Reverse-time migration (RTM) represents one of the most accurate, but also one of the costliest algorithms available forimaging complex media. The RTM computational cost can be reduced by shot encoding, i.e. by imaging simultaneously alarge number of shots. Different encoding strategies are possible, including linear and random phase encoding, whichrepresent end members of a more general family of encodings. For a fixed maximum delay (i.e. computational cost), wecan trade spatial bandwidth for crosstalk noise. Linear phase encoding is characterized by low bandwidth and high SNR,while random phase encoding by high bandwidth and low SNR. Mixed encodings allow us calibrating the amount ofresolution desired in the migrated image, given an acceptable level of noise.

SUMMARY

Comparison of shot encoding functions for reverse-time migrationFrancesco Perrone* and Paul Sava, Center for Wave Phenomena, Colorado School of Mines

WAVE-EQUATION MIGRATIONWave-Equation migration consists of two steps:

• wavefield reconstruction

• Imaging condition

The imaging condition is a nonlinear operation: zero-timecrosscorrelation of source and receiver wavefield

The summation is over all the shots/experiments.

The wavefield reconstruction extrapolates the source andreceiver wavefields, respectively, forward and backward intime in the velocity model:

• source wavefield:

• receiver wavefield:

The index k indicates the shot number.

s

k= s

k(x,t)

r

k= r

k(x,t)

I(x) = sk(x,t)r

k(x,t) dt!

k

"

SHOT ENCODING MIGRATION

S(x,t,!) = sk(x,t) *" (t - f (x

k,!))

k

#

R(x,t,!) = rl(x,t) *" (t - f (x

l,!))

l

#

Synthetic source and receiver wavefields:

Imaging condition

I(x) = Ik+

k

! sk(x," )r

l

*(x," ) ei" ( f (x

l,# )$ f (x

k,# ))

#

! d"%k& l

!

CROSSTALK

Comparison of shot encoding functions for reverse-time migrationFrancesco Perrone* and Paul Sava, Center for Wave Phenomena, Colorado School of Mines

ENCODING EFFECTS APPROXIMATION TO THE IDENTITY

CROSSTALK MATRIX:

ei! ( f (x

l," )# f (x

k," ))

"

$LINEAR SHOT ENCODING:

RANDOM SHOT ENCODING:

LINEAR RANDOM

Linear-shot encoding migration(L-SEM) produces low artifacts butrecovers a limited number ofcomponents of the spatial spectrum ofthe image.

Random-shot encoding migration(R-SEM) recovers more spatialcomponents but the crossterms in theimaging condition spread over theentire image.

Point scatterer in constant velocity

OBSERVATIONS

• Trade off crosstalk VS spatial bandwidth.

• The equivalence between shot-profile migration andencoding migration can be achieved only as a limit.

• We can combine these two extrema, sharingadvantages and drawbacks.

f (x

k,!) =U (x

k,!)

f (x

k,!) =

sin("!)

v(x

k# x

0)

GOOD

Comparison of shot encoding functions for reverse-time migrationFrancesco Perrone* and Paul Sava, Center for Wave Phenomena, Colorado School of Mines

MIXED SHOT ENCODING:

MIXED SHOT ENCODING

APPROXIMATION TO THE IDENTITY

CROSSTALK MATRIX:

ei! ( f (x

l," )# f (x

k," ))

"

$

+ σ =

LINEAR RANDOM MIXED

Combining the small crosstalk noiseof Linear shot encoding and thehigh spatial bandwidth of Randomshot encoding we can move towardan intermediate scheme thatincreases the spatial bandwidthwithout introducing anoverwhelming level of crosstalk.

MIXED SHOT ENCODINGPlane wave dithering

f (x

k,!) =

sin("!)

v(x

k# x

0) +U (x

k,!)

• A bigger number of plane-wavecomponents is considered with respectto standard L-SEM, even though withsmaller weight.

• Reduced crosstalk from differentexperiments.

• The random-like nature of crosstalkis preserved.

CONCLUSIONS

Comparison of shot encoding functions for reverse-time migrationFrancesco Perrone* and Paul Sava, Center for Wave Phenomena, Colorado School of Mines

ACKNOWLEDGMENTS

REFERENCESLINEAR

RANDOM

NUMERICAL RESULTS

SRM

• Clara Andreoletti and Nicola Bienati

• Eni

MIXED

• SRM: 50 shots

• L-SEM: 50 plane-waves

• R-SEM: 50 randomdelays realizations

• M-SEM: 50 ditheredplane-waves

• Comparison of Linear andRandom Shot Encoding andanalysis with respect tocrosstalk and spatial bandwidthin the image.

• Combination of L-SEM and R-SEM is effective at increasingspatial bandwidth andcontrolling crosstalk.

• M-SEM reduces thecomputational cost for RTM forspecified bandwidth and SNR.

• M-SEM achieves spatialresolution comparable to SRMwithout suffering fromundersampling problems.

• Romero L. A. et al., 2000, Phaseencoding of shot records inprestack migration, Geophysics, 65

• Whitmore N., 1995, An imaginghierarchy for common-angle planewave seismograms, PhD thesis,University of Tulsa.

• Liu F. et al., 2006, Toward aunified analysis for source plane-wave migration, Geophysics, 71

• Etgen J., 2005, How manyangles do we really need fordelayed-shot migration?, 75thAnn. Internat. Mtg., Soc. of Expl.Geophys.

• Soubaras R., 2006, Modulated-shot migration, 76th Ann. Internat.Mtg., Soc. of Expl. Geophys.