Brad Artman1, Deyan Draganov2, Kees Wapenaar2, Biondo Biondi1

1Stanford Exploration Project, Geophysics, Stanford University, 94305, USA2Department of Applied Earth Sciences, Delft University of Technology, Mijnbouwstraat 120, 2628 RX, Delft, The Netherlands

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 artefacts, 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 for more advanced imaging conditions in the future.

Introduction

Intuitive approach via incident plane-wave

Panels (a-c) depict a plane wave traveling left to right through a subsurface with a single layer. Panel (d) is the transmission response 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.

(d)

(b)(a)

(c)

Reflector

Up-coming incident plane waveDownward reflection

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.

Surface

Down-goingfree-surface reflection

Up-coming subsurface reflection

dept

hde

pth

dept

hti

me

Hyperbolas by correlation

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 cross-correlation to reflections at all receiver locations.Cross-correlation of each trace, acting as a source location, with every other trace results in N2 traces of output. This volume of data represents N simulated shot-gathers, each with N traces. While transmission recordings are very long, the output of the cross-correlations need only as many lags (multiplied t) to include the two-way travel-time of the deepest reflector of interest.

tim

e

r1 r2r1 r2r1 r1 r1 r2

transmissionto

reflection

lag/

twt

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.

Shot-gathers from noise

Fig. 3: (a) Modelled shot-gather over a multi-layered earth. (b) Cross-correlation of modelled transmission data to simulate the same shot-gather using 6 minute input. (c) Simulated gather using 20 second input.

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.

The short records used in panel (c) result in almost loosing the reflection hyperbola. Even the longer record in panel (b) is poor compared to the real shot-gather in panel (a).

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.

(a) (b) (c)

Signal enhancement from migration

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.

(a) (b)

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 530 meters, respectively. The power of the modeled direct arrival (0 depth) masks the second reflector. A surface related multiple from the shallow reflector almost touches the bottom of the deeper syncline at 500 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.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.

Direct Migration of Passive Seismic Data

Brad Artman1, Deyan Draganov2, Kees Wapenaar2, Biondo Biondi1

1Stanford Exploration Project, Geophysics, Stanford University, 94305, USA2Department of Applied Earth Sciences, Delft University of Technology, Mijnbouwstraat 120, 2628 RX, Delft, The Netherlands

Direct Migration of Passive Seismic Data

Wavefield datuming Direct migration

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 performing the second step, maintaining the dimensionality of the wavefields, what do, 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.

U= Receiver wavefield (up-going)D= Source wavefield (down-going)R= Total refection dataT= Transmission wavefield

Fig. 4: Flow diagram for datuming seismic data to a deeper subsurface level. Sign associated with the vertical arrows indicates the causality of extrapolation. Circle with x’s represent cross-correlation.

Rz+1

Tz TzRz

+

Uz Dz

Uz+1 Dz+1

+

Tz+1 Tz+1

+

SP Datuming SR Datuming Passive Datuming

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.

Mathematical Rigour

,dd*)},,(W){,,(R)},,(W{),,(R BABBBA*

AAD

BA0

xxξxxxxξξξ ωωωω

.dd*)},,(W{)},(T){,(T)},,(W{),,(R BABB*

BobsAobs*

AAD

BA0

xxξxxxxξξξ ωωωωω

-

.)},(T){,(T),,(R *BobsAobsBA ωωω xxxx

The WRW description of seismic datuming (Wapenaar, 1989) is

where superscripts indicate one-way direction, and the {W}* operators are matched inverse-extrapolators that forward extrapolate data, R, from a position x to the deeper location

Neglecting acausal terms and the delta function at zero-lag, the correlation of traces from a transmission wavefield gives the reflection seismic wavefield

Substitution of the RHS of this relation into the datuming equation above yields

This relation holds for any depth level pair (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

ω

ω )z,,(R)z,(I xx

for source receiver migration. For shot profile migration substitute the correlation of U and D and then sum the images from each individual shot

).z(D)z(U)z(I * ,s,,,s,,, s

ωωω

xxx

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).

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.

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.

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

We thank the Stanford Exploration Project the Dutch Science Foundation STW and the Dutch Research Center for Integrated Solid Earth Sciences (ISES).