, the delays of events in the cross correlogram space are given
by
p
x2s + zr,2 2
x2s + zr,1 2
−
;
τDD =
c
c
p
p
p
x2s + zr,1 2
(xs − xd )2 + zd2 + x2d + (zr,2 − zd )2
τDR =
−
;
c p
p c
p
(2)
x2s + zr,2 2
(xs − xd )2 + zd2 + x2d + (zr,1 − zd )2
−
;
τRD =
c p
c
p
x2d + (zr,2 − zd )2
x2d + (zr,1 − zd )2
−
.
τRR ≡
c
c
We can use these formulas to predict an event in the cross-correlogram
p
based on an assumed subsurface structure. By comparing our prediction
to the actual cross-correlogram, we will confirm either the validity of the
prediction or its error. We propose the notion of a curvilinear stack as a
tool to detect the presence of an event with a given moveout in the crosscorrelogram. The formal definition of the curvilinear stack is as follows.
Parameterize the cross-correlogram C(xs , t) by the source offset xs and the
lag t. Assume further that a family of curves τ (xs , zℓ ) is available. For each
zℓ , the curve τ (xs , zℓ ) represents an expected moveout of an event produced
by a reflector lying at depth zℓ . In what follows, we will always consider
the DR-component of the cross-correlogram, which is also used in standard
interferometry. Other components could conceivably be used but it has not
been done in this report. Integrating the cross-correlogram along these curves
Z
SC (zℓ ) = C xs , τDR (zℓ , xs ) dxs ,
(3)
11
we obtain a stack, which is a function of depth. A coherent peak in the
stack around a specific value of zℓ indicates the presence of an event with a
moveout τ (zℓ , ·), while a noise-like behavior suggests its absence.
The curvilinear stack defined above can also be written in a time domain.
For a reflector located at depth zℓ , define the travel time t(zℓ ) from the virtual
source to the reflector and back to the virtual receiver. Then
SC (t) = SC t(zℓ ) .
(4)
We have shown that a curvilinear stack can serve as a detection tool for
events in the cross-correlogram space that possess a given moveout. Note
that it does not suffer from some of serious shortcomings that its classical
counterpart suffers from. The curvilinear stack does not rely on the presence
of a stationary phase point, and it will register an event that a classical stacking mechanism will miss entirely. Also, spatial aliasing described above is not
an issue as curvilinear integration provides a necessary moveout correction,
which avoids having to sum uncompensated parts of the wavelet.
Despite these advantages, a word of caution is in order. The amplitude
of the stack being an integral over all offsets depends on many factors. Using the stack with the expressed purpose of inverting the amplitude for the
reflectivity may be problematic particularly in practical situations where the
assumed velocity is subject to errors. The curvilinear stack should therefore
be regarded as a tool for investigating the cross-correlogram space but not
as the final product.
12
4
Semblance analysis
In previous sections, we analyzed the dependence of events in the crosscorrelogram space on location and/or orientation of plane reflectors and point
diffractors. As is clear from Equations (1-2), the event shape also depends
on the velocity v in the medium. For a fixed source, the lag in the DRcomponent of the cross-correlogram is the difference between the travel time
of the reflected wave to the virtual receiver and the time of a direct arrival
to the virtual source.
The average velocity along the path of a direct wave can be easily obtained
by dividing the known distance between the physical source and the virtual
source by the time of the first break in the common receiver gather. We
will threfore assume that this velocity is known. Of interest is the typically
unknown velocity along the path of a reflected wave. For a setup with flat
reflectors, it is the same as the average velocity between the virtual receiver
and the reflector. We use the previously defined curvilinear stack to construct
a semblance functional aimed at recovering this velocity.
We pose the problem as follows. Assuming the presence of several reflectors and/or point diffractors in the medium, we would like to identify their
location and orienation, and to estimate the velocity above each structure.
We use a modification of a standard velocity semblance functional adapted
to our problem Kimball and Marzetta (1984). For a plane reflector, assume
without loss of generality that the offset xℓ = 0. Then the reflector is completely defined by a triplet (zℓ , θℓ , v), where v is the average velocity above the
13
reflector. We define semblance (or fidelity) function of this setup as follows:
2
RT R
C xs , τDR (zℓ , xs ) + t dxs dt
−T S
̺(v, θℓ ; zℓ ) =
,
(5)
RT R
|S |
C 2 xs , τDR (zℓ , xs ) + t dxs dt
−T S
where |S | is the length of the source line (the total number of sources in the
discrete case). The semblance is proportional to the energy in the curvilinear
stack along the predicted DR-component of the cross-correlogram. In a highfrequency regime, this dependence becomes explicit, since
RT
SC2 (zℓ ) dt
−T
̺(v, θℓ ; zℓ ) →
|S |
RT
, as f → ∞,
(6)
SC 2 (zℓ ) dt
−T
Infinite frequency allows simultenous resolution of all depths. For a bandlimited pulse, each depth must be probed separately due to the resolution limit
imposed by the spatial wavelength of the pulse.
If the velocity and the structure parameters are identified correctly, the
value of the semblance will be close to 1. Otherwise it is close to zero. The
result of this velocity semblance analysis applied to all possible values of
the velocity, depth and dip is a library of objects in the subsurface along
with their complete description. Each event in the cross-correlogram space is
tagged with a physical object as well as with an analytic formula describing
its moveout. This moveout in turn can be used as needed for extrapolation
of filling gaps in the cross-correlogram, and to consequently improve the
standard interferometric stack.
14
5
Numerical examples
In this section, we apply the analysis developed above to three numerical
setups. In all cases, 50 sources are placed on the surface equidinstantly
between −2500 and 2500 m. Two receivers are fixed 5 m apart at the depth
of 1500 m. The source is a Ricker wavelet of central frequency 20 Hz. In
each case, we compute the semblance cube and look for the location of high
values. We see that in every case the complete structure of the subsurface is
recovered very well.
5.1
Example 1
The first medium consists of three horizontal layers, which form two flat
reflectors.
Figure 6: Model setup for Example 1
15
The correct values of the dips, depths and average velocities above each
reflector are fully recovered and they show in red (Fig. 8). The resolution
of the semblance in all directions is good. The physical velocities inside each
layer are easily computed from the average values prvided by the semblance
analysis.
Figure 7: Cross-correlogram for Example 1
5.2
Example 2
In the second example, the layers and hence the reflectors are dipping at
an angle 30 ◦ (Fig. 10). Recall that our analysis assumes that a medium
velocity can be well approximated with a constant. In the case of dipping
reflectors this approximation becomes invalid as soon as the dip is large. For
an extreme case of vertically layered medium, waves emanating from different
16
Figure 8: Semblance cube for Example 1
Velocity semblance for dip = 0 deg
4500
0.9
0.8
4000
Velocities (m/s)
0.7
0.6
3500
0.5
0.4
3000
0.3
2500
0.2
0.1
2000
2500
3000
3500
4000
4500
Depths (m)
5000
5500
Figure 9: Semblance slice for Example 1
17
0
sources travel in completely different velocities. For a reasonably low dip the
analysis produces just small errors.
The semblance cube (Fig. 12) contains two local maxima. Parameters of
the top layer are extracted exactly. The depth of the second layer is estimated
to be 4.35 km while the correct depth is 4.2 km. Its estimated dip is also
slightly incorrect at 34 ◦ . The overestimate is due to the fact that the average
velocity to the left of the borehole is slower than average, while it is larger
than average to the right of the borehole.
Figure 10: Medium setup for Example 2
5.3
Example 3
In the final example, the medium contains two flat reflectors and one point
diffractor (Fig. 15). As the predicted shape of correlogram events for a
18
Figure 11: Cross-correlogram for Example 2
Figure 12: Semblance cube for Example 2
19
Velocity semblance for dip = 30 deg
4500
0.8
0.7
4000
Velocities (m/s)
0.6
3500
0.5
0.4
3000
0.3
0.2
2500
0.1
2000
2500
3000
3500
4000
4500
Depths (m)
5000
5500
0
Figure 13: Semblance slice 1 for Example 2
reflector is different from any curve corresponding to a diffractor, it is expected that the semblance performed with reflector moveouts will ignore any
new information infused by the diffractor. This can indeed be seen in the
semblance cube (Fig. 17).
If diffractors are of interest, a separate semblance analysis (Fig. 19)
should be performed with the help of Eq. (2). That reveals the location
of the diffractor as well as the average velocity between it and the virtual
receiver.
Combining the two semblance analyses, we conclude that the entire structure of the subsurface is recovered exactly with good resolution.
20
Velocity semblance for dip = 34 deg
0.9
4500
0.8
Velocities (m/s)
4000
0.7
0.6
3500
0.5
0.4
3000
0.3
0.2
2500
0.1
2000
2500
3000
3500
4000
4500
Depths (m)
5000
5500
0
Figure 14: Semblance slice 2 for Example 2
6
Results and Conclusions
Seismic interferometry is a novel and promising venue of research with many
imaging applications. A cross-correlogram of two common receiver gathers
is the fundamental object for interferometric analysis.
A reflection event in a conventional CDP gather is earliest for receivers
near the shot and is later for receivers further from the shot. The geometry
of events in a cross-correlogram is not as obvious. Under a VSP setup, the
stationary phase point occurs at the latest time for the event and all other
conventionally unwanted portions of the event occur earlier.
We have developed analytical expressions for the moveout of reflections
and diffractions on a cross-correlogram. We propose to use these moveout
expressions to create a velocity analysis methodology using semblance as the
21
Figure 15: Medium setup for Example 3
statistic, to analyze a cross-correlogram.
Our preliminary results show that our velocity analysis is very sensitive to
the location and dip of the reflection. The velocity semblance for reflections
is quite insensitive to diffracted events, and vice versa. This type of velocity
sensitivity may allow for the enhancement or reduction of events produced
by certain features in the subsurface.
The development of this type of filtering will form the scope for additional
research and development.
22
Figure 16: Cross-correlogram for Example 3
Figure 17: Reflector-based semblance cube for Example 3
23
Velocity semblance for dip = 0 deg
4500
0.9
0.8
4000
Velocities (m/s)
0.7
0.6
3500
0.5
0.4
3000
0.3
0.2
2500
0.1
2000
2500
3000
3500
4000
4500
Depths (m)
5000
5500
0
Figure 18: Reflector-based semblance slice for Example 3
Figure 19: Diffractor-based semblance cube for Example 3
24
Velocity semblance for offset = 1000 m
3000
0.7
2800
0.6
2600
Depth (m)
2400
0.5
2200
0.4
2000
0.3
1800
1600
0.2
1400
0.1
1200
1000
2000
2500
3000
3500
Velocity (m/s)
4000
4500
Figure 20: Diffractor-based semblance slice for Example 3
7
Acknowledgements
This work was supported by the Earth Resources Laboratory Founding Member Consortium. The authors would like to thank Prof. Alison Malcolm and
Maria Gabriela Melo Silva for many useful joint discussions of this work.
25
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