Frequency-dependent phase coherence for noise suppression ...diapiro.ictja.csic.es/gt/mschi/SCIENCE/2007-psfreq.pdf · lateral phase coherence as function of distance, time, frequency,

Frequency-dependent phase coherence for noise suppression in seismic

array data

M. Schimmel1 and J. Gallart1

Received 7 August 2006; revised 28 November 2006; accepted 27 December 2006; published 10 April 2007.

[1] We introduce a coherence measure for seismic signal enhancement throughincoherent noise attenuation. Our processing tool is designed for densely spaced arraysand identifies signals by their coherent appearance. The approach is based on thedetermination of the lateral phase coherence as function of distance, time, frequency, andslowness. The coherence is derived from the local phases of neighboring stations whichwe obtain from analytic signals through the S-transform. The coherence is used toattenuate incoherent components in the time-frequency representations of the seismicrecords. No waveforms are averaged in our approach to maintain local amplitudeinformation. This way we construct a data-adaptive filter which enhances coherentsignals using the frequency-dependent and slowness-dependent phase coherence. Weexplain the method and show its abilities and limitations with theoretical test data.Furthermore, we have selected an ocean bottom seismometer (OBS) record section fromNW-Spain and a teleseismic event registered at Spanish broadband stations to showthe filter performance on real array data. Incoherent noise has been attenuated in all casesto enable a less ambiguous signal detection. In our last example, the filter alsoreveals weak conversions/reflections at the 410-km and 660-km discontinuities whichare hardly visible in the unfiltered input data.

Citation: Schimmel, M., and J. Gallart (2007), Frequency-dependent phase coherence for noise suppression in seismic array data,

J. Geophys. Res., 112, B04303, doi:10.1029/2006JB004680.

1. Introduction

[2] The signal variability, the presence of noise whichshares signal characteristics, the complexity of wave propa-gation in complex media, and the multitude of differentinterfering signals all inhibit the understanding and theroutine use of entire seismograms. Signal processingapproaches are therefore important tools to detect, enhance,or extract certain signals from the data which carry infor-mation we want to use in subsequent studies. The improve-ment of these tools is motivated by the steady improvementof data quality, increased station density that better samplesthe seismic wavefield, increasing computer power, and theneed to extract more information to constrain the finestructure of the Earth.[3] Many signal detection tools exist to separate signals

from noise based on different attributes (for example,polarization, coherence, instantaneous frequency) and ondata representations in different domains (for example, time,frequency, slowness, wavenumber, or their combinations).However, there is generally no clear separation betweensignal and noise, and often it is difficult to establishobjective criteria for noise suppression in any domain. Inall cases, the choice of the most effective approach depends

on the application as well as on the signal and noisecharacteristics themselves.[4] In this paper, the signal is defined as that which is

coherent on nearby traces over a range of frequencies andslownesses, while random noise is not. This is a widespreaddefinition which has led to many different approaches basedon different coherence measures under various approxima-tions. For instance, f-x (frequency-offset) and t-x (time-offset) prediction (and projection) filters use the propertythat coherent signals can be predicted from nearby traces[e.g., Hornbostel, 1991]. These techniques keep signalswhich can be predicted while suppressing those that cannot.[5] Other methods expand the two-dimensional (two-

dimensional in time and offset) data matrix using orthogonaltransforms such as the principal component, Karhunen–Loeve transform, and eigenimage approaches [e.g., Trickett,2003]. These methods are related in that they exploitthe property that coherent energy is mainly mapped ontothe first eigenvectors or eigenimages, corresponding to thelargest eigenvalues. Therefore a truncated (or weighted)eigenvector or eigenimage decomposition can recover mostof the coherent signal energy while the omitted part of theexpansion retains most of the noise. The prediction andeigenanalysis filters are mostly used with large data volumessuch as multifold data in seismic exploration. A signalsubspace method which finds more application in globalarray seismology is the multiple signal classification tech-nique (MUSIC) [e.g., see Bokelmann and Baisch, 1999;Almendros et al., 2001].

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112, B04303, doi:10.1029/2006JB004680, 2007ClickHere

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1Institute of Earth Sciences, CSIC, Barcelona, Spain.

Copyright 2007 by the American Geophysical Union.0148-0227/07/2006JB004680$09.00

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http://dx.doi.org/10.1029/2006JB004680

[6] Frequency-wavenumber ( f-k) and traveltime inter-cept-slowness (t-p) (Radon or slant stack) filters are possi-bly the most common filters to identify and separatecoherent waves. These filters take advantage of the factthat aligned features map in a distinguishable way in thecorresponding data representation following their apparentvelocity, frequency, or intercept time. The f-k representationcan be obtained by a two-dimensional Fourier transformwhich means that noise and signal are represented as planewaves traveling through the array. The t-p representation isachieved through a slant stack or Radon transform. Both f-kand t-p filters are often useful to suppress correlated noisesuch as ground roll or water column reverberations in largedata volumes. Time- and space-variant f-k [Duncan andBeresford, 1994] and t-p filters [e.g., van der Baan andPaul, 2000] are more sensitive to signal variations and canbe employed in a data-adaptive manner.[7] In array seismology, f-k analyses and beam forming

are important to detect and track signals by their slownessand back azimuth [e.g., Rost and Thomas, 2002; Koper etal., 2004; Kruger and Ohrnberger, 2005; Tanaka, 2005].Beam forming is a stacking approach as a function ofazimuth and slowness. The stacks are often additionallyweighted by coherence measures to lower the detectionthreshold by attenuating noise. The vector character of thewavefield can be exploited by three-component (3-C)stacking [e.g., Kennett, 2000] or through inclusion ofpolarization attributes.[8] Data-adaptive coherence filters can also be designed

using the signal localization capabilities of time-frequencyapproaches. Carrozzo et al. [2002] filters seismic wide-angle data by employing a wavelet decomposition intodifferent detail levels. Portions of the details of adjacenttraces are cross-correlated to obtain the filtered tracesthrough a weighted reconstruction. Pinnegar and Eaton[2003] employ the S-transform by Stockwell et al. [1996]and use the average of the amplitude S-spectra of adjacenttraces to weight the S-spectra of each trace.[9] Our filter belongs to the last filter class in that it takes

advantage of signal localization in time, frequency, andspace. We utilize the S-transform with the inverse transformby Schimmel and Gallart [2005]. The coherence weights aredetermined with instantaneous phase stacks [Schimmel andPaulssen, 1997] to weight signal components followingtheir spatial phase coherence in a physically allowableslowness range. In the following, the filter and its ingre-dients are explained, and its capability and limitations areillustrated with test data. The filter can be used for thedifferent types of data and arrays. We have selected anocean bottom seismometer (OBS) record section fromNW-Spain and a teleseismic event registered at Spanishbroadband stations to show the filter performance on realarray data with very different characteristics. The filter caneasily be adapted to other situations.

2. Methodology

[10] Our approach is based on the determination of thelateral phase coherence as function of distance, time,frequency, and slowness. The local phases are obtainedfrom local spectra achieved by time-frequency analysis.The coherence is quantified by the phase stack concept

[Schimmel and Paulssen, 1997] and serves to attenuateincoherent components in the local spectra. Finally, thefiltered traces are obtained after a back transform to thetime domain. This way, we construct a data-adaptivecoherence filter. Before delving into the filter constructionitself, we review the concept of each of its ingredients.

2.1. Time-Frequency Representation

2.1.1. The S-Transform[11] We use the S-transform by Stockwell et al. [1996] to

obtain for each time series a time-frequency representationwith absolute referenced phase which allows the compa-rison of phases. The S-transform is based on a slidingwindow Fourier analysis and is similar to the Gabor orshort time Fourier transform [Gabor, 1946]. But in contrastto the short time Fourier transform, the S-transform employsfrequency-dependent windows in analogy to the wavelettransform. This permits a better resolution of low frequencycomponents and enables a better time resolution of highfrequency signals.[12] For the following, we adopt the Fourier transform

U fð Þ ¼Z1

�1

u tð Þe�i 2p f tdt; ð1Þ

to determine the spectrum U( f ) of a time series u(t). Thecorresponding inverse transform is

u tð Þ ¼Z1

�1

U fð Þe i 2p f tdf : ð2Þ

[13] The S-transform of u(t) is, following Stockwell et al.[1996],

S t; fð Þ ¼Z1

�1

u tð Þw t � t; fð Þe�i 2p f t dt; f 6¼ 0; ð3Þ

S t; f ¼ 0ð Þ ¼ limT!1

1

T

ZT2

�T2

u tð Þ dt; ð4Þ

with a Gaussian function w(t � t, f ) centered at time t anda standard deviation proportional to j1/f j

w t � t; fð Þ ¼ j f jk

ffiffiffiffiffiffi2p

p e�f 2 t�tð Þ2

2k2 ; k > 0; ð5Þ

to obtain the time localized spectra. Here f stands forfrequency and t, t are time variables. t is the center time ofthe Gaussian window and k is a scaling factor whichcontrols the number of oscillations in each window. Thefactor k permits control of the time-frequency resolution.The Gaussian window can be replaced by other functions[McFadden et al., 1999; Pinnegar and Mansinha, 2003],but we prefer Gaussian windows since these functionspermit one to achieve the lower bound of the uncertainty

B04303 SCHIMMEL AND GALLART: FREQUENCY-DEPENDENT PHASE COHERENCE

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relationship for optimal signal localization in time andfrequency. The transform of the data into its time-frequencyrepresentation is illustrated in Figure 1. The figure showsthe moving-analysis window which is scaled by the periodand the signal localization in the time-frequency spectrum.In the following, we abbreviate time-frequency spectrum/representation to S-spectrum in analogy to S-transform.[14] From equation (3), it is visible that the phasor e�i2p f t

does not translate with the Gaussian window. The phasor isfixed to the time series, and the local phase spectra aretherefore absolutely referenced. Without this phasor theS-transform could be represented as a continuous wavelettransform [Stockwell et al., 1996].[15] Equation (3) can also be expressed [Stockwell et al.,

1996, 1999] as following the multiplication of spectra

S t; fð Þ ¼Z1

�1

U aþ fð Þe�2p2

k2 f 2a2

ei2pat da; f 6¼ 0; ð6Þ

which simplifies the implementation on computers. Its deri-vation [Stockwell, 1999] involves writing the S-transform asa convolution, transforming both functions into thefrequency domain, and annulling the Fourier transformby a back transform. In equation (6), U(a) is the Fouriertransform of u(t) (equation (1)) and is calculated only

once. U(a) is shifted by f before it is multiplied with thewindow function. Next, the calculation of S(t, f) is thenachieved applying the inverse Fourier transform from a tot. The Gaussian function in the integrand of equation (6)has a frequency-dependent bandwidth and acts on theshifted spectrum U(a + f ). This procedure is equivalent tobandpassing the phase-shifted input signal u(t) with aGaussian function centered at frequency f with standarddeviation s =

kjf j2p as can be seen when substituting b = a + f

to obtain

S t; fð Þ ¼Z1

�1

U bð Þe�2p2

k2 f 2b�fð Þ2

ei2p b�fð Þtdb: ð7Þ

2.1.2. The Inverse S-Transform[16] Stockwell et al. [1996] show that the S-spectrum can be

back transformed to the time domain. If the window satisfiesthe condition (which, defined as in equation (5), it does)

Z1

�1

w t � t; fð Þ dt ¼ 1; ð8Þ

then the time averaging of the S-spectrum S(t, f ) yields thespectrum U( f ) of the input signal u(t):

Z1

�1

S t; fð Þdt ¼ U fð Þ: ð9Þ

[17] This procedure permits to freely move between thetime, frequency, and time-frequency domains, which invitesto use weight functions F(t, f ) to construct data-adaptivefilters:

Ufilt1 fð Þ ¼Z1

�1

S t; fð ÞF t; fð Þdt: ð10Þ

[18] Here F(t, f ) 2 [0,1] is the time-frequency weightfunction which permits to attenuate undesired features inthe S-spectra, for example, for signal extraction and/ornoise attenuation. However, such manipulation can causespurious signals [Schimmel and Gallart, 2005]. The mainproblem is that due to the time integration in equation (9),the time localization imposed through F(t, f) is not directlytranslated to the time domain. Schimmel and Gallart [2005]point to this problem and propose an alternative inversetransform (equation (11)) for applications where timelocalization is important. Their approach avoids time aver-aging since it is based on the individual consideration ofthe local spectra as a function of time. Consequently, thesignal localization in time translates directly to the timeseries. Our inverse S-transform is expressed as

ufilt2 tð Þ ¼ kffiffiffiffiffiffi2p

p Z1

�1

S t; fð ÞF t; fð Þj f j ei2�ft df : ð11Þ

Figure 1. Illustration of the determination of the time-frequency representation through the S-transform whichforms part of our approach. The size of the moving-analysiswindow is scaled by the period to account for the intrinsichigher time resolution at shorter periods.


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[19] We show in the Appendix A that the input data areretrieved in case there is no manipulation of the S-spectrum,i.e., F(t, f) = 1. The inverse approaches are further analyzedby Simon et al., [2007].

2.2. Analytic Signal Approximation

2.2.1. Analytic Signal Theory[20] The complex or analytic signal theory [Gabor, 1946;

Taner et al., 1979] is an approach which uniquely asso-ciates a phase to a real signal. This phase information isvery important for the quantitative determination of phasesynchronization, coherence, time shifts, and instantaneousfrequencies, among others. The analytic signal is defined asthe sum of the real-time series u(t) and its Hilbert transformH[u(t)] as imaginary part and is expressed as

us tð Þ ¼ u tð Þ þ iH u tð Þ½ ¼ A tð ÞeiF tð Þ: ð12Þ

Here A(t) and F(t) are the local amplitude (envelope) andthe local or instantaneous phase of the signal. The Hilberttransform only affects the phase of the spectral componentsof the real signal u(t). This operation shifts the positive andnegative frequency components by �p/2 and p/2, respec-tively. The signal and its Hilbert transform are orthogonal,and the transformation to the complex domain is a linearand unique operation. The analytic signal us(t) has onlypositive frequency components in its spectrum. In practice,us(t) is often achieved by the inverse Fourier transform ofthe doubled signal spectrum U( f ) with suppressed negativefrequency components.[21] The Hilbert transform H[u(t)] can be expressed as

the convolution 1pt * u(t), where 1

pt is the function whichheavily weights the time series. The instantaneous phase isa very valuable attribute. It depends implicitly on theneighboring samples, i.e., on the local signal waveform,to permit instantaneous frequency analysis [Taner et al.,1979] and the amplitude-unbiased quantitative comparisonof signals [Schimmel and Paulssen, 1997; Schimmel, 1999].The signal comparison is based on the fact that coherentsignals should have identical phases. We want to apply thisconcept to the complex S-spectrum (equation (3)). Follow-ing Stockwell et al. [1996], the phase of the S-spectrum canbe understood as the instantaneous phase of the analyticsignal and is therefore suitable for our filter.2.2.2. When is the S-Spectrum Analytic?[22] The following consideration shows when the S-

spectrum can be considered to be analytic at a fixedfrequency f. In analogy to equation (6), we can write theS-spectrum of the analytic signal us(t) = u(t) + i H[u(t)]as

Ss t; fð Þ ¼Z1

�1

U aþ fð Þe�2p2

k2 f 2a2

1� ieip2sgn aþf½

� �ei2pat da;

f 6¼ 0 and k > 0:

ð13Þ

The factor eip2sgn½aþf comes from the complex part of the

analytic signal and shifts the phases as a function of the signof the frequencies to obtain the Hilbert transform. Now we

substitute b = a + f into equation (13) and use 1�ieip2sgn½b =

0, 8b < 0 and 1�ieip2sgn½b = 2, 8b > 0 to obtain

Ss t; fð Þ ¼ 2

Z1

0

U bð Þe�2p2

k2 f 2b�fð Þ2

ei2p b�fð Þtdb: ð14Þ

[23] A comparison with equation (7) shows that there is aclose relation between the S-transform S(t, f) of a real-timeseries and the S-transform Ss(t, f) of an analytic signal. Bothexpressions (equations (7) and (14)) can be understood asband pass filters of the time series u(t), where the Gaussianfunction is centered at frequency f with a standard deviations =

kjf j2p . Given that 99.7% of the area of the Gaussian

function is located within f � 3 s � b � f + 3s, theGaussian is approximately zero outside the 3s interval. Wecan determine the minimum frequency of the 3s interval tobmin = f � 3 s = f(1�3k

2p) and therefore obtain k � 2p3� 2.1

for bmin � 0. Since the Gaussian function is approximatelyzero outside the 3s interval, the integrand of equation (7) iszero for b < 0 and we can write for any positive frequency fand k � 2.1:

Ss t; fð Þ ’ 2S t; fð Þ: ð15Þ

[24] We see that if we assume that the Gaussian functionin the integrand of equation (7) is narrow to concentrateenergy well at the positive frequencies (b > 0), then the S-spectra of the real and analytic signal are in close agreementfor f � 0. The approximation improves for smaller k whichdepends on the choice of the s interval. For instance, the99% or 99.9% limits of the area of the Gaussian function aregiven by a 2.58 or 3.29 s interval which results into k � 2.7or k � 1.9, respectively.[25] Finally, we show under which conditions the S-

transform of an analytic signal is also analytic. Usingequation (3), the S-transform of the analytic signal us(t) isgiven by

Ss t; fð Þ ¼Z1

�1

us tð Þw t � t; fð Þe�i 2p f t dt: ð16Þ

[26] Multiplication of exp{i2pft} on both sides of theequation gives

Ss t; fð Þei 2p f t ¼Z1

�1

us tð Þw t � t; fð Þei 2p f t�tð Þ dt: ð17Þ

[27] The right-hand side of equation (17) is a convolu-tion of an analytic signal us(t) with another function. Theresult is also analytic, since the convolution reduces to amultiplication in the frequency domain, where Us( f < 0) = 0.That is, S(t, f ) ei2pft is analytic for small k values accord-ing to equation (15). Therefore equation (17) can be writtenas

S t; fð Þei 2p f t ¼ A t; fð ÞeiF t; fð Þ; ð18Þ


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where A(t, f ) and F(t, f ) are the envelope and theinstantaneous phase of an analytic signal at frequency fand for 0 < k � 2. We conclude that under above conditions,the S-transform S(t, f) for any real signal is approximatelyanalytic at a fixed frequency after the application of at-dependent phase shift.

2.3. Phase Coherence Measure

[28] We employ the phase stack concept by Schimmel andPaulssen [1997] to measure the coherence as a function offrequency and slowness. The phase stack cps(t) is defined asthe absolute value of the complex summation of theamplitude normalized analytic traces

cps tð Þ ¼1

N

XNj¼1

eiFj tð Þ

��n

; ð19Þ

where j and N are the trace index and the total number oftraces, respectively. cps(t) does not depend explicitly onamplitudes and is therefore an amplitude-unbiased measurewhich is robust to signal amplitude changes and/oroutliers. cps(t) takes advantage of the principles ofconstructive and destructive interferences due to thecomplex summation. The summation for coherent signalsis visualized by the summation of unit vectors which allpoint to the same direction in the complex plane. Owing tothe employed normalization by N, cps(t) ranges between 0and 1, with 1 being achieved for perfectly coherentsignals. The vector summation is destructive for incoherentsignals which results into a small value for cps(t). Thepower v is used to enhance the sharpness of the transitionbetween coherence and incoherence. v = 2 or 3 is generallya good choice. In the work of Schimmel and Paulssen[1997], cps(t) is used to weight linear slowness stacks bytheir coherence. Here individual traces rather than wave-form stacks will be weighted to retain site-dependent signalcharacteristics.[29] Frequency-slowness-dependent phase stacks can be

determined in analogy to equation (19) using the phasesFj(t, f ) of S-spectra (equation (18)). The local phase stacksare expressed as a function of time t, location vector ~rj of

station j, horizontal slowness~shor, and a window of 2N + 1traces for the local lateral-coherence determination

~cps t;~rj; f ;~shor� �

¼ 1

2N þ 1

XjþN

k¼j�N

eiFk t� ~rk�~rjð Þ�~shor ; fð ÞþiwDtjk

��n

:

ð20Þ

[30] The phase correction wDtjk is due to the phase shift(equation (18)) to make the S-transform analytic. We canconsider it as a phase shift due to the absolutely referencedphases of the S-spectra. The phases are corrected for therelative time shift Dtjk = (~rk �~rj)~shor. For linear arrays orrecord sections, (~rk �~rj)~shor reduces to Dxjk p with Dxjk =j~rk �~rjj being the interstation distance and p = j~shorj.[31] Here it is assumed that the seismic signals have a

plane wavefront with apparent velocity p�1 within thewindow defined by 2N + 1 traces. The summations areperformed along straight traveltime trajectories with gra-dient p. In practice, this is a good approximation for densearrays, small windows, and long wavelengths. Othertrajectories such as hyperbolas can be implemented.[32] We often favor the phasors from neighboring stations

using weights in the phase stack. This increases the impor-tance of nearby stations in relation to stations which arefurther away. The weights are given by the Gaussian

functiong(Dxjk, f ) = exp{� Dx2jk

2� fð Þ2} as function of interstation

distance Dxjk and frequency f. The standard deviation s( f )is an empirical linear function and permits to use weightswhich are small at large distances and high frequencies toaccount for the larger signal variability at the higherfrequencies. The determination of the weights is picturedin Figure 2.

2.4. Filter Construction

[33] The lateral phase-coherence filter is designed usingthe just outlined coherence measure ~cps (t, ~rj, f, p). Thismeasure provides for each seismic trace at each time andfrequency the lateral phase coherence as a function ofslowness. This information can be reduced to weights ofthe local time-frequency representations S(t, ~rj, f) because

Figure 2. Local phase stack components are weighted as a function of distance and frequency toaccount for wave propagation phenomena. (a) The triangles mark the receivers of a two-dimensionalstation array. Dxij is the distance between stations i and j. The circle with radius s(f ) marks the standarddeviation of the Gaussian weight function (b) at station i. (b) The Gauss curves are the distance andfrequency-dependent weight functions used to compute the local phase stacks for station i.


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we do not need the explicit slowness dependence of ~cps. Itis sufficient to know the maximum coherence values(equation (21)) as function of time, frequency, and distance

cps t;~rj; f� �

¼ max ~cps t;~rj; f ; p� �

p: ð21Þ

[34] These values are the data-adaptive weights in ourfilter. The procedure is summarized in Figure 3. With S(t,~rj, f )being the S-spectrum of the jth trace, the filtered S-spectrumbecomes

Sfilt t;~rj; f� �

¼ cps t;~rj; f� �

S t;~rj; f� �

: ð22Þ

[35] The filtered time series are obtained after the appli-cation of the inverse S-transform (equation (11)).[36] The main tunable filter parameters are 2N + 1 (window

width), v (power of local phase stack), k (resolution of theS-transform), and the range and number of physically allowableslowness values and frequencies. Other processing steps can beincorporated if data and/or application require further refine-ment. For instance, we keep the option to smooth the coherencecps(t,~rj, f ) by using amean ormedian filter similar to Schimmeland Gallart [2004]. These filters replace the center value of asmall volume which is centered on each element of cps by itsmean or median value. This procedure is applied before theweighting and can remove isolated coherent features or canprevent the attenuation of isolated incoherent components.Furthermore, we permit to control the width of the timewindow (DT in Figure 3) in the determination of theslowness-dependent ~cps (t,~rj, f, p) to account for slight signalmisalignments. These and other settings do not stronglyinfluence the filter output but can provide refined/modifieddata images.

3. Examples With Synthetic Data

[37] In what follows, we apply the method to syntheticdata to test the filter and to demonstrate its ability and

limitations to suppress incoherent noise. We show theadvantages of a frequency-dependent approach and pointto some important aspects of the theory.

3.1. Frequency-Dependent Noise Contamination

3.1.1. White Noise in Signal Frequency Band[38] The test data of the first example are shown in

Figure 4a. We use four coherent signals which havedominant frequencies ranging between 1 and 8 Hz. Thesignals are contaminated by random noise. Signals andnoise amplitude spectra are shown in Figure 4c. Here wecan see that the noise is white in the signal frequencyband. The record section has been filtered using thefollowing parameters: sliding data window N = 10 tracesand DT = 3 time samples at all frequencies and dis-tances, with 45 different slowness values between �0.03and 0.03 s/trace, power v = 4, and S-transform windowwidth k = 1 (standard deviation equals 1 period). Thelocal phase stacks have been computed at every secondtrace, second frequency, and second time sample. Theintermediate values are interpolated, and the filter outputis shown in Figure 4b. It can be seen from this figurethat the filter reduced the noise and increased the signal-to-noise ratio (S/N).[39] Figures 4d and 4e show the same test data, but with

larger noise contamination, and the corresponding filteroutput. The filter parameters have not been modified, andthe noise amplitude spectrum is shown in Figure 4c. Thenoise amplitudes are large and strongly influence the signalwaveforms. Nevertheless, the filter attenuates the noise andreveals the laterally coherent signals. The reconstructedwaveforms in Figure 4e are not as good as in Figure 4b.This is expected since signal and noise share the samefrequencies. In fact, a signal detection has only beenpossible since a mean coherence could be detected throughthe local phase stacks at different frequencies. Furtherincrease of the noise level would cause a larger impacton the filtered signal waveforms until the signal coherencehas been completely destroyed which inhibits the signaldetection.

Figure 3. Sketch outlines the determination of the local coherence at a selected distance, time, andfrequency. To the left, we see the sliding data window which is moved through the frequency-dependentenvelope-normalized complex traces. Slowness-dependent phase stacks are performed on the windoweddata to obtain the local coherence (phase stack amplitude) shown to the right. The maximum coherencevalue is ascribed to the window center and used as filter weight.


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3.1.2. Colored Noise[40] The coherent signals 1 to 4 shown in Figure 5a are

now contaminated with colored noise. The noise is largeand hides the signals as can be seen in Figure 5b. Thesetest data are filtered using the parameters from theprevious example. The filter output and the signal andnoise amplitude spectra are shown in Figures 5c and 5d,respectively. The filter attenuates most of the noise andreveals all signals in spite of the large amplitude noise.Also, in this example, the amplitude spectra in Figure 5dare crucial to understand the filter performance on the testdata. The noise and signal amplitude spectra overlap at thehigher frequencies. Note that the noise amplitude spec-trum (Figure 5d) has been multiplied by 0.25 for visualpurposes which may give the misleading impression oflittle spectral overlap. Nevertheless, the dominant fre-quency components of the signals are not strongly cor-rupted which explains why the signals are revealed fromthe large amplitude noise. We observe from the filteroutput that the reconstructed signals 1 and 2 are of lowerfrequency than the original signals. This is because thecoherence has been diminished or destroyed by the strongnoise contamination at about 5 to 8 Hz.[41] Note that no frequency-dependent noise attenuation

is possible with a time domain approach where theweights reduce to ~cps (t, ~rj), for example, by usingequation (19) in analogy to equation (21). At best, suchan approach can provide the noise contaminated signalwaveforms which would not at all resemble the inputsignals in this example.

3.1.3. Curved and Discontinuous Signal Trajectories[42] Signals appear often on curved and/or discontinuous

signal trajectories. The discontinuous signal trajectories areusually caused by lateral heterogeneities or data problems.This has been simulated in our data from Figure 6a. Signal 1contains gaps with widths of 6, 2, and 1 traces (marked byshort arrows) and ends abruptly at trace number 80. Signals2 and 3 have a curved signal trajectory, and signal 2 hasdiscontinuous amplitudes. The amplitudes of this signalhave been multiplied by 0.5 on every second trace untiltrace number 40 and on each trace for trace numbers 50 to60. The amplitudes of signal 3 change continuously; that is,they are modulated by a sine function. The traces have beencontaminated with random noise in the signal frequencyband. We filter these data employing a short Gaussianwindow (2 * s = 7 traces at all frequencies) with widthDT = 40 ms. The window is short to approximate thecurved signal trajectory. The filter output is shown inFigure 6b and well reflects all original signals.[43] It can be seen from Figure 6b that the gaps in signal 1

have not been filled with new signals; that is, the signals areseen as in the filter input. This is important and means thatno signals have been added where there are no signals. Thecoherence is a weight which ranges between 0 and 1 andcan therefore only attenuate amplitudes following the mea-sured signal coherence. The signals are less coherent nearthe gaps, and this can lead to decreased signal amplitudestoward the gaps. Local waveform averaging would haveaveraged the amplitudes, filled the small gaps, and blurredthe signals into traces beyond trace number 80.

Figure 4. (a) Synthetic record section contains four coherent signals which are labeled by numbers andcontaminated with random noise. (b) shows filter output for the data from (a). The incoherent noise isattenuated, and the signals are reconstructed. (c) This part shows the signal and noise amplitude spectra.The noise is white in the signal amplitude band. (d) Same as (a), but with large noise contamination.(e) shows the filter output for the data from (d). Noise has been attenuated, but the signal reconstruction isnot as good due to the increased corruption of signal components. Signals are still detected due to therandomness of noise contamination.


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[44] The coherence is amplitude-unbiased which meansthat the coherent signals with discontinuous or continuousvarying amplitudes should not be attenuated due to theiramplitude variations. Therefore also signals 2 and 3 do notdiffer much from their original. The lower amplitudesignals, however, are more sensitive to noise which imprintsin the filter performance. Another example with continuousamplitude variations is signal 4 in Figure 5.[45] Concerning the curved signal trajectories, we have

used a short rectangular (almost linear) coherence windowto follow approximately the traveltime curve. This is usually

sufficient to provide satisfactory results. Other more adap-tive windows might be more suitable and can be used toimprove the determination of the signal coherence alongnonlinear traveltime curves.

4. Real Data Examples

[46] Here we are showing two different examples whichillustrate the successful application of our data-adaptivelateral-coherence filter. We use ocean bottom seismometer(OBS) data as an example for a linear array and an

Figure 5. (a) and (b) show the record section with the coherent signals 1 to 4 and the noisecontaminated signals, respectively. (c) This panel contains the filter output for the data from (b). Noise isattenuated and all signals can be identified. (d) shows the signal and noise amplitude spectra of the testdata. The noise amplitude spectrum has been multiplied by 0.25 for visual purposes.

Figure 6. (a) Synthetic test data with discontinuous, curved, and continuously changing signaltrajectory. (b) Filter output for a short window (2 * s = 7 traces). The small arrows at signal 1 (figure a)mark gaps with a width of 6, 2, and 1 traces. These gaps are not filled in the filter output; that is, no newsignals are generated. Also, the signals with curved continuous and discontinuous trajectory have beenpassed by the filter.


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earthquake recorded at Spanish broadband stations as two-dimensional station array.

4.1. Linear Array: AWide-Angle Reflection/RefractionProfile

[47] Figure 7 shows the OBS data used for this example.The data belong to the MARCONI project (OBS 16,profile 3) which has as its main objective the study of thecontinental margin in the Bay of Biscay at the transition ofthe Eurasian and Iberian Plates. The main targets in thesedata are reflections and refractions from the lower crust. Theseismic energy is generated by air guns which have beenshot in 40-s intervals, and the record section is formed by alinear shot array rather than a linear station array.[48] As usual for underwater acquisition, the strongest

arrivals are the direct wave from the source to the OBS,the first water column reverberation, and even the higherorder multiples from previous shots. The latter phasesstart to appear at offsets of about 60 km. These arrivalsare undesired since they obscure the weak signals fromthe crust. Our preprocessing consists of band-passing(3–12 Hz) and of suppressing the disturbing part ofthese ‘‘water waves’’ with a frequency-wavenumber (f-k)filter. The result is shown in Figure 7a and is the inputof our lateral-coherence filter.[49] The coherence-filtered record section is shown in

Figure 7b. The coherence has been determined using aGaussian window with a standard deviation of 3 km at allfrequencies. The time, frequency, and distance windowshave been centered at every 3rd trace, 3rd time sample, and3rd frequency sample. Furthermore, we have employed39 slowness values ranging between �0.18 and 0.24 s/km(at offsets smaller than 20 km) and between �0.14 and0.1 s/km (at offsets larger than 20 km).[50] A clear noise reduction is visible from Figure 7. The

lower crust and Moho reflections/refractions which appearat about 5 to 7 s and 30 to 100 km and their multiple (about3 s later) have clearly been enhanced. The filter helps tofollow these phases to offsets at about 110 km and facilitatesthe picking due to their distinct onsets.[51] Our filter is computationally demanding, and it

requires some care to keep processing time within reason-able limits when working with large record sections. Forinstance, filtering the 1015 traces (501 samples per trace) ofFigure 7 took 2 hours and 30 min on a Pentium 4 (3.4 GHz,1 Gb RAM) computer. We first apply the filter on a smallportion of the data set to find satisfactory filter parametersand then run the filter on the entire data. Strategies todecrease computation time include data decimation, limita-tion of frequency range, limitation of window size, increaseof step size in time, distance, and frequency.

4.2. Two-Dimensional Array: A Japan EarthquakeRecorded in North Iberia

[52] Now we consider data of an earthquake in Japan(26 May 2003, 39�N, 141�E, 69 km) recorded at broadbandstations in North Iberia. The epicenter, stations, and part ofthe great circle arc are shown in Figure 8. The circles anddiamonds mark permanent stations from the CartographicInstitute of Catalonia (ICC) and the National GeographicInstitute (IGN), respectively. The triangles mark portablestations from the Institute Jaume Almera. The great circle

arc shows that the waves should enter the array mainly fromthe North (back azimuth �20�).[53] Figures 9a and 10a show the 0.04–0.6 Hz band-

passed Z components recorded west and east of the�2� meridian, respectively. This is for visual purposes only.A record section of the entire data set would mix therecordings from different areas. However, there is no needto process the data of NE and NW Iberia independently dueto our frequency-dependent nearest-station weightingscheme in Figure 2. The seismic recordings have beenaligned with respect to the P phases at 0 s. The pP andPP phases are visible as strong arrivals at about 20 and220 s.[54] The filter outputs in Figures 9b and 10b show

cleaned records with attenuated noise and clear seismicphases. In this example, we account for the generally shortercorrelation length at higher frequencies through a frequency-dependent coherence window. Thus the standard deviationsof the Gaussian window change linearly between s( f =0.04 Hz) = 2� and s( f = 1.3 Hz) = 0.7� as a function offrequency. The slowness window ranges from �3 to 4s/�.Signal amplitudes between 50 and 230 s have been stronglyattenuated due to the little coherence of the coda phases incomparison with P and pP. Nevertheless, one can discernweak coda signals by increasing the coda amplitudes.Figures 9c, 9d, 10c, and 10d show the filter output aftermultiplying the amplitudes between 29 and 230 s by a factorof 5 and 4, respectively. Figures 9c and 9d differ only by theincluded traveltime curves which likely explain the detectedsignals. We use ak135 [Kennett et al., 1995] as the velocitymodel.[55] The raypaths of the mantle phases are sketched in

Figure 11. d stands for depth to discontinuity. With theexception of PPdp, these phases are near-source reflections/conversions. Note that PdpP and PPdp have the sametraveltime. These phases have an opposite polarity withrespect to the remaining phases. It seems that the filter revealsPdpP, SdP, and SdpP for the 410- and 660-km discontinuities.These discontinuities are global polymorphic phase changescaused by the denser crystal phase at increased pressure,i.e., depth. We placed the 410- and 660-km discontinuities at390- and 670-km depth to better adjust the observed signals.It is not our purpose to determine the exact depth, but wenoticed that a shallower 410-km and deeper 660-km discon-tinuity consistently better fit our observation. This isexpected for these near-source reflections/conversions whichoccur in a colder than average ambient temperature due to thepresence of a descending slab. The 410-km discontinuity hasa positive Clapeyron slope, i.e., an exothermic behavior[e.g., Bina and Helffrich, 1994] which causes the phasetransition to occur at lower pressures for lower temperatures.The 660-km discontinuity is endothermic, and the phasetransition moves to greater depth for colder material.

5. Discussion

[56] The principal ingredients of our processing tool arethe S-transform [Stockwell et al., 1996] and the phase stack[Schimmel and Paulssen, 1997] of neighboring traces tocompute the local amplitude-unbiased phase coherence as afunction of frequency and slowness. The main tunable filter


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Figure

7.

(a)showstheverticalcomponentsofan

ocean

bottom

seismometer

(OBS16,profile3,MARCONIproject,NW

Spain)foraseries

ofairgunshotsseparated

byabout100m

(40s).Maindataprocessingconsistsoftheapplicationofa

bandpassfilter

(frequency

plateau

from

3to

12Hz)

andafrequency-w

avenumber

(f-k)filter.Thef-kfilter

attenuated

the

directwater

wave,thefirstwater

columnreverberation,andthehigher

order

multiplesfrom

previousshots.(b)containsthe

filter

output.Thestandarddeviationforthecoherence

determinationiss=3km

atallfrequencies

andthepower

isv=3.

Incoherentnoiseisattenuated

atalldistanceswhichpermitsan

improved

signal

detection,even

atlargeoffsets.


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parameters control the time-frequency resolution, the sensi-tivity of the phase stack, the physically allowable slownessand frequency range, and the distance (frequency-dependent)for which signals should show spatial coherence. In princi-ple, our filter can be applied on data from any source orreceiver array as long as signals can be characterized by theircoherence. The signal-to-noise ratio is increased when thenoise is less coherent than the signals. In the presence ofcoherent noise, this is achieved when the spatial window islarger than the noise coherence interval or by excluding the

corresponding slowness values from the slowness interval.Alternatively, other tools such as global or time- and space-variant f-k filters [Duncan and Beresford, 1994] can effec-tively be applied to suppress correlated noise.[57] In our examples, we apply our coherence filter on

data from a linear source array (OBS record section) andfrom a two-dimensional broadband station array. It can beseen that the filter successfully enhances signals by inco-herent noise attenuation. The lower crust and Moho reflec-tions/refractions in the MARCONI wide-angle data are now

Figure 8. Event and station map for the data shown in Figures 9 and 10. The circles and diamonds markpermanent broadband stations from the Cartographic Institute of Catalonia (ICC) and the NationalGeographic Institute (IGN), respectively. The triangles mark portable broadband stations from the JaumeAlmera Institute. The inlet shows the epicenter (star) in Japan and the great circle arc to North Iberia.

Figure 9. (a) Z component seismograms for an earthquake in Japan recorded in NW Iberia (Figure 8).The data have been 0.04 to 0.6 Hz bandpassed and have been clipped at 98% for visual purposes.(b) Coherence filter output. A frequency-dependent Gaussian window (s < 2�) has been employed inthe determination of the lateral coherence. In (c) and (d), we amplify the amplitudes between 39 and230 s (marked by the thick gray lines) by a factor of 5.


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visible to larger offsets. These data come from a complexand interesting geographic area due to the transition of theEurasian and Iberian plates. It is used together with otherprofiles for the characterization of the crust in the conti-nental Cantabrian margin (Gallart et al., 2004). Weaksignals have also been revealed for the teleseismic Japanevent. The detected upper mantle reverberations/conversionswere originally obscured by noise and are now observed inNE and NW Iberia. These observations are independent dueto our nearest-station weighting (Figure 2) which avoids theusage of data from distant stations.[58] A plausible and consistent explanation is that the

reverberations are caused by the 410- and the 660-kmdiscontinuities beneath Japan. The signals are better adjustedby placing the discontinuities at about 390 and 670 km depthwhich can be explained by the vicinity of a cold region inthe mantle due to the subducting Pacific plate. The 410-and 660-km discontinuities have a positive and a negativeClapeyron slope, respectively, and occur at smaller andhigher pressures in the presence of cold material. FollowingKirby et al. [1996], the 410-km discontinuity may be elevatedto 350 km while the 660-km discontinuity can warp down toabout 720 km due to subducting slabs. A systematic obser-vation of these phases can provide temperature estimates andlimits the fate of subducted material at the base of the uppermantle [Helffrich et al., 1989; Bina and Helffrich, 1994].[59] The interpretation is based on traveltime and polarity

which is not sufficient to give full evidence to our interpre-tation. Other plausible explanations may exist and a detailedstudy is required to reduce possible ambiguities. Neverthe-less, our examples served to show the abilities of the filterand the importance of using tools which can reveal impor-tant signals hidden by incoherent noise.[60] The main advantage of the time-frequency analysis is

the time-varying spectral representation; that is, signals and

noise can be localized in time and frequency. Therefore thetime series need not be stationary and instantaneous attrib-utes can be designed. This is an attractive property which isused in different data-adaptive filters [e.g., Carrozzo et al.,

Figure 11. Sketch of possible raypaths for the detectedcoda signals. ‘‘d’’ stands for depth to discontinuity, i.e., 410-and 660-km. With the exception of PPdP, all phases are nearsource-site reflections/conversions. Note that PPdP andPdpP are phases with the same traveltime.

Figure 10. Same as Figure 9, but for the stations in NE Iberia. The amplitudes between 39 and 230 shave been amplified by 4 in (c) and (d).


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2002; Pinnegar and Eaton, 2003; Schimmel and Gallart,2004; Diallo et al., 2005]. The latter two filters use theS-transform and the Continuous Wavelet transform, respec-tively, to enhance signals by their instantaneous polarization.Pinnegar and Eaton [2003] utilize ratios of amplitudeS-spectra while Carrozzo et al. [2002] perform waveletdecompositions to correlate portions of details of adjacenttraces.[61] Other powerful strategies are based on eigenanalyses

and are mostly applied to large data volumes to extractenergetic coherent signals. In contrast to expansions withharmonic functions (for example, Fourier series), the basisfunctions are data-adaptive. However, traveltime variations,such as signals arriving on curved trajectories, require a largernumber of eigenimages to reproduce the increased details.This may imply the inclusion of noisier images.[62] In contrast to these methods, our processing tool has

been designed with focus onto weak but coherent signalswhich enable to constrain fine structure. This differs frommethods which favor the extraction of main features, i.e.,the energetic signals. Weak signals are more sensitive tonoise than the large amplitude signals and can only bedetected by their repeated appearance due to the multitudeof other signals and noise. Small traveltime perturbationsand small interferences already make the weak signaldetection difficult. Our filter strategy permits to detectsignals independent of their amplitude as long as there is amean coherence at the different frequency components. Thefiltered waveforms are reconstructed using the coherence-weighted signal components.

6. Conclusions

[63] We present a data-adaptive lateral-coherence filter forarray data. This filter attenuates incoherent noise as a functionof frequency and slowness. The lateral-coherent signals areenhanced through the incoherent signal attenuation.We showthat weak but coherent signals can be detected with ourprocessing tool. The quality of the signal reconstructiondepends on the corruption of the frequency-dependent coher-ence by noise. If noise and signals share the same frequencycomponents, then depending on the degree of corruption,coherent signals may still be detected due to the randomnessof noise. If there is no mean coherence over differentfrequency components, then no signals are detected.[64] Our approach is not restricted to a particular array

configuration and can serve as a detection tool for differentdata types in many different scenarios. The concept of themethod is simple, and adaptations for special data orapplications remain easy. In our examples from passiveand active seismology, the filter successfully attenuatesincoherent noise and reveals signals originally hidden innoise. To improve the detection of coda signals, the datashould be aligned with respect to a reference phase toincrease the lateral coherence.[65] Finally, there exist many signal detection tools due to

the variability of signal and noise characteristics, differentdata configurations, and tasks. There exists no best method,and the selection of processing tools depends much on theproblem at hand. Our method is not an all-round method,and, such as with any other method, one should be awareabout its limitations and functioning for an adequate use. In

any case, we recommend not to restrict to only one methodto obtain the most from the data and to start filtering withmoderate settings.

Appendix A: The New Inverse S-Transform

[66] Here we show that the application of the newinverse S-transform [Schimmel and Gallart, 2005] on theS-spectrum of a time series u(t) yields u(t). For furtherdiscussions on the inverse S-transform and its discretization,see work by Simon et al., [2007]. The inverse S-transform isgiven by using the weight function F(t,f) = 1 in equation (11):

~u �ð Þ ¼ kffiffiffiffiffiffi2�

p Z1

�1

S �; fð Þjf j ei2�f � df : ðA1Þ

[67] Substituting S(t, f) (equations (3) and (5)) intoequation (A1), canceling out the normalization factorsk


p/jfj, and rearranging the order of integration gives

~u �ð Þ ¼Z1

�1

u tð ÞZ1

�1

e� f 2 ��tð Þ2

2k2 ei2�f ��tð Þ df

24

35dt: ðA2Þ

Equation (A2) can be written as the convolution ~u(t) = u *I(t) with

I �ð Þ ¼Z1

�1

e� f 2�2

2k2þ i2�f �

df : ðA3Þ

[68] One can see from the convolution that ~u(t) = u(t) ifI(t) is the Dirac delta function d(t). Hence I(t) is nowconsidered with more detail. Note that the integral of equation(A2) is not the Fourier transform of a Gaussian function dueto its dependence on t. We now rearrange I(t) to

I �ð Þ ¼Z1

�1

e� f 2�2

2k2 cos 2�f �ð Þ þ i sin 2�f �ð Þ½ df ; ðA4Þ

to simplify its integration. The imaginary part of the integral isodd and becomes zero. Therefore equation (A4) reduces to

I �ð Þ ¼Z1

�1

e� f 2�2

2k2 cos 2�f �ð Þdf ; ðA5Þ

[69] For t = 0, we obtain

I t ¼ 0ð Þ ¼Z1

�1

df ¼ 1; ðA6Þ

and for t 6¼ 0 we get

I t 6¼ 0ð Þ ¼ffiffiffiffiffiffi2p

pke�2k2p2 1

j t j ; ðA7Þ

using an integral table [e.g., Gradshteyn and Ryzhik, 2000].Equation (A7) reduces to I(t 6¼ 0) =

c kð Þjtj with c(k) =


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pke

�2k2p2

. It can be seen that I(t 6¼ 0) becomes quicklyzero for k� 1 (for example, for realistic values of k: c(k = 1)’10�8, c(k = 1.5)’ 10�19, c(k = 2) ’ 10�34). This means thatwith I(0) =1, I(t) approaches a Dirac delta function d(t) andwe obtain ~u(t) ’ u(t) using equation (A2).

[70] Acknowledgments. We thank S. Figueras from the CartographicInstitute of Catalonia (ICC), R. Anton from the National GeographicInstitute (IGN), and J. Diaz and M. Ruiz from the Institute of EarthSciences Jaume Almera for providing us with their data. We are gratefulto C. Simon and S. Ventosa for fruitful and valuable discussions on theinverse S-transform. The Associate Editor D. Toomey, F. Simons, D. Eaton,and an anonymous reviewer provided constructive reviews which improvedour manuscript. Plots were made using Generic Mapping Tools (GMT) byWessel and Smith [1991] and Seismic Unix by Cohen and Stockwell [1999].This research has been enabled by the Spanish Ministry of Education andScience through a Ramon and Cajal Fellowship and the Consolider-Ingenio2010 program CSD2006-00041, Topo-Iberia.

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��J. Gallart and M. Schimmel, Institute of Earth Sciences, CSIC, c/ Lluis

Sole i Sabaris, s/n, 08028, Barcelona, Spain. ([email protected])


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Frequency-dependent phase coherence for noise suppression ...diapiro.ictja.csic.es/gt/mschi/SCIENCE/2007-psfreq.pdf · lateral phase coherence as function of distance, time, frequency,