Brad Artman1, Deyan Draganov2, Kees Wapenaar2, Biondo Biondi1
1Stanford
Exploration Project, Geophysics, Stanford University, 94305, USA
2Department of Applied Earth Sciences, Delft University of Technology, Mijnbouwstraat 120, 2628 RX, Delft, The Netherlands
Shot-gathers from noise
Introduction
General relations between the reflection and the transmission responses of a 3-D
inhomogeneous medium (Wapenaar, 2004) have been developed to forward model the
transmission coda, suppress multiples, provide the basis of seismic interferometry, and
forward model the reflection response from the transmission response. The final
relation, exploited for acoustic daylight imaging, shows that by cross-correlating traces
of the transmission response of a medium one can synthesize the reflection response
collected in a conventional active source experiment.
Having generated shot-gathers in this manner, they can be processed with conventional
techniques to enhance signal, remove artifacts, or create a migrated image. Here, we
present the theory of direct migration of the measured transmission response wavefields
without the need to first generate the reflection response wavefields by cross-correlation.
Removing this step allows for significant time savings when dealing with these
inherently large data sets and produces an equivalent image. Apart from computational
advantages, moving the modelling of the reflection response from the transmission
response down to the image point during migration opens the possibility of more
advanced imaging conditions in the future.
Intuition via incident plane-wave
Figure 3 shows how the quality of modeled shot-gathers decreases (by (Nt )1/2 ) when
shorter transmission records are processed. Eleven sub-surface impulses convolved with
long random sequences were summed to generate the transmission wave-fields.
Surface
Reflector
Downward reflection
Up-coming incident plane wave
depth
depth
(a)
(b)
(c)
Fig. 3: (a) Modelled shot-gather over a multi-layered earth. (b) Cross-correlation of
modelled transmission data to simulate same shot-gather using 100 minute input. (c)
Simulated gather using 20 second input.
(a)
Down-going
free-surface
reflection
Both modeled gathers will improve with increased coverage of the bottom of the earth
model with source locations. In a real experiment increased recording time results in
both capture of increasing quantities of ambient noise as well as the entire wave-train of
long-coda incident energy.
(b)
Up-coming
subsurface
reflection
Signal enhancement from migration
time
depth
The short records used in panel (c) results in almost loosing the reflection hyperbola.
Even the longer record in panel (b) is poor compared to the real shot-gather in panel (a).
(c)
(d)
Fig. 1: Incident plane wave source passing through single reflector earth model at
successive time steps (a-c). Panel d shows transmission wavefield recorded at the surface.
Panels (a-c) depict a plane wave traveling left to right through a subsurface with a single
layer. Panel (d) is the transmission record recorded at the surface from the plane wave
energy. The sum of many such records, or a very long time axis that captures many such
arrivals is the observed transmission wavefield, Tobs , for the passive seismic experiment.
Hyperbolas by correlation
r1
r1
r2
r1 r1
r2
r1 r2
(a)
(b)
Fig. 4: (a) Modelled shot-gather over a multi-layered earth with short transmission
wavefield input. (b) Shot-profile migration image of modelled reflection data.
lag/twt
time
transmission
to
reflection
Fig. 2: (a) Incident plane wave source passing through single reflector earth model with
rays showing the propagation paths. (b) Idealized traces from a transmission wavefield.
(c) Shot-gather (reflection wavefield) modelled using the r1 trace as source location.
Correlating all the recorded traces has two important functions: 1) Establish time zero of the
modeled reflection experiment, and 2) removing the unknown, complicated source wavelets.
The direct arrival acts as a source wavelet at every location that is compared via the crosscorrelation to reflections at all receiver locations.
The almost unintelligible modeled gather in panel (a) highlights the ramifications of not collecting
enough passive seismic data before processing. Panel (b) however shows that despite the poor
quality of the input shot gathers, the migrated image of all of the shot-gathers recovers subsurface
structure quite well. The earth model used here has two layers, both with synclinal structure, at
depths 300 and 500 meters. The power of the modeled direct arrival (0 depth) masks the second
reflector.
Cross-correlation of each trace, acting as a source location, with every other trace results in
N2 traces of output. This volume of data is N simulated shot-gathers each with N traces.
A surface related multiple from the shallow reflector almost touches the bottom of the deeper
syncline at 5000 meters. Because the correlation of the traces from the transmission wavefeild
models exactly the reflection data, all the customary problems facing the processing of traditional
seismic data resurface after correlation and migration.
While transmission recordings are very long, the output of the cross-correlations need only as
many lags (multiplied Dt) to include the two-way travel-time of the deepest reflector of interest.
After the raw transmission data are correlated all of the tools of modern seismic imaging are
readily applicable to handle features now observable in the seismic section.
Brad Artman1, Deyan Draganov2, Kees Wapenaar2, Biondo Biondi1
1Stanford
Exploration Project, Geophysics, Stanford University, 94305, USA
2Department of Applied Earth Sciences, Delft University of Technology, Mijnbouwstraat 120, 2628 RX, Delft, The Netherlands
Direct migration
Wavefield datuming
Migration by wavefield extrapolation can be separated into two distinct operations.
First is the propagation of a seismic wavefield to a deeper level within the subsurface.
Second, we collapse the time/frequency axis with an imaging condition. Without the
performing the second step, maintaining the dimensionality of the wavefields, however
is datuming. Algorithms to migrate/datum seismic data with a wavefield extrapolation
operation fall into two categories that produce equivalent results: 1) Shot-profile (SP),
which propagates individual source and receiver wavefields, and 2) source-receiver
(SR), which propagates the entire seismic wavefield sorted into midpoint-offset
coordinates.
SR Datuming
Rz
SP Datuming
Dz
Uz
+
Uz+1
Dz+1
Inserting the transmission wavefield into the shot-profile migration algorithm thus produces an identical image
to one produced by first modeling the reflection response and then migrating. To capitalize on the signal
enhancement provided by prestack imaging, we can save significant computer time by migrating the passive
data directly when correlation (and the necessary FFT's) of the inherently long traces is costly.
Passive Datuming
Tz
+
+
Rz+1
Tz+1
Tz
Tz+1
U= Receiver wavefield (up-going)
D= Source wavefield (down-going)
R= Total refection data
T= Transmission wavefield
Fig. 4: Flow diagram for datuming seismic data to a greater
subsurface depth level. Sign associated with the vertical arrows
indicates the causality of extrapolation. Circle with x’s represent
cross-correlation.
Cross-correlation of the up-going and down-going wavefields during SP datuming gives
exactly the wavefield generated by down-ward continuation of the total seismic
wavefield, R, using SR datuming. Previously, we explained that cross-correlation of
traces from a recorded transmission wavefield, T, yields the wavefield R. We presented
the argument assuming that correlation happen at the surface of the earth. However,
using T as both the U and D wavefields in a SP style algorithm, performing the
correlation at depth after propagation yields identical results.
Fig. 5: The modelled shot-gather of panel (b), is indicative of the data that was migrated
directly to produce the image in panel (b). Migrating the shot-gathers produces an identical
image, though uses almost two orders of magnitude less computer time.
Mathematical Rigour
The WRW description of seismic datuming (Wapenaar, 1989) is
R (ξ A , ξB , ω) 


{ W (ξ A , x A , ω)} R (x A , xB , ω){ W (xB , ξ B , ω)} * dx A dxB ,
*



D0
where superscripts indicate one-way direction, and the W operators are matched inverseextrapolators that forward extrapolate data, R, from a position x to the deeper location x.
Neglecting acausal terms and the delta function at zero-lag, the correlation of traces from
a transmission wavefield gives the reflection seismic wavefield
R (x A , xB , ω)  T


obs
(x A , ω){ T

obs
(xB , ω)} .
*
Substitution of the RHS of this relation into the datuming equation above yields
R (ξ A , ξ B , ω)  

{ W (ξ A , x A , ω)} T
*


obs
(x A , ω){ T

obs
(xB , ω)} { W (xB , ξ B , ω)} * dx A dxB .
*

D0
This relation holds for any depth level pair (x,x) just as the diagram above is valid for
any depth level z to z+1.
Datuming to imaging
The motivation to image modeled reflection data from transmission data presented in
the Signal enhancement section can be incorporated into the datuming framework
above. Beginning with a datumed wavefield, to produce an image I(x,z), we evaluate an
imaging condition
I(x, z) 
 R(x, ω, z)
ω
for source receiver migration. For shot profile migration substitute the correlation of U
and D and then sum the images from each individual shot
I(x, z)   U(x, s, ω, z)D (x, s, ω, z).
*
s
References
J.F. Claerbout. Toward a unified theory of reflector mapping. Geophysics, 36:467-481, 1971.
C.P.A.Wapenaar and A.J. Berkhout. Elastic wavefield extrapolation. Elsevier, 2 1989.
C.P.A.Wapenaar, Jan Thorbecke and Deyan Draganov. Relations between reflection and
transmission responses of 3-D inhomogeneous media. Geophys. J. Intern., 2004.
Acknowledgments
ω
The sum over the frequency axis in both relations extracts energy from the wavefields at
the zero time-lag of the correlation while simultaneously performing an IFT to produce the
desired image (Claerbout, 1971).
We thank the Stanford Exploration Project AND WHO ELSE DEYAN??? for financial support.