Anti-aliasing Condition and Filter for Reciprocity Equations ofCorrelation Type

Weiping Cao and Gerard Schuster

ABSTRACT

An anti-aliasing formula is derived for interferometricredatuming of seismic data. More generally, this for-mula is valid for numerical implementation of the reci-procity equations of correlation type, which is used forredatuming, extrapolation, interpolation, and migra-tion. The anti-aliasing condition can be, surprisingly,more tolerant of a coarser trace sampling compared tothe standard anti-aliasing condition. Numerical resultswith synthetic data show that interferometry artifactsare effectively reduced when the anti-aliasing conditionis applied to interferometric redatuming.

INTRODUCTION

Redatuming wavefields using Green’s theorem, also knownas backward extrapolation of wavefields, has long beenused in the oil industry to remove elevation statics or tomitigate the defocusing effects of certain geologic features,such as weathering zones or salt bodies (Berryhill, 1979;Wapenaar and Berkhout, 1989; Bevc, 1997; Schneider,1978). The key idea is to multiply the shot gather dataD(A|x) by the conjugate of a weighted Green’s functionG(B|x) and sum along the source position x to estimatethe wavefield for a source along a deeper datum level de-fined by the position vector B (see Figure 1). Mathemati-cally this operation is performed in the frequency domainas

A,BεV0 D(A|B) ≈∫

S0

w(A,B,x)G(B|x)∗D(A|x)d2x (1)

where A is the trace position, S0 is the surface area alongwhich the shots are located, w(A,B,x) represents theweighting function that depends on the source-receiver co-ordinates and frequency ω, and D(A|B) is the redatumeddata at the new datum surface with new source position

at B. Here, the Green’s function G(B|x) is model-basedand is typically computed by using a ray-based method ora numerical solution to the acoustic wave equation.

The rigorous mathematical foundation for this reda-tuming equation was initially developed by Porter (1970)and Bojarski (1983) and is also known as the reciprocityequation of correlation type (de Hoop, 1995). A gener-alization of the reciprocity equations to acoustic, electro-magnetic, elastodynamic, poroelastic, and electroseismicwaves is described by Wapenaar (2007) who used a unifiedmatrix-vector wave equation. Artifacts associated withredatuming wavefields measured on a coarse or irregulargrid can be mitigated by a least squares approach (Liuand Sacchi, 2004; Ferguson, 2007).

Redatuming wavefields is one of two steps1 in seismicmigration (Schneider, 1978; Claerbout, 1992) which esti-mates the subsurface reflectivity distribution from reflec-tion data. Similar to the integrand in equation 1, themigration kernel consists of the product of the data andthe conjugated Green’s function computed for an assumedvelocity model.

In the latter part of the 20th century, a new type ofredatuming was introduced such that the model-basedGreen’s function in equation 1 is replaced by a data-basedGreen’s function (Claerbout, 1968). Applications for reda-tuming wavefields included helioseismology (Duvall et al.,1993; Rickett and Claerbout, 1999) , earthquake seismol-ogy (Scherbaum, 1987a,b; Shapiro and Campillo, 2004;Gerstoft et al., 2006; Larose et al., 2006), physical acous-tics (Fink, 1993, 1997, 2006; Lobkis and Weaver, 2001),and exploration geophysics (Schuster et al., 2004; Calvertet al., 2004). These redatuming procedures were denotedby a variety of names such as reverse time acoustics, day-light imaging, interferometric imaging, and virtual sourceimaging, but they were eventually unified under the name

1Migration is defined as extrapolationg the wavefield recorded atthe surface to a deeper depth level followed by the application of animaging condition (Claerbout, 1992).

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174 Cao and Schuster

”seismic interferometry” by the discovery that their com-mon mathematical foundation is the reciprocity equationof correlation type (Wapenaar, 2004). Typically, seis-mic interferometry is applied to traces recorded on a dis-crete grid of geophones, but until now the interferometricNyquist sampling criterion for avoiding aliasing artifactswas not known.

This paper provides the anti-aliasing condition for inter-ferometric datuming. For some configurations of receiversand sources, the anti-aliasing condition is less stringentthan that of standard seismic recording experiments. Thispaper is organized into four sections: the introduction, thetheory part, a numerical result section, and finally conclu-sions.

ANTI-ALIASING CONDITION FORINTERFEROMETRIC REDATUMING

For simplicity of exposition, we will use the far-field ap-proximation to the reciprocity equation of correlation type(Wapenaar, 2004; Schuster, 2009) such that the Green’sfunction G(A|B) is given by

A,BεV0 Im[G(B|A)] = k

∫S0

G(B|x)∗G(x|A)d2x, (2)

where k is approximated by the wavenumber at the re-ceiver location, the integration is over the surface denotedby S0 and the points A and B are within the volume V0

bounded by S0 and the lower half-space. Here we assumethat the source signature has been deconvolved and theabove equation can be estimated from equation 1 by re-placing D(A|x) by G(A|x), D(A|B) by Im[D(A|B)], andw(A,B,x) by k. Note, reciprocity allows for the inter-change of the source and receiver positions in the Green’sfunctions so G(A|B) = G(B|A).

Approximating the integral in equation 2 by a discretesum over evenly spaced grid points along the source lineS0 yields the following equation:

A,BεV0 Im[G(B|A)] ≈∑

i

Γ(B,A,xi), (3)

where

Γ(B,A,xi) = kG(B|xi)∗G(xi|A)∆x2,

= |Γ(B,A,xi)|eiφ(B,A,xi). (4)

Here, ∆x is the spatial sampling interval along the sourceline denoted by So, xi is the ith position vector of thesource along S0 and φ(B,A,xi) represents the phase ofthe integrand in equation 2.

Sampling a signal too sparsely in space causes spatialaliasing (Maeland, 2004); equivalently, sampling the in-tegrand in equation 2 too coarsely in space will result inunrecoverable errors in estimating G(A|B). Therefore,the general anti-aliasing condition for equation 3 is thatthe phase variation of Γ(B,A,xi) between adjacent geo-

phones is less than π. For equation 4 this means

|φ(B,A,xi)− φ(B,A,xi−1)| < π, (5)

where it is assumed that the amplitude term |Γ(B,A,x)|spatially varies more slowly than the phase spectrum φ(B,A,x)and xi−1 denotes the xi−1 source position adjacent to thexi source position. This criterion is similar to the Nyquistsampling criterion for properly sampling a sinusoidal func-tion sin(φ(B,A,x)), and prevents aliasing artifacts in thesummation.

Physical Meaning

The physical meaning of the aliasing condition in equa-tion 5 is determined by decomposing the Green’s functionsinto a summation of direct waves, primary reflections, andother events:

G(B|A) = G(B|A)direct + G(B|A)primary + ... (6)

Figure 1a illustrates the direct and ghost reflection raysfor these Green’s functions, which can be asymptoticallyrepresented as

G(B|x)direct ≈ β(B,x)eiωτdirectxB ;

G(x|A)ghost ≈ α(A,x)eiωτghostxBA . (7)

Here, τdirectxx′ is the specular traveltime for a direct arrival

from x to x′, and τghostxBA is the specular ghost reflection for

a source at x and receiver at A. The terms α(A,x) andβ(B,x) account for geometrical spreading in the reflectionand direct waves, respectively.

Redatuming the receiver from x to B (see Figure 1)is accomplished by temporally correlating the direct ar-rival with the the ghost reflection and summing the resultover the source coordinates as described in equation 3.Mathematically this is accomplished in the frequency do-main by the replacements: G(B|x) → G(B|x)direct andG(x|A) → G(x|A)ghost in equation 4 to give

Γ(B,A,x) ≈ k∆x2α(A,x)β(B,x)eiω(τghostxBA

−τdirectxB ), (8)

where the phase is denoted as φ(A,B,x) = ω(τghostxBA −

τdirectxB ). Plugging this phase into the anti-aliasing condi-

tion of equation 5 yields

|τghostxiBA − τdirect

xiB − τghostxi−1BA + τdirect

xi−1B | < T/2, (9)

where T = 2π/ω is the period. Assuming an horizon-tal line of sources along the x coordinate and expandingτghostxi−1A and τdirect

xi−1B about xi in a Taylor series yields

τdirectxi−1B = τdirect

xiB −∂τdirect

xiB

∂x∆x + O(∆x2);

τghostxi−1BA = τghost

xiBA −∂τghost

xiBA

∂x∆x + O(∆x2), (10)

where ∆x = |xi − xi−1|. Inserting these terms into equa-

Anti-alias Condition and Filter for Reciprocity Equations of Correlation Type 175

tion 9 and assuming a linear moveout in the events be-tween adjacent geophones (i.e., neglect 2nd-order termsin the Taylor expansion) yields

∆x <T

2|∂τghost

xiBA

∂x −∂τdirect

xiB

∂x |, (11)

which compares to the anti-aliasing condition for properlysampling either a direct wave or a primary reflection:

∆x <T

2|∂τdirect

xiB

∂x |; ∆x <

T

2|∂τghost

xiBA

∂x |. (12)

For positive moveout velocities, the denominator in equa-tion 11 is less than those in equation 11, so we concludethat the anti-aliasing condition described by equation 11is more tolerant of coarse sampling intervals than theconditions described by equations 12. Note events withmoveout velocities opposite in sign require more stringentgeophone sampling criterion in equation 11. In contrast,the reciprocity equations of convolution type (Wapenaar,2007) results in the anti-aliasing condition of

∆x <T

2|∂τghost

xiBA

∂x +∂τdirect

xiB

∂x |, (13)

which is less tolerant of coarse geophone sampling formoveout velocities with the same polarity.

The physical meaning of the interferometric anti-aliasingcondition in equation 11 is discovered by recognizing itssimilarity to applying a local linear moveout (LLM) to re-flection data2. In this case the linear moveout velocity isthat of the direct arrival with apparent velocity

∂τdirectxiB

∂x ,where the associated time shift lessens the liklihood ofaliasing the reflection event with a coarse source spacing.

The product of two Green’s functions described by equa-tion 6 yields many combinations of events that lead to,e.g., a primary event with the receiver redatumed to posi-tion B. For example a source side ghost in Figure 2a cor-related with the related ghost reflection in Figure 2b yieldsthe redatumed primary reflection in Figure 2c. Thereforethe anti-aliasing condition in equation 11 can be general-ized to

∆x <T

2|∂τevent1

xiA

∂x −∂τevent2

xiB

∂x |, (14)

where event1 at A and event2 at B correspond to thetwo distinguished events in the recorded data that are

2That is, assume a shot gather d(x, t) = δ(t− x/v) that containsa linear event with moveout velocity v. Applying a LLM time cor-rection (with moveout velocity v0) to these traces yields the timeshifted traces d(x, t) = δ(t − x/v + x/v0). The spectrum of thesetraces is proprtional to ei(k−k0)x, where k0 = ω/v0 and k = ω/v.Hence, the effective apparent wavelength of the shifted traces hasbeen lengthened from 2π/k to 2π/(k − k0) and so can enjoy largertrace spacing without being aliased. Applying a linear moveout timecorrection to a shot gather is a commonly used processing step tomitigate aliasing artifacts in coarsely sampled traces.

transformed into a redatumed event under correlation andsummation of trace pairs at A and B.

NUMERICAL TESTS

We test the interferometric anti-aliasing condition on syn-thetic traces generated for an Vertical Seismic Profile (VSP)experiment where the VSP gathers are transformed intovirtual surface seismic profile (SSP) data. Figure 4 depictsthe recorded VSP data generated by a finite-difference so-lution to the acoustic wave equation for the 3-layer modelshown in Figure 3. 100 receivers are deployed in the wellwith a recording spacing of 20 m, and 200 shots are ex-cited on the surface with an interval of 20 m. VSP tracesare transformed into virtual SSP gathers using equation 2.Similar to the scheme of interferometric migration of VSPghost in He et al. (2007), only the direct arrival of G(B|x)is included in the correlation redatuming.

Figures 5a-c depict the virtual SSP traces redatumedwith VSP gathers for different recording spacings. Herethe minimum apparent wavelength λz

min is estimated tobe 164 m. Comparing the virtual SSP gathers with thetrue SSP gather in Figure 6 shows increasing artifacts forcoarser recording spacing in the well. Similar to the anti-aliasing filter used in migration (Gray, 1992; Lumley et al.,1994), we can low-pass filter the aliased data to avoid vi-olation of the anti-aliasing condition in equation 14. Thisprocedure consists of following steps:

1. The LLM∂τdirect

xiB

∂x of G(B|x) is estimated from thelocal slope of the direct wave traveltime curve of theVSP shot gather.

2. For each time sample of G(A|x), the LLM∂τghost

xiA

∂xis calculated using local slant stack in the VSP shotgather, and equation 13 is applied to check if currentsample is aliased. The value of aliased samples in thetraces is set to zero, and a filtered trace Gf (A|x) isobtained.

3. G(B|x) and Gf (A|x) are input to equation 2 and theredatuming result with the anti-aliasing condition isobtained.

Figures 5d-f depicts the results of this anti-aliasing op-eration where many of the aliasing artifacts are removed.

To reduce the interferometry artifacts, we also applieddip filtering to the VSP gathers so that only the down-going arrivals remain. Figure 7 shows the VSP gatherafter a dip filter, to give mainly the down-going events.Figures 8a-c depict the virtual SSP gathers obtained withthe VSP gathers after dip filtering. Comparing Figures 8a-c with Figures 5a-c indicates fewer artifacts in the virtualSSP gathers when only the down going arrivals are em-ployed in the interferometric redatuming. However, whenthe anti-aliasing condition is applied, many of the remain-ing artifacts are removed from the virtual SSP gathers asshown in Figures 8d-f.

176 Cao and Schuster

CONCLUSIONS

An anti-aliasing condition is derived for interferometricredatuming of seismic data, and should be applicable toany numerical implementation of the reciprocity equationof correlation type. This formula can be used for inter-ferometry, redatuming, extrapolation, interpolation, andmigration.

The anti-aliasing formula is equivalent to the one ob-tained by locally time shifting the event of interest bya linear moveout correction, where the moveout velocityis that of an event with a shorter raypath. Honoring theanti-aliasing condition allows unaliased interferometric re-datuming and the design of an anti-aliasing filter that mit-igates aliasing artifacts.

ACKNOWLEDGEMENTS

We thank the members of the 2009 University of Utah To-mography and Modeling/Migration (UTAM) Consortiumfor the financial support of this research.

Figure 3: The velocity model and VSP recording geome-try. Both the recording spacing in the well and the shoot-ing interval on the surface are 20 m.

Figure 4: A VSP shot gather. The surface shot is deployedat (300 m, 10 m).

Figure 6: The true SSP gather simulated with finite dif-ference modeling.

Figure 7: The downgoing arrivals from the VSP shotgather in Figure 4.

Anti-alias Condition and Filter for Reciprocity Equations of Correlation Type 177

Figure 1: Goal of redatuming here is to extrapolate receiver positions from xεSo along surface to the position at B.Traces in the frequency domain are denoted by D(A|x) and recorded by geophones excited by a source at x. a) and b)depict original rays, and c). depicts the rays of the ghost reflection arrival after datuming the receiver from x to B.

Figure 2: Correlation of a a) source-side ghost with a b) related ghost reflection followed by summation over sourcelocations at x yields the redatumed primary reflection in c).

178 Cao and Schuster

Figure 5: SSP gathers redatumed using standard and anti-aliased interferometry from the VSP gathers with differentrecording spacings in the well. The comparison shows the anti-aliasing condition effectively removes many of the artifactsin the interferometry results.

Anti-alias Condition and Filter for Reciprocity Equations of Correlation Type 179

Figure 8: SSP gathers redatumed using standard and anti-aliased interferometry from the down-going arrivals of theVSP gathers with different recording spacings in the well. The comparison shows the anti-aliasing condition effectivelyremoves many of the artifacts in the interferometry results.

180 Cao and Schuster

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