Phase and amplitude tracking for seismic event separation

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Citation Li, Yunyue Elita, and Laurent Demanet. “Phase and AmplitudeTracking for Seismic Event Separation.” GEOPHYSICS 80, no. 6(November 2015): WD59–WD72. © 2015 Society of ExplorationGeophysicists

As Published http://dx.doi.org/10.1190/GEO2015-0075.1

Publisher Society of Exploration Geophysicists

Version Final published version

Citable link http://hdl.handle.net/1721.1/100553

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Phase and amplitude tracking for seismic event separation

Yunyue Elita Li1 and Laurent Demanet1

ABSTRACT

We have developed a method to decompose seismic recordsinto atomic events, each defined by a smooth phase function anda smooth amplitude function. This decomposition is intrinsi-cally nonlinear and calls for a nonconvex least-squares optimi-zation formulation, along the lines of full-waveform inversion.To overcome the lack of convexity, we have developed an iter-ative refinement-expansion scheme to initialize and track thephase and amplitude for each atomic event. For short, we calledthe method phase tracking. The initialization is carried out byapplying multiple signal classification to a few seed traces inwhich events can be separated and identified by their arrivaltimes and amplitudes. We then construct the initial solutionat the seed traces using linear phase functions from the

arrival times and constant amplitude functions, assuming themedium is mostly dispersion free. We refine this initial solutionto account for dispersion and imperfect knowledge of the wave-let at the seed traces by fitting the observed data using a gradientdescent method. The resulting phase and amplitude func-tions are then carefully expanded across the traces in anadequately smooth way to match the whole data record. Wehave evaluated the proposed method on two synthetic recordsand a field record. Because the parametrization of the seismicevents is physically meaningful, it also enables a simple form ofbandwidth extension of the observed shot record to unobservedlow and high frequencies. We tested this procedure on the sameshot records. Bandwidth extension is in principle helpful to ini-tialize full-waveform inversion with frequency sweeps and en-hanced its resolution.

INTRODUCTION

In this paper, we address the problem of decomposing a seismicrecord into elementary, or atomic, components corresponding toindividual wave arrivals. Letting t stand for time and x represent thereceiver location, we seek to decompose a shot profile d into a smallnumber r of atomic events vj as

dðt; xÞ ≃Xr

j¼1

vjðt; xÞ: (1)

Each vj should consist of a single wavefront — narrow yet bandlimited in t, but coherent across different x — corresponding to anevent of direct arrival, reflection, refraction, or corner diffraction.In the simplest convolutional model, we would write vjðt; xÞ ¼

ajðxÞwðt − τjðxÞÞ for some wavelet w, amplitude ajðxÞ, and timeshift τjðxÞ. In the Fourier domain, this model would read

vjðω; xÞ ¼ wðωÞajðxÞeiωτjðxÞ: (2)

This model fails to capture frequency-dependent dispersion andattenuation effects, phase rotations, inaccurate knowledge of thesource wavelet, and other distortion effects resulting from near res-onances. To restore the flexibility to encode such effects withoutexplicitly modeling them, we consider instead throughout this paperthe following expression:

vjðω; xÞ ¼ wðωÞajðω; xÞeibjðω;xÞ; (3)

where the amplitudes aj and the phases bj are smooth in x and ω,and bj deviates little from an affine (linear + constant) function ofω. If the guess wðωÞ for the wavelet is not entirely inaccurate, theamplitude ajðω; xÞ is meant to compensate for it.Finding physically meaningful, smooth functions aj and bj to fit

a model such as equations 1 and 3 is a hard optimization problem.Its nonconvexity is severe: It can be seen as a remnant, or cartoon, ofthe difficulty of full-waveform inversion from high-frequency data.We are unaware that an authoritative solution to either problem hasbeen proposed in the geophysical imaging community.

Manuscript received by the Editor 8 February 2015; revised manuscript received 2 June 2015; published online 1 September 2015.1Massachusetts Institute of Technology, Cambridge, Massachusetts, USA. E-mail: yunyueli@mit.edu; demanet@gmail.com.© 2015 Society of Exploration Geophysicists. All rights reserved.

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Many methods have been proposed to pick individual seismicevents, such as autoregression (AR) filters (Leonard and Kennett,1999) (close in spirit to the matrix pencil method [Hua and Sarkar,1990]), crosscorrelations (Cansi, 1995), wavelets (Zhang et al.,2003), neural networks (Gentili and Michelini, 2006), etc. Thesepapers mostly address the problem of picking isolated arrivals time,not parameterizing interfering events across traces. Some dataprocessing methods operate by finding local slope events, suchas plane-wave annihilation (Fomel, 2002). This idea have been usedto construct prediction filters for localized wavelet-like expansionmethods (Fomel and Liu, 2010), which in turn allow us to solveproblems such as trace interpolation in a convincing manner. Itis plausible that concentration or clustering in an appropriate wave-letlike domain could be the basis for an algorithm of event separa-tion. Separation of variables in moveout-corrected coordinates hasalso been proposed to identify dipping events, such as in Raoult(1983) and Blias (2007). Smoothness criteria along reflection eventshave been proposed for separating them from diffraction events,such as in Fomel et al. (2007). These traditional methods fail forthe event decomposition problem as previously stated:

• Because of cycle-skipping, the gradient descent quickly con-verges to uninformative local minimums.

• Because data sets do not often have useful low frequencies,multiscale sweeps cannot be seeded to guide gradient descentiterations toward the global minimum.

• Because the events are intertwined by possibly destructive in-terference, simple counterexamples show that greedy “eventremoval” methods such as matching pursuit cannot be ex-pected to succeed in general.

• Because wavefront shapes are not known in advance, lineartransforms such as the slant stack (Radon), velocity scan,wavelets/curvelets, or any other kind of nonadaptive filters,do not suffice by themselves. The problem is intrinsically non-linear.

The contribution of this paper is the observation that tracking in xand ω, in the form of careful growth of a trust region, can satisfac-torily mitigate the nonconvexity of a simple nonlinear least-squarescost function, yielding favorable decomposition results on some syn-thetic shot profiles and some field data. We have not been able to dealwith the nonconvexity of this cost function in any other way than bytracking.Seismic records from field experiments contain many types of

events resulting from the interaction between the source and thecomplex subsurface. Separating these events has been a long-stand-ing challenge in seismic data processing. The successful resolutionof this question would have many implications:

• It would improve our ability to discriminate events based onthe different physical processes that generated them: P-Swave separation, primary-multiple separation, reflection-dif-fraction separation, simultaneous recording separation, etc.

• It automates the “traveltime picking” operation; hence, thismay help prepare a data set for traveltime tomography.

• A description of the data set in terms of phases and ampli-tudes is adequate for interpolation of missing samples. It isalso the proper domain in which model reduction should beperformed in the high-frequency regime.

• Perhaps most importantly, it enables a not entirely inaccurateextrapolation to high and low frequencies not present in the

data set, with a possible application to seeding frequencysweeps in full-waveform inversion.

This article is organized as follows: We first define the objectivefunction and derive its gradient. We point out that the objective func-tion is severely nonconvex, and we explain an explicit initializationscheme with multiple signal classification (MUSIC) and the expan-sion and refinement scheme for phase and amplitude tracking. Finally,we demonstrate our separation algorithm on two synthetic records anda field record. We illustrate the potential of event separation forextrapolation to unobserved frequencies in the last two examples.

METHOD

Cost function and its gradient

We consider the nonlinear least-squares optimization formulationwith a Tikhonov-like cost function:

Jðfaj; bjgÞ ¼1

2kuðω; xÞ − dðω; xÞk22

þ λXj

k∇2ωbjðω; xÞk22 þ μ

Xj

k∇xbjðω; xÞk22

þ γXj

k∇ω;xajðω; xÞk22; (4)

where d is the measured data in the frequency domain, ∇k and ∇2k,

with k ¼ ω, x, respectively, denote first-order and second-order par-tial derivatives, and∇ω;x denotes the full gradient. The prediction u is

uðω; xÞ ¼Xr

j¼1

wðωÞajðω; xÞeibjðω;xÞ; (5)

with the wavelet wðωÞ assumed known to a certain level of accuracy.(Mild phase and amplitude inaccuracies in wðωÞ are, respectively,compensated by bjðω; xÞ and ajðω; xÞ.) The constants λ, μ, and γare chosen empirically.It is important to regularize with ∇2

ωbjðω; xÞ rather than∇ωbjðω; xÞ, so as to penalize departure from dispersion-free linearphases rather than to penalize large traveltimes.The cost function is minimized using a gradient descent method

within a growing trust region. The gradients of equation 4 with re-spect to aj and bj are computed as follows:

∂J∂aj

¼ 1

2

�∂u∂aj

δu� þ δu∂u�

∂aj

�þ 2γ∇2

ω;xaj;

∂J∂bj

¼ 1

2

�∂u∂bj

δu� þ δu∂u�

∂bj

�þ 2λ∇2

ω · ∇2ωbj þ 2μ∇2

xbj; (6)

where

∂u∂aj

¼ we−ibj ;∂u�

∂aj¼ w�eibj ;

∂u∂bj

¼ −iwaje−ibj ;∂u�

∂bj¼ iw�ajeibj ;

δu ¼ u − d: (7)

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The notation ∇2ω;x refers to the Laplacian in ðω; xÞ and * denotes

complex conjugation. All the derivatives in the regularization termsare discretized by centered second-order accurate finite differences.The inverse problem defined by the objective function in equa-

tion 4 is clearly nonconvex, in large part due to the oscillatory natureof seismic data. Matching oscillations pointwise generically runsinto the cycle-skipping problem. Lack of convexity is also foundin phase-retrieval problems (Demanet and Hand, 2014; Gholami,2014), although in a less severe form.

Initialization

We initialize the iterations by making use of an explicit solutionof the minimization problem in a very confined setting where

• we pick a single seed trace x in which the events of interestsare well-separated;

• we pick a subset of seed frequencies ω around the dominantfrequency of the source wavelet;

• after deconvolving the source wavelet, we assume a simpli-fied model in which the amplitudes are constant, and thephases are linear in ω. As a result, we momentarily returnto the convolutional model,

dðωÞ ≃Xr

j¼1

ajeiωτj ; (8)

of equation 2 to locally approximate equation 3.

In this situation, the problem reduces to a classical signal processingquestion of identification of sinusoids, i.e., identification of the trav-eltimes τj and amplitudes aj. There exist at least two high-qualitymethods for this task: the matrix pencil method of Hua and Sarkar(1990) and the MUSIC algorithm (Schmidt, 1986; Biondi and Kos-tov, 1989; Kirlin and Done, 1999). We choose the latter for its sim-plicity and robustness. (It would be a mistake to use either l1

minimization, or iterative removal pursuit algorithms, for this sinus-oid identification subproblem.)Assume for the moment that the number r of events is known,

although we address its determination later in the paper. The MUSICalgorithm only needs data fdðωiÞg at m ¼ 2rþ 1 frequencies to de-termine the arrival times and amplitudes for r different events. In prac-tice, the number m of frequencies may be taken to be larger than2rþ 1 if robustness to noise is a more important concern than thelack of linearity of the phase in ω. In either case, we sample the datadðωiÞ on a grid of spacing Δω around the dominant frequency ωf ofthe source wavelet, where the signal-to-noise ratio (S/N) is rela-tively high.The variant of the MUSIC algorithm that we use in this article

requires building a Toeplitz matrix, whose columns are constructedby translations of the data samples as

T ¼

266664

dðωfÞ dðωf−1Þ · · · dðωf−kþ1Þdðωfþ1Þ dðωfÞ · · · dðωf−kþ2Þ

..

. ... . .

. ...

dðωfþk−1Þ dðωfþk−2Þ · · · dðωfÞ

377775; (9)

where k ¼ ðm − 1Þ∕2. After a singular value decomposition (SVD)of T, we separate the components relative to the r largest singularvalues, from the others, to get

T ¼ UsΣsVTs þ UnΣnVT

n : (10)

We interpret the range space of Us as the signal space and the rangespace of Un as the noise space, hence the choice of indices. Theorthogonal projector onto the noise subspace can be constructedfrom Un as

Pn ¼ UnUTn : (11)

We then consider a quantitative measure of the importance of anygiven arrival time t ∈ ½0; 2π

Δω�, via the estimator function

αðtÞ ¼ 1

kPneiωtk: (12)

In the exponent, ω is a vector with k consecutive frequencies on agrid of spacing Δω.In practice, the number of individual events is a priori unknown.

The determination of this number is typically linked to the extra-neous knowledge of the noise level, by putting to zero the eigen-values of T below some threshold ϵ, chosen so that the resultingerror vector has a magnitude that matches the predetermined level.In the noiseless case, the estimator function in equation 12 has r

sharp peaks that indicate the r arrival times τj for the r events. In thenoisy case, or in the case when the phases bj are nonlinear in ω, thenumber of identifiable peaks is a reasonable estimator for r, and thelocations of those peaks are reasonable estimators of τj, such thatthe signal contains r phases locally of the form ωτjþ a constant. Infield applications, this procedure should be applied in a trust regionΩ, which contains a handful of nearby traces. The consistent arrivaltimes τjðxÞ across traces provide a more robust estimation for thecoherent events. Once the traveltimes τj are found, the amplitudesaj follow from solving the small, overdetermined system in equa-tion 8. Each complex amplitude is further factored into a positiveamplitude and a phase rotation factor, and the latter is absorbed intothe phase function. We summarize the initialization procedure inAlgorithm 1.

Algorithm 1. Initialization with MUSIC.

Select a trust region Ω and a threshold level ϵ

for x ∈ Ω do

Build the Toeplitz matrix T and SVD

Build the projection matrix Pn with threshold ϵ

Compute the estimator αðtÞPick the peaks of αðtÞ to determine rðxÞ arrival times τjðxÞSolve for the amplitude ajðxÞ

end for

Find the consistent τjðxÞ, ajðxÞ and determine r

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Tracking by expansion and refinement

The phases and amplitudes generated by the initialization givelocal seeds that need to be refined and expanded to the wholerecord:

• The refinement step is the minimization of equation 4 withthe data misfit restricted to the current trust regionΩ. In otherwords, we use the result of initialization in the form of equa-tion 2 as an initial guess and upgrade it to take the more ac-curate form (equation 3).

• The expansion step consists in growing the region Ω to in-clude the neighboring frequency samples and extending thesolution smoothly to the neighboring traces.

These steps are nested rather than alternated: The inner refinementloop is run until the value of J levels off, before the algorithm re-turns to the outer expansion loop. The expansion loop is itself splitinto an outer loop for (slow) expansion in x and an inner loop for(rapid) expansion in ω. The nested ordering of these steps is crucialfor convergence to a meaningful minimizer.A simple trick is used to speed up the minimization of J in the

complement of Ω, where only the regularization terms are active:ajðω; xÞ is extended constantly in x and ω, whereas bjðω; xÞ is ex-tended constantly in x and linearly in ω. These choices correspondto the exact minimizers of the Euclidean norms of the first and sec-ond derivatives of aj and bj, respectively, with zero-boundary con-ditions on the relevant derivatives at the endpoints. This trick maybe called preconditioning the regularization terms.

At the conclusion of this main algorithm, the method returns rphases bjðω; xÞ that are approximately linear in ω and approxi-mately constant in x; and r amplitudes ajðω; xÞ that are approxi-mately constant in x and ω such that

dðω; xÞ ≃Xr

j¼1

wðωÞajðω; xÞeibjðω;xÞ: (13)

We summarize the procedure of the refinement and expansion inAlgorithm 2.Further in the paper, we refer to the respective plots of bj and aj

as functions of ω and x as “phase spectra” and “amplitude spectra.”

Application: Frequency extrapolation

Frequency extrapolation, also known as bandwidth extension,is a tantalizing test of the quality of a representation such as equa-tion 8. A least-squares fit is first performed to find the best constantapproximations ajðω; xÞ ≃ αjðxÞ and the best affine approxima-tions bjðω; xÞ ≃ ωβjðxÞ þ ϕjðxÞ from values of ω within a usefulfrequency band. These phase and amplitude approximations can beevaluated at values of ω outside this band, to yield synthetic flat-spectrum atomic events of the following form:

αjðxÞeiðωβjðxÞþϕjðxÞÞ. (14)

These synthetic events can be further multiplied by a high- or low-pass wavelet, and summed up, to create a synthetic data set. Thisoperation is the seismic equivalent of changing the pitch of a speechsignal without speeding it up or slowing it down.Extrapolation to high frequencies should benefit high-resolution

imaging, whereas extrapolation to low frequencies should helpavoid the cycle-skipping problem that full-waveform inversion en-counters when the low frequencies are missing from the data.Notice that extrapolation to zero frequency is almost never accu-

rate using such a simple procedure: More physical information isrequired to accurately predict zero-frequency wave propagation.The exception is free-space wave propagation in 3D, where theGreen’s function has a linear phase and constant amplitude in ω,hence completely predictable from two samples in ω.

NUMERICAL EXAMPLES

In this section, we demonstrate the tracking method on two syn-thetic data records and one field data record. The first synthetic ex-ample illustrates the workflow behind the iterations. In the secondsynthetic example, we test our algorithm on a noisy seismic recordobtained by finite-difference modeling, with application to fre-quency extrapolation.

Synthetic example: Three-event separation

In this example, we create an artificial three-event seismic recordby convolving a Ricker wavelet along the traveltime curves of onedirect arrival event, one water-bottom reflection event, and one re-flection event from a deeper layer. The dominant frequency of theRicker wavelet is 20 Hz. The amplitude of the Ricker wavelet de-creases with increasing offset. After generating the data in the timedomain, we bring the data to the frequency domain and use the bandbetween 5 and 35 Hz for inversion.

Algorithm 2. Phase and amplitude tracking algorithm.

Input: observed data ~dðω; xÞ, estimated wavelet wðωÞ, trust regionΩ ¼ ½ xbeg xend �MUSIC: run Algorithm 1 to obtain r, ajðxÞ, and τjðxÞ

initialize a0jðω; xÞ←ℜðajðxÞÞ, andb0jðω; xÞ←ωτjðxÞ þℑðajðxÞÞ

for i ¼ 0; 1; : : : N − 1 do

Forward modeling: obtain ~uðω; xÞObjective function evaluation: compute J within the range of Ωif J < ϵ then

if Ω covers the whole record

Converge: Output ajðω; xÞ and bjðω; xÞelse

Expand: Ω←½xbeg − Δxxend þ Δx�ajðω;xbeg−ΔxÞ←ajðω;xbegÞ, bjðω;xbeg−ΔxÞ←bjðω;xbegÞajðω;xendþΔxÞ←ajðω;xendÞ, bjðω;xendþΔxÞ←bjðω;xendÞ

end if

else

Compute gradient:∂J∂aj

and∂J∂bj

Update model: aiþ1j ←aij − β

∂J∂aj

, biþ1j ←bij − β

∂J∂bjend if

end for

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The reference seismic profile is shown in Figure 1a. The eventsare well-separated when the offset is small. As the offset grows, thethree events cross each other. Moreover, the water-bottom reflectionevent approaches the water-column direct arrival as its asymptote.Therefore, both events become almost perfectlyaligned at large offsets over 2.5 km. Separatingthese crossing and overlapping events is a some-what challenging task to any automatic eventtracker.To initialize the method, we choose three seed

traces where the events are well-separated. Wetrain the MUSIC algorithm with only sevenfrequencies around the dominant frequency ofthe Ricker wavelet for each trace. Figure 2a showsone seismic trace at x ¼ 200 m. The MUSICestimator with many trial arrival times is shownin Figure 2b. The peaks of the MUSIC estimatoraccurately pinpoint the reference arrival times ofthe events.With these three seed traces, we run the expan-

sion-refinement tracking algorithm explainedearlier. The inverted seismic profile is shown inFigure 1b. The kinematics (phases) and the am-plitudes are almost perfectly resolved.Figure 3 shows the comparison between the

inverted atomic events in Figure 3a–3c with thereference atomic events in Figure 3d–3f. Theatomic events were separated with high accuracy.Figure 4 shows the phase spectra (Figure 4a–4c)and the amplitude spectra (Figure 4d–4f) for thereference atomic events. In comparison, we showthe phase spectra (Figure 5a–5c) and the ampli-tude spectra (Figure 5d–5f) for the inverted atomicevents in Figure 5. The phase spectra for all theevents are accurately recovered. However, the am-plitude spectra for the first two events are not per-fectly recovered due to the ambiguities at largeoffsets.

Synthetic example: Shallow Marmousimodel

In this example, we test our algorithm ona more realistic shot gather. The shot gather isgenerated from a shallow part of the Marmousimodel using finite-difference modeling. The fi-nite-difference scheme is second-order accuratein time and fourth-order accurate in space. Weuse a 20-Hz Ricker wavelet as the source wave-let. The receiver spacing is 40 m.As a preprocessing stop, we remove the direct

arrival from the shot record because it has thestrongest amplitude that would overwhelm therecord. We apply an automatic gain control tothe remaining events so that the amplitudes onthe record is more balanced. We mute the laterarrivals in the data so that we resolve a limitednumber of events at a time. Finally, we add30% independent and identically distributed

Gaussian noise to the data within the bandwidth between 7

and 40 Hz.Figure 6a shows the shot record with six clear events after pre-

processing. For inversion of the atomic events, we consider the data

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Figure 1. Comparison of the reference seismic profile is shown in panel (a) and theinverted seismic profile is shown in panel (b). Figures are clipped to the same value.The inversion perfectly reconstructs the seismic data.

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Figure 2. (a) Comparison of the seismic trace at x ¼ 200 m and (b) its MUSIC esti-mator (blue curve) and the picked traveltimes (red diamonds). The MUSIC estimatoraccurately identifies the arrival time of each event.

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only between 7 and 40 Hz. We choose three seed traces atapproximately x ¼ 1.5 km where the events are well-separated.We apply the MUSIC algorithm using 13 frequencies around thedominant frequency. Figure 7a shows one example of the seed traceat x ¼ 1.5 km. The MUSIC estimator accurately determines thearrival times of the six different events, as shown in Figure 7b.Figure 6b shows the inverted shot profile using the proposed

tracking algorithm. The inversion has clearly separated the six strongreflection events from the record, removed the severe random noise,and improved the coherence of the amplitude along the events. Thisdemonstrates the robustness of the method. Figure 8 shows the cor-responding well-separated atomic reflection events.Figures 9 and 10 show the phase and amplitude spectra for each

atomic event. As expected, the phase and amplitude spectra are

smooth in ω and x. We allow the amplitudes of an event to dropto zero when the average estimated energy of this event at thattrace is less than 1% of that at the seed traces. Hence, we allowthe events to be terminated in the middle of the section. At eachtrace x, we estimate the parameters αjðxÞ; βjðxÞ, and ϕjðxÞ and de-termine the approximated phase spectrum for each event. With thisparametrization, we now extrapolate the data to frequencies within[1, 90] Hz.The atomic events after frequency extrapolation are shown in

Figure 11. Compared with the atomic events in Figure 8, the wave-form is much more compact thanks to the increased bandwidth,whereas the traveltime information is kept intact. Figure 7 comparesthe seed trace after frequency extrapolation (Figure 7c) with theoriginal seed trace (Figure 7a). The waveforms are more compact.

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Figure 4. (a-c) Phase spectra and (d-f) amplitudespectra of the true atomic events.

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Figure 3. Comparison of (a-c) the inverted atomicevents with (d-f) the true atomic events. Figuresare clipped to the same value. The inversion accu-rately reconstructs the atomic events.

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The high-resolution data could be used for broadband high-resolu-tion seismic imaging (Fu et al., 2014).To evaluate the accuracy of low-frequency extrapolation, we

model the seismic record using a broadband source wavelet, whoseamplitude spectrum is mostly flat between 1 and 7 Hz. We thencompare the modeled data with the data obtained by frequencyextrapolation within the same bandwidth in Figure 12. Althoughthe amplitudes differ between the two records, the phase functionfrom the frequency extrapolation represents a reasonable estimationof the modeled phase function.There are a few reasons to explain the mismatch between the

extrapolated low-frequency data and the modeled low-frequencydata. First, the unmodeled scattering events contribute to the low

frequencies and overlap with the low-frequency signal from themodeled reflection events. Second, there are numerical dispersioneffects on the modeled low-frequency data, whereas the extrapolatedrecord is constructed using a nondispersive assumption. Finally, thenoise in the original record may introduce extra amplitude and phaseperturbations, which do not appear on the noiseless modeled low-fre-quency record. All of these effects will manifest themselves in thefield applications.

Field data example

In this example, we test our algorithm on a shallow field recordacquired onshore in China (Wang et al., 2013). Figure 13a shows

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Figure 6. (a) Comparison of the noisy shot recordand (b) the inverted shot record. The inverted datahave clearly reconstructed the strong reflectionevents and removed the severe random noise inthe data. Amplitude and phase are more coherentalong each reflection event.

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Figure 5. (a-c) Phase spectra and (d-f) amplitudespectra of the inverted atomic events. Plots in thisfigure are clipped at the same value as the plots inFigure 4, respectively. Notice the amplitudes arewell-reconstructed before the three events cross(offset less than 2 km). After the crossing point,the direct arrival and the water-bottom reflectionare perfectly overlapped, causing the amplitudespectra (panels d and e) to be not perfectly resolveddue to the ambiguity. Nonetheless, the correspond-ing phase spectra (panels a and b) are well-recon-structed.

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early events of the record. Strong receiver statics can be observedfrom the shifting phases across nearby traces. Strong amplitudevariations are also observed from this record. Figure 13b showsthe average power spectrum of the record. The data have a largeS/N between [16, 85] Hz. However, the low-frequency components(less than 10 Hz) are completely missing from the data.For the inversion of event separation, we further applied an inner

mute to the data and band-pass the data between [16, 70] Hz. Fig-ure 14a shows the resulting shot record and we only use this band-

limited record to invert for the atomic events. We choose three seedtraces at x ¼ −0.01; 0; 0.01, and we identify seven events from thecoherent phase picks after applying MUSIC on each trace. Clearly,the algorithm does not identify all the events in the record (aspointed out by the arrows in Figure 15a). Instead, only the strongcoherent events have been selected.The phase and amplitude tracking algorithm resolves seven

atomic events (Figure 16), and the reconstructed shot record isshown in Figure 14b. The inverted record has been cleaned up

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Figure 8. The six well-separated atomic events.Severe noise has been removed from each event.The amplitude and phase are more coherent foreach event.

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Figure 7. (a) Comparison of a narrow-band ([7,40] Hz) seismic trace at x ¼ 1.5 km, (b) its MUSICestimator (blue curve) and the picked traveltimes(red diamonds), and (c) the trace after frequencyextrapolation to [1, 90] Hz.

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significantly with fewer weak events and overlapping events. Withthe smoothness constraints, the amplitude and phase of each eventacross traces are more coherent.Figure 17 compares the shot record in its full bandwidth with the

inverted shot record after frequency extrapolation to [0.5, 120] Hz.Compared with the field record, the arrival times of the resolvedevents are preserved in the extrapolated record. However, the re-corded data appear to contain more events than the extrapolated rec-ord. We can explain this observation from two aspects. First, ouralgorithm indeed does not model all events in the recorded data.Moreover, due to the missing low frequencies, each event has side-lobes in the recorded data, whereas in the extrapolated data, eachselected event has a more compact support in time.

Figure 18a shows the shot record after a low-pass filter at 16 Hz.To make a fair comparison, we band-pass the extrapolated shot rec-ord (Figure 17b) between 10 and 16 Hz and plot the resulting recordin Figure 18b. The field record appears very noisy due to the low S/N,the receiver statics, and the strong amplitude variations acrosstraces. In contrast, the inverted record after extrapolation reservesthe phase information with improved amplitude coherence acrosstraces. Figure 15b compares one extrapolated trace with the corre-sponding recorded trace within the low-frequency bandwidth. Afteramplitude normalization, the amplitude and phase are matchedquite well when most events are successfully identified in the laterarrival time. However, the box highlights the discrepancy betweenthe extrapolated and the recorded trace in earlier time due to the

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Figure 9. Phase spectrum of each atomic event issmooth along frequency and space axes, as re-quired by the optimization.

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Figure 10. Amplitude spectrum of each atomicevent is smooth along frequency and space axes, asrequired by the optimization.

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unmodeled events. The extrapolated data can be further improvedwith a second pass of phase-and-amplitude tracking on the dataresidual to pick up the remaining coherent events. Figure 18c showsthe inverted shot record after extrapolation to [0.5, 10] Hz. The re-corded data within this bandwidth have little meaningful signal,whereas the extrapolated record may provide reliable phase infor-mation at these low frequencies.

ASSUMPTIONS AND LIMITATIONS

The fundamental assumption for our tracking algorithm to defineseismic events is that the acoustic or elastic wave equations are

mostly dispersion free, i.e., that they give rise to solutions withmostly linear phases in the frequency domain. Therefore, in thecases of surface waves and body waves in a severely dispersivemedium, our algorithm will have difficulty identifying and trackingthe events.Obviously, frequency extrapolation to the unobserved bandwidth

is purely based on the dispersion-free assumption. We have ob-served phase errors at the extrapolated frequencies due to numericaldispersion in the synthetic test, which naturally indicates potentialphase errors due to real dispersion effects in the earth. However, aslong as the dispersion is mild, the extrapolated low-frequency datacould be a close approximation to the true low-frequency response.

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Figure 11. The same six atomic events as in Fig-ure 8 after frequency extrapolation. The waveformis much more compact thanks to the increasedbandwidth, whereas the traveltime information iskept intact.

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Figure 12. (a) Comparison of the shot record mod-eled with a low-frequency broadband ([1, 7] Hz)wavelet and (b) the shot record after frequencyextrapolation to [1, 7] Hz. Although the amplitudesdiffer between the two records, the phase functionfrom the frequency extrapolation represents a rea-sonable estimation of the modeled phase function.

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Hence, fitting the extrapolated low-frequency waveforms couldbring full-waveform inversion closer to the global minimum.Extrapolation to low frequencies is also limited by the inability of

the phase-amplitude model to capture the physics of zero-frequency

radiation due to the generic presence of a scattering pole in Green’sfunction at ω ¼ 0.The success of the proposed iterative method for the nonconvex

optimization formulation hinges on a delicate interplay between therefinement of phases and amplitudes within a trust region, versusslowly growing this trust region. If the region is grown too fast, theiterations will converge to an undesirable local minimum.The refinement and expansion depend nonlinearly on the regu-

larization parameters, the determination of which is empirical andcan be tedious. We replace the slow converging regularization term,λP

jk∇2ωbjðω; xÞk22, by implicitly solving the regularization prob-

lem. The rest of the regularization terms are solved explicitly. We letthe partial gradient of bj at the kth iteration be

δbkj ¼1

2

�∂u∂bj

δu� þ δu∂u�

∂bj

�þ 2μ∇2

xbj. (15)

We solve the following implicit system to obtain the phase functionsfor the ðkþ 1Þth iteration:

bkþ1j ¼ bkj −

λ

β∇T∇bkþ1

j − βδbkj : (16)

Hence, the updated phase functions are

bkþ1j ¼

�1þ λ

β∇T∇

�−1ðbkj − βδbkjÞ: (17)

This trick allows us to use large λ values during the inversion.Numerical tests suggest that the inversion results remain stable withrespect to the regularization parameters when/where the S/N is high.In generally noisy cases, the solutions can be significantly differentwith different regularization parameters. The values for each param-eter in different numerical tests are shown in Table 1.

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Figure 13. A field data record is shown in panel (a), and its averagepower spectrum is shown in panel (b). The data show strong receiverstatics (shifty phases across traces) and amplitude variations. The datahave a large S/N between [16, 85] Hz. The low-frequency compo-nents (less than 10 Hz) are missing from the data.

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Figure 14. (a) Comparison of the band-limited([16, 70] Hz) shot record and (b) the inverted shotrecord. The inverted data have clearly recon-structed the strong reflection events.

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Figure 15. High-frequency ([16, 70] Hz) band-limited seismic trace in (a) and low-frequency (10, 16 Hz) band-limited seismic trace in (b). Inthe top plot, the inverted trace (red) matches the recorded trace (blue) quite well for the selected events. The arrows point out two events thathave not been selected by the inversion at the early arrival times. In the bottom plot, we compare the extrapolated trace (red) with the recordedtrace (blue). The (normalized) amplitude and phase are matched quite well when most events are successfully identified in the later arrival time.However, the box highlights the discrepancy between the extrapolated and the recorded traces in earlier time due to the unmodeled events.

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Figure 16. Seven well-separated atomic events.

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The most important limitation of the tracking method is thedeterioration of the accuracy in phase (hence the traveltime) estima-tion in the presence of unmodeled or splitting events. Our solutionfor the determination of the number of events further depends on theavailability of a few seed traces where the MUSIC estimator cor-rectly categorizes them. We could potentially analyze the variabilityof the solutions with different regularization parameters to deter-mine the presence of unmodeled or splitting events, which requiresfurther investigation.

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Figure 17. Comparison of the full bandwidth shotrecord in panel (a) and the inverted shot record afterfrequency extrapolation to [0.5, 120] Hz in panel(b). Kinematic information of the selected eventsis well-preserved by extrapolation. The recordeddata appear to contain more events than the extrapo-lated data. The reason can be twofold: First, ourtracking algorithm indeed does not model all theevents in the recorded data. Moreover, due to themissing low frequencies, each event has sidelobesin the recorded data, whereas in the extrapolateddata, each selected event has a more compact sup-port.

Table 1. Regularization parameters used in each test.

Case λ μ γ

Three-event separation 1.0 1.0 2e − 3

Shallow Marmousi 1.0 1eþ 1 2e − 3

Field data 1.0 4eþ 2 3e − 3

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Figure 18. (a) Comparison of the shot record after a low-pass filter at 16 Hz, (b) the inverted record after frequency extrapolation to [10,16] Hz, and (c) the inverted record after frequency extrapolation to [0.5, 10] Hz. The field record appears to be very noisy due to low S/N, thereceiver statics, and the strong amplitude variations across traces. In contrast, the extrapolated record within the same bandwidth reserves thephase information with improved amplitude coherence across traces. The recorded data less than 10 Hz have little meaningful signal, whereasthe extrapolated record may provide reliable phase information at these low frequencies.

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CONCLUSIONS

We propose a data-driven method for decomposing seismic re-cords into individual, atomic events that correspond to isolated arriv-als. The only piece of physical information needed to define theseevents is the fact that the acoustic or elastic wave equations are mostlydispersion free. The explicit handle on each atomic event enables usto approximate the high- and low-frequency seismic response beyondthe observed frequency bandwidth. The numerical examples demon-strate the robustness of the method and the potential of using theextrapolated low-frequency data for the initialization of full-wave-form inversion.

ACKNOWLEDGMENTS

This project was funded by Total S.A. L Demanet is also gratefulto Air Force Office of Scientific Research, Office of Naval Re-search, and National Science Foundation for funding.

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