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

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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 through

incoherent noise attenuation. Our processing tool is designed for densely spaced arrays and identifies signals by their coherent appearance. The approach is based on the determination of the lateral phase coherence as function of distance, time, frequency, and slowness. The coherence is derived from the local phases of neighboring stations which we obtain from analytic signals through the S-transform. The coherence is used to attenuate incoherent components in the time-frequency representations of the seismic records. No waveforms are averaged in our approach to maintain local amplitude information. This way we construct a data-adaptive filter which enhances coherent signals using the frequency-dependent and slowness-dependent phase coherence. We explain the method and show its abilities and limitations with theoretical test data. Furthermore, we have selected an ocean bottom seismometer (OBS) record section from NW-Spain and a teleseismic event registered at Spanish broadband stations to show the filter performance on real array data. Incoherent noise has been attenuated in all cases to enable a less ambiguous signal detection. In our last example, the filter also reveals weak conversions/reflections at the 410-km and 660-km discontinuities which are 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 which shares signal characteristics, the complexity of wave propagation in complex media, and the multitude of different interfering signals all inhibit the understanding and the routine use of entire seismograms. Signal processing approaches are therefore important tools to detect, enhance, or extract certain signals from the data which carry information we want to use in subsequent studies. The improvement of these tools is motivated by the steady improvement of data quality, increased station density that better samples the seismic wavefield, increasing computer power, and the need to extract more information to constrain the fine structure 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 on data representations in different domains (for example, time, frequency, slowness, wavenumber, or their combinations). However, there is generally no clear separation between signal and noise, and often it is difficult to establish objective criteria for noise suppression in any domain. In all cases, the choice of the most effective approach depends 1

Institute of Earth Sciences, CSIC, Barcelona, Spain.

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

on the application as well as on the signal and noise characteristics themselves. [4] In this paper, the signal is defined as that which is coherent on nearby traces over a range of frequencies and slownesses, while random noise is not. This is a widespread definition which has led to many different approaches based on different coherence measures under various approximations. For instance, f-x (frequency-offset) and t-x (timeoffset) prediction (and projection) filters use the property that coherent signals can be predicted from nearby traces [e.g., Hornbostel, 1991]. These techniques keep signals which can be predicted while suppressing those that cannot. [5] Other methods expand the two-dimensional (twodimensional in time and offset) data matrix using orthogonal transforms such as the principal component, Karhunen – Loe`ve transform, and eigenimage approaches [e.g., Trickett, 2003]. These methods are related in that they exploit the property that coherent energy is mainly mapped onto the first eigenvectors or eigenimages, corresponding to the largest eigenvalues. Therefore a truncated (or weighted) eigenvector or eigenimage decomposition can recover most of the coherent signal energy while the omitted part of the expansion retains most of the noise. The prediction and eigenanalysis filters are mostly used with large data volumes such as multifold data in seismic exploration. A signal subspace method which finds more application in global array seismology is the multiple signal classification technique (MUSIC) [e.g., see Bokelmann and Baisch, 1999; Almendros et al., 2001].

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SCHIMMEL AND GALLART: FREQUENCY-DEPENDENT PHASE COHERENCE

[6] Frequency-wavenumber ( f-k) and traveltime intercept-slowness (t-p) (Radon or slant stack) filters are possibly the most common filters to identify and separate coherent waves. These filters take advantage of the fact that aligned features map in a distinguishable way in the corresponding data representation following their apparent velocity, frequency, or intercept time. The f-k representation can be obtained by a two-dimensional Fourier transform which means that noise and signal are represented as plane waves traveling through the array. The t-p representation is achieved through a slant stack or Radon transform. Both f-k and t-p filters are often useful to suppress correlated noise such as ground roll or water column reverberations in large data volumes. Time- and space-variant f-k [Duncan and Beresford, 1994] and t-p filters [e.g., van der Baan and Paul, 2000] are more sensitive to signal variations and can be 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 slowness and back azimuth [e.g., Rost and Thomas, 2002; Koper et al., 2004; Kru¨ger and Ohrnberger, 2005; Tanaka, 2005]. Beam forming is a stacking approach as a function of azimuth and slowness. The stacks are often additionally weighted by coherence measures to lower the detection threshold by attenuating noise. The vector character of the wavefield can be exploited by three-component (3-C) stacking [e.g., Kennett, 2000] or through inclusion of polarization attributes. [8] Data-adaptive coherence filters can also be designed using the signal localization capabilities of time-frequency approaches. Carrozzo et al. [2002] filters seismic wideangle data by employing a wavelet decomposition into different detail levels. Portions of the details of adjacent traces are cross-correlated to obtain the filtered traces through 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 adjacent traces 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, and space. We utilize the S-transform with the inverse transform by Schimmel and Gallart [2005]. The coherence weights are determined with instantaneous phase stacks [Schimmel and Paulssen, 1997] to weight signal components following their spatial phase coherence in a physically allowable slowness range. In the following, the filter and its ingredients are explained, and its capability and limitations are illustrated with test data. The filter can be used for the different types of data and arrays. We have selected an ocean bottom seismometer (OBS) record section from NW-Spain and a teleseismic event registered at Spanish broadband stations to show the filter performance on real array data with very different characteristics. The filter can easily be adapted to other situations.

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

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[Schimmel and Paulssen, 1997] and serves to attenuate incoherent components in the local spectra. Finally, the filtered traces are obtained after a back transform to the time domain. This way, we construct a data-adaptive coherence filter. Before delving into the filter construction itself, 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 representation with absolute referenced phase which allows the comparison of phases. The S-transform is based on a sliding window Fourier analysis and is similar to the Gabor or short time Fourier transform [Gabor, 1946]. But in contrast to the short time Fourier transform, the S-transform employs frequency-dependent windows in analogy to the wavelet transform. This permits a better resolution of low frequency components and enables a better time resolution of high frequency signals. [12] For the following, we adopt the Fourier transform Z1 Uð f Þ ¼

uðt Þei 2 p f t dt;

ð1Þ

1

to determine the spectrum U( f ) of a time series u(t). The corresponding inverse transform is Z1 uðt Þ ¼

U ð f Þe i 2 p f t df :

ð2Þ

1

[13] The S-transform of u(t) is, following Stockwell et al. [1996], Z1 S ðt; f Þ ¼

uðt Þwðt t; f Þei 2 p f t dt; f 6¼ 0;

ð3Þ

1 T

1 S ðt; f ¼ 0Þ ¼ lim T !1 T

Z2 uðt Þ dt;

ð4Þ

T2

with a Gaussian function w(t t, f ) centered at time t and a standard deviation proportional to j1/f j j f j f 2 ðttÞ2 wðt t; f Þ ¼ pffiffiffiffiffiffi e 2k 2 ; k > 0; k 2p

ð5Þ

to obtain the time localized spectra. Here f stands for frequency and t, t are time variables. t is the center time of the Gaussian window and k is a scaling factor which controls the number of oscillations in each window. The factor 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 functions permit one to achieve the lower bound of the uncertainty

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once. U(a) is shifted by f before it is multiplied with the window function. Next, the calculation of S(t, f) is then achieved applying the inverse Fourier transform from a to t. The Gaussian function in the integrand of equation (6) has a frequency-dependent bandwidth and acts on the shifted spectrum U(a + f ). This procedure is equivalent to bandpassing the phase-shifted input signal u(t) with a Gaussian function centered at frequency f with standard j deviation s = kjf 2p as can be seen when substituting b = a + f to obtain Z1 S ðt; f Þ ¼

U ðbÞe

2p2 ðbf Þ2 k2 f 2

ei2pðbf Þt db:

ð7Þ

1

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 satisfies the condition (which, defined as in equation (5), it does) Z1 wðt t; f Þ dt ¼ 1;

ð8Þ

1

then the time averaging of the S-spectrum S(t, f ) yields the spectrum U( f ) of the input signal u(t): Figure 1. Illustration of the determination of the timefrequency representation through the S-transform which forms part of our approach. The size of the moving-analysis window is scaled by the period to account for the intrinsic higher time resolution at shorter periods. relationship for optimal signal localization in time and frequency. The transform of the data into its time-frequency representation is illustrated in Figure 1. The figure shows the moving-analysis window which is scaled by the period and 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 ei 2 p f t does not translate with the Gaussian window. The phasor is fixed to the time series, and the local phase spectra are therefore absolutely referenced. Without this phasor the S-transform could be represented as a continuous wavelet transform [Stockwell et al., 1996]. [15] Equation (3) can also be expressed [Stockwell et al., 1996, 1999] as following the multiplication of spectra Z1 S ðt; f Þ ¼

U ða þ f Þe

2p2 2 a k2 f 2

ei2pat da; f 6¼ 0;

ð6Þ

1

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

Z1 S ðt; f Þdt ¼ U ð f Þ:

ð9Þ

1

[17] This procedure permits to freely move between the time, frequency, and time-frequency domains, which invites to use weight functions F(t, f ) to construct data-adaptive filters: Z1 Ufilt1 ð f Þ ¼

S ðt; f ÞF ðt; f Þdt:

ð10Þ

1

[18] Here F(t, f ) 2 [0,1] is the time-frequency weight function which permits to attenuate undesired features in the S-spectra, for example, for signal extraction and/or noise attenuation. However, such manipulation can cause spurious signals [Schimmel and Gallart, 2005]. The main problem is that due to the time integration in equation (9), the time localization imposed through F(t, f) is not directly translated to the time domain. Schimmel and Gallart [2005] point to this problem and propose an alternative inverse transform (equation (11)) for applications where time localization is important. Their approach avoids time averaging since it is based on the individual consideration of the local spectra as a function of time. Consequently, the signal localization in time translates directly to the time series. Our inverse S-transform is expressed as

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1 pffiffiffiffiffiffi Z S ðt; f ÞF ðt; f Þ i2ft ufilt2 ðt Þ ¼ k 2p df : e jf j 1

ð11Þ


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[19] We show in the Appendix A that the input data are retrieved in case there is no manipulation of the S-spectrum, i.e., F(t, f) = 1. The inverse approaches are further analyzed by 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 associates a phase to a real signal. This phase information is very important for the quantitative determination of phase synchronization, coherence, time shifts, and instantaneous frequencies, among others. The analytic signal is defined as the sum of the real-time series u(t) and its Hilbert transform H[u(t)] as imaginary part and is expressed as us ðt Þ ¼ uðt Þ þ iH ½uðt Þ ¼ Aðt ÞeiFðtÞ :

ð12Þ

Here A(t) and F(t) are the local amplitude (envelope) and the local or instantaneous phase of the signal. The Hilbert transform only affects the phase of the spectral components of the real signal u(t). This operation shifts the positive and negative frequency components by p/2 and p/2, respectively. The signal and its Hilbert transform are orthogonal, and the transformation to the complex domain is a linear and unique operation. The analytic signal us(t) has only positive frequency components in its spectrum. In practice, us(t) is often achieved by the inverse Fourier transform of the doubled signal spectrum U( f ) with suppressed negative frequency components. [21] The Hilbert transform H[u(t)] can be expressed as the convolution pt1 * u(t), where pt1 is the function which heavily weights the time series. The instantaneous phase is a very valuable attribute. It depends implicitly on the neighboring samples, i.e., on the local signal waveform, to permit instantaneous frequency analysis [Taner et al., 1979] and the amplitude-unbiased quantitative comparison of signals [Schimmel and Paulssen, 1997; Schimmel, 1999]. The signal comparison is based on the fact that coherent signals should have identical phases. We want to apply this concept to the complex S-spectrum (equation (3)). Following Stockwell et al. [1996], the phase of the S-spectrum can be understood as the instantaneous phase of the analytic signal and is therefore suitable for our filter. 2.2.2. When is the S-Spectrum Analytic? [22] The following consideration shows when the Sspectrum can be considered to be analytic at a fixed frequency f. In analogy to equation (6), we can write the S-spectrum of the analytic signal us(t) = u(t) + i H[u(t)] as

S s ðt; f Þ ¼

Z1 U ða þ f Þe

2p2 2 a k2 f 2

p 1 iei2sgn½aþf ei2pat da;

ð13Þ

1

f 6¼ 0 and k > 0: p

substitute b = a + f into equation (13) and use 1iei2sgn½b = ip2sgn½b = 2, 8b > 0 to obtain 0, 8b < 0 and 1ie Z1

s

S ðt; f Þ ¼ 2

U ðbÞe

2p2 ðbf Þ2 k2 f 2

ei2pðbf Þt db:

ð14Þ

0

[23] A comparison with equation (7) shows that there is a close relation between the S-transform S(t, f) of a real-time series and the S-transform Ss(t, f) of an analytic signal. Both expressions (equations (7) and (14)) can be understood as band pass filters of the time series u(t), where the Gaussian function is centered at frequency f with a standard deviation j s = kjf 2p . Given that 99.7% of the area of the Gaussian function is located within f 3 s b f + 3s, the Gaussian is approximately zero outside the 3s interval. We can determine the minimum frequency of the 3s interval to 3k ) and therefore obtain k 2p bmin = f 3 s = f(12p 3 2.1 for b min 0. Since the Gaussian function is approximately zero outside the 3s interval, the integrand of equation (7) is zero for b < 0 and we can write for any positive frequency f and k 2.1: S s ðt; f Þ ’ 2S ðt; f Þ:

ð15Þ

[24] We see that if we assume that the Gaussian function in the integrand of equation (7) is narrow to concentrate energy well at the positive frequencies (b > 0), then the Sspectra of the real and analytic signal are in close agreement for f 0. The approximation improves for smaller k which depends on the choice of the s interval. For instance, the 99% or 99.9% limits of the area of the Gaussian function are given by a 2.58 or 3.29 s interval which results into k 2.7 or k 1.9, respectively. [25] Finally, we show under which conditions the Stransform of an analytic signal is also analytic. Using equation (3), the S-transform of the analytic signal us(t) is given by Z1

s

S ðt; f Þ ¼

us ðt Þwðt t; f Þei 2 p f t dt:

ð16Þ

1

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

S s ðt; f Þei 2 p f t ¼

Z1

us ðt Þwðt t; f Þei 2 p f ðttÞ dt:

ð17Þ

1

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

The factor ei2sgn½aþf comes from the complex part of the analytic signal and shifts the phases as a function of the sign of the frequencies to obtain the Hilbert transform. Now we 4 of 14

S ðt; f Þei 2 p f t ¼ Aðt; f ÞeiFðt; f Þ ;

ð18Þ

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

where A(t, f ) and F(t, f ) are the envelope and the instantaneous phase of an analytic signal at frequency f and for 0 < k 2. We conclude that under above conditions, the S-transform S(t, f) for any real signal is approximately analytic at a fixed frequency after the application of a t-dependent phase shift. 2.3. Phase Coherence Measure [28] We employ the phase stack concept by Schimmel and Paulssen [1997] to measure the coherence as a function of frequency and slowness. The phase stack cps(t) is defined as the absolute value of the complex summation of the amplitude normalized analytic traces cps ðt Þ ¼

n N 1 X iFj ðt Þ e ; N j¼1

ð19Þ

where j and N are the trace index and the total number of traces, respectively. cps(t) does not depend explicitly on amplitudes and is therefore an amplitude-unbiased measure which is robust to signal amplitude changes and/or outliers. cps(t) takes advantage of the principles of constructive and destructive interferences due to the complex summation. The summation for coherent signals is visualized by the summation of unit vectors which all point to the same direction in the complex plane. Owing to the employed normalization by N, cps(t) ranges between 0 and 1, with 1 being achieved for perfectly coherent signals. The vector summation is destructive for incoherent signals which results into a small value for cps(t). The power v is used to enhance the sharpness of the transition between coherence and incoherence. v = 2 or 3 is generally a good choice. In the work of Schimmel and Paulssen [1997], cps(t) is used to weight linear slowness stacks by their coherence. Here individual traces rather than waveform stacks will be weighted to retain site-dependent signal characteristics. [29] Frequency-slowness-dependent phase stacks can be determined in analogy to equation (19) using the phases Fj(t, f ) of S-spectra (equation (18)). The local phase stacks are expressed as a function of time t, location vector ~ rj of

station j, horizontal slowness ~ shor , and a window of 2N + 1 traces for the local lateral-coherence determination n jþN X 1 iFk ðtð~ rk ~ rj Þ~ shor ; f ÞþiwDtjk ~cps t;~ rj ; f ;~ shor ¼ e : 2N þ 1 k¼jN ð20Þ

[30] The phase correction wDtjk is due to the phase shift (equation (18)) to make the S-transform analytic. We can consider it as a phase shift due to the absolutely referenced phases of the S-spectra. The phases are corrected for the rk ~ rj ) ~ shor . For linear arrays or relative time shift Dtjk = (~ rj ) ~ shor reduces to Dxjk p with Dxjk = record sections, (~ rk ~ rj j being the interstation distance and p = j~ shor j. j~ rk ~ [31] Here it is assumed that the seismic signals have a plane wavefront with apparent velocity p1 within the window defined by 2N + 1 traces. The summations are performed along straight traveltime trajectories with gradient p. In practice, this is a good approximation for dense arrays, small windows, and long wavelengths. Other trajectories such as hyperbolas can be implemented. [32] We often favor the phasors from neighboring stations using weights in the phase stack. This increases the importance of nearby stations in relation to stations which are further away. The weights are given by the Gaussian Dx2 function g(Dxjk, f ) = exp{2ð fjkÞ2 } as function of interstation distance Dxjk and frequency f. The standard deviation s( f ) is an empirical linear function and permits to use weights which are small at large distances and high frequencies to account for the larger signal variability at the higher frequencies. The determination of the weights is pictured in Figure 2. 2.4. Filter Construction [33] The lateral phase-coherence filter is designed using rj , f, p). This the just outlined coherence measure ~cps (t, ~ measure provides for each seismic trace at each time and frequency the lateral phase coherence as a function of slowness. This information can be reduced to weights of the local time-frequency representations S(t, ~ rj , f) because

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Figure 3. Sketch outlines the determination of the local coherence at a selected distance, time, and frequency. To the left, we see the sliding data window which is moved through the frequency-dependent envelope-normalized complex traces. Slowness-dependent phase stacks are performed on the windowed data to obtain the local coherence (phase stack amplitude) shown to the right. The maximum coherence value is ascribed to the window center and used as filter weight. we do not need the explicit slowness dependence of ~cps . It is sufficient to know the maximum coherence values (equation (21)) as function of time, frequency, and distance

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

cps t;~ rj ; f ¼ max ~cps t;~ rj ; f ; p p :

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 have dominant frequencies ranging between 1 and 8 Hz. The signals are contaminated by random noise. Signals and noise amplitude spectra are shown in Figure 4c. Here we can see that the noise is white in the signal frequency band. The record section has been filtered using the following parameters: sliding data window N = 10 traces and DT = 3 time samples at all frequencies and distances, with 45 different slowness values between 0.03 and 0.03 s/trace, power v = 4, and S-transform window width k = 1 (standard deviation equals 1 period). The local phase stacks have been computed at every second trace, second frequency, and second time sample. The intermediate values are interpolated, and the filter output is shown in Figure 4b. It can be seen from this figure that the filter reduced the noise and increased the signalto-noise ratio (S/N). [39] Figures 4d and 4e show the same test data, but with larger noise contamination, and the corresponding filter output. The filter parameters have not been modified, and the noise amplitude spectrum is shown in Figure 4c. The noise amplitudes are large and strongly influence the signal waveforms. Nevertheless, the filter attenuates the noise and reveals the laterally coherent signals. The reconstructed waveforms in Figure 4e are not as good as in Figure 4b. This is expected since signal and noise share the same frequencies. In fact, a signal detection has only been possible since a mean coherence could be detected through the local phase stacks at different frequencies. Further increase of the noise level would cause a larger impact on the filtered signal waveforms until the signal coherence has been completely destroyed which inhibits the signal detection.

ð21Þ

[34] These values are the data-adaptive weights in our filter. The procedure is summarized in Figure 3. With S(t, ~ rj , f ) being the S-spectrum of the jth trace, the filtered S-spectrum becomes Sfilt t;~ rj ; f ¼ cps t;~ rj ; f S t;~ rj ; f :

ð22Þ

[35] The filtered time series are obtained after the application 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 the S-transform), and the range and number of physically allowable slowness values and frequencies. Other processing steps can be incorporated if data and/or application require further refinement. For instance, we keep the option to smooth the coherence rj , f ) by using a mean or median filter similar to Schimmel cps(t,~ and Gallart [2004]. These filters replace the center value of a small volume which is centered on each element of cps by its mean or median value. This procedure is applied before the weighting and can remove isolated coherent features or can prevent the attenuation of isolated incoherent components. Furthermore, we permit to control the width of the time window (DT in Figure 3) in the determination of the rj , f, p) to account for slight signal slowness-dependent ~cps (t, ~ misalignments. These and other settings do not strongly influence the filter output but can provide refined/modified data images.

3. Examples With Synthetic Data [37] In what follows, we apply the method to synthetic data to test the filter and to demonstrate its ability and

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Figure 4. (a) Synthetic record section contains four coherent signals which are labeled by numbers and contaminated with random noise. (b) shows filter output for the data from (a). The incoherent noise is attenuated, 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 is not as good due to the increased corruption of signal components. Signals are still detected due to the randomness of noise contamination.

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 large and hides the signals as can be seen in Figure 5b. These test data are filtered using the parameters from the previous example. The filter output and the signal and noise amplitude spectra are shown in Figures 5c and 5d, respectively. The filter attenuates most of the noise and reveals all signals in spite of the large amplitude noise. Also, in this example, the amplitude spectra in Figure 5d are crucial to understand the filter performance on the test data. The noise and signal amplitude spectra overlap at the higher frequencies. Note that the noise amplitude spectrum (Figure 5d) has been multiplied by 0.25 for visual purposes which may give the misleading impression of little spectral overlap. Nevertheless, the dominant frequency components of the signals are not strongly corrupted which explains why the signals are revealed from the large amplitude noise. We observe from the filter output that the reconstructed signals 1 and 2 are of lower frequency than the original signals. This is because the coherence has been diminished or destroyed by the strong noise contamination at about 5 to 8 Hz. [41] Note that no frequency-dependent noise attenuation is possible with a time domain approach where the rj ), for example, by using weights reduce to ~cps (t, ~ equation (19) in analogy to equation (21). At best, such an approach can provide the noise contaminated signal waveforms which would not at all resemble the input signals 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 are usually caused by lateral heterogeneities or data problems. This has been simulated in our data from Figure 6a. Signal 1 contains gaps with widths of 6, 2, and 1 traces (marked by short arrows) and ends abruptly at trace number 80. Signals 2 and 3 have a curved signal trajectory, and signal 2 has discontinuous amplitudes. The amplitudes of this signal have been multiplied by 0.5 on every second trace until trace number 40 and on each trace for trace numbers 50 to 60. The amplitudes of signal 3 change continuously; that is, they are modulated by a sine function. The traces have been contaminated with random noise in the signal frequency band. We filter these data employing a short Gaussian window (2 * s = 7 traces at all frequencies) with width DT = 40 ms. The window is short to approximate the curved signal trajectory. The filter output is shown in Figure 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 are seen as in the filter input. This is important and means that no signals have been added where there are no signals. The coherence is a weight which ranges between 0 and 1 and can therefore only attenuate amplitudes following the measured signal coherence. The signals are less coherent near the gaps, and this can lead to decreased signal amplitudes toward the gaps. Local waveform averaging would have averaged the amplitudes, filled the small gaps, and blurred the signals into traces beyond trace number 80.

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

[44] The coherence is amplitude-unbiased which means that the coherent signals with discontinuous or continuous varying amplitudes should not be attenuated due to their amplitude variations. Therefore also signals 2 and 3 do not differ much from their original. The lower amplitude signals, however, are more sensitive to noise which imprints in the filter performance. Another example with continuous amplitude variations is signal 4 in Figure 5. [45] Concerning the curved signal trajectories, we have used a short rectangular (almost linear) coherence window to follow approximately the traveltime curve. This is usually

sufficient to provide satisfactory results. Other more adaptive windows might be more suitable and can be used to improve the determination of the signal coherence along nonlinear traveltime curves.

4. Real Data Examples [46] Here we are showing two different examples which illustrate the successful application of our data-adaptive lateral-coherence filter. We use ocean bottom seismometer (OBS) data as an example for a linear array and an

Figure 6. (a) Synthetic test data with discontinuous, curved, and continuously changing signal trajectory. (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 new signals are generated. Also, the signals with curved continuous and discontinuous trajectory have been passed by the filter. 8 of 14

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earthquake recorded at Spanish broadband stations as twodimensional station array. 4.1. Linear Array: A Wide-Angle Reflection/Refraction Profile [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 the continental margin in the Bay of Biscay at the transition of the Eurasian and Iberian Plates. The main targets in these data are reflections and refractions from the lower crust. The seismic energy is generated by air guns which have been shot in 40-s intervals, and the record section is formed by a linear 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 higher order multiples from previous shots. The latter phases start to appear at offsets of about 60 km. These arrivals are undesired since they obscure the weak signals from the crust. Our preprocessing consists of band-passing (3 – 12 Hz) and of suppressing the disturbing part of these ‘‘water waves’’ with a frequency-wavenumber (f-k) filter. The result is shown in Figure 7a and is the input of our lateral-coherence filter. [49] The coherence-filtered record section is shown in Figure 7b. The coherence has been determined using a Gaussian window with a standard deviation of 3 km at all frequencies. The time, frequency, and distance windows have been centered at every 3rd trace, 3rd time sample, and 3rd frequency sample. Furthermore, we have employed 39 slowness values ranging between 0.18 and 0.24 s/km (at offsets smaller than 20 km) and between 0.14 and 0.1 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 appear at about 5 to 7 s and 30 to 100 km and their multiple (about 3 s later) have clearly been enhanced. The filter helps to follow these phases to offsets at about 110 km and facilitates the picking due to their distinct onsets. [51] Our filter is computationally demanding, and it requires some care to keep processing time within reasonable limits when working with large record sections. For instance, filtering the 1015 traces (501 samples per trace) of Figure 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 small portion of the data set to find satisfactory filter parameters and then run the filter on the entire data. Strategies to decrease computation time include data decimation, limitation of frequency range, limitation of window size, increase of step size in time, distance, and frequency. 4.2. Two-Dimensional Array: A Japan Earthquake Recorded in North Iberia [52] Now we consider data of an earthquake in Japan (26 May 2003, 39°N, 141°E, 69 km) recorded at broadband stations in North Iberia. The epicenter, stations, and part of the great circle arc are shown in Figure 8. The circles and diamonds mark permanent stations from the Cartographic Institute of Catalonia (ICC) and the National Geographic Institute (IGN), respectively. The triangles mark portable stations from the Institute Jaume Almera. The great circle

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arc shows that the waves should enter the array mainly from the North (back azimuth 20°). [53] Figures 9a and 10a show the 0.04– 0.6 Hz bandpassed 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 the recordings from different areas. However, there is no need to process the data of NE and NW Iberia independently due to our frequency-dependent nearest-station weighting scheme in Figure 2. The seismic recordings have been aligned with respect to the P phases at 0 s. The pP and PP phases are visible as strong arrivals at about 20 and 220 s. [54] The filter outputs in Figures 9b and 10b show cleaned records with attenuated noise and clear seismic phases. In this example, we account for the generally shorter correlation length at higher frequencies through a frequencydependent coherence window. Thus the standard deviations of the Gaussian window change linearly between s( f = 0.04 Hz) = 2° and s( f = 1.3 Hz) = 0.7° as a function of frequency. The slowness window ranges from 3 to 4s/°. Signal amplitudes between 50 and 230 s have been strongly attenuated due to the little coherence of the coda phases in comparison with P and pP. Nevertheless, one can discern weak coda signals by increasing the coda amplitudes. Figures 9c, 9d, 10c, and 10d show the filter output after multiplying the amplitudes between 29 and 230 s by a factor of 5 and 4, respectively. Figures 9c and 9d differ only by the included traveltime curves which likely explain the detected signals. We use ak135 [Kennett et al., 1995] as the velocity model. [55] The raypaths of the mantle phases are sketched in Figure 11. d stands for depth to discontinuity. With the exception of PPdp, these phases are near-source reflections/ conversions. Note that PdpP and PPdp have the same traveltime. These phases have an opposite polarity with respect to the remaining phases. It seems that the filter reveals PdpP, SdP, and SdpP for the 410- and 660-km discontinuities. These discontinuities are global polymorphic phase changes caused by the denser crystal phase at increased pressure, i.e., depth. We placed the 410- and 660-km discontinuities at 390- and 670-km depth to better adjust the observed signals. It is not our purpose to determine the exact depth, but we noticed that a shallower 410-km and deeper 660-km discontinuity consistently better fit our observation. This is expected for these near-source reflections/conversions which occur in a colder than average ambient temperature due to the presence of a descending slab. The 410-km discontinuity has a positive Clapeyron slope, i.e., an exothermic behavior [e.g., Bina and Helffrich, 1994] which causes the phase transition to occur at lower pressures for lower temperatures. The 660-km discontinuity is endothermic, and the phase transition moves to greater depth for colder material.

5. Discussion [56] The principal ingredients of our processing tool are the S-transform [Stockwell et al., 1996] and the phase stack [Schimmel and Paulssen, 1997] of neighboring traces to compute the local amplitude-unbiased phase coherence as a function of frequency and slowness. The main tunable filter

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Figure 7. (a) shows the vertical components of an ocean bottom seismometer (OBS 16, profile 3, MARCONI project, NW Spain) for a series of air gun shots separated by about 100 m (40 s). Main data processing consists of the application of a bandpass filter (frequency plateau from 3 to 12 Hz) and a frequency-wavenumber (f-k) filter. The f-k filter attenuated the direct water wave, the first water column reverberation, and the higher order multiples from previous shots. (b) contains the filter output. The standard deviation for the coherence determination is s = 3 km at all frequencies and the power is v = 3. Incoherent noise is attenuated at all distances which permits an improved signal detection, even at large offsets.

B04303 SCHIMMEL AND GALLART: FREQUENCY-DEPENDENT PHASE COHERENCE

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Figure 8. Event and station map for the data shown in Figures 9 and 10. The circles and diamonds mark permanent broadband stations from the Cartographic Institute of Catalonia (ICC) and the National Geographic Institute (IGN), respectively. The triangles mark portable broadband stations from the Jaume Almera Institute. The inlet shows the epicenter (star) in Japan and the great circle arc to North Iberia. parameters control the time-frequency resolution, the sensitivity of the phase stack, the physically allowable slowness and frequency range, and the distance (frequency-dependent) for which signals should show spatial coherence. In principle, our filter can be applied on data from any source or receiver array as long as signals can be characterized by their coherence. The signal-to-noise ratio is increased when the noise is less coherent than the signals. In the presence of coherent noise, this is achieved when the spatial window is larger 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 spacevariant f-k filters [Duncan and Beresford, 1994] can effectively 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) and from a two-dimensional broadband station array. It can be seen that the filter successfully enhances signals by incoherent noise attenuation. The lower crust and Moho reflections/refractions in the MARCONI wide-angle data are now

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 in the determination of the lateral coherence. In (c) and (d), we amplify the amplitudes between 39 and 230 s (marked by the thick gray lines) by a factor of 5. 11 of 14

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Figure 10. Same as Figure 9, but for the stations in NE Iberia. The amplitudes between 39 and 230 s have been amplified by 4 in (c) and (d). visible to larger offsets. These data come from a complex and interesting geographic area due to the transition of the Eurasian and Iberian plates. It is used together with other profiles for the characterization of the crust in the continental Cantabrian margin (Gallart et al., 2004). Weak signals have also been revealed for the teleseismic Japan event. The detected upper mantle reverberations/conversions were originally obscured by noise and are now observed in NE and NW Iberia. These observations are independent due to our nearest-station weighting (Figure 2) which avoids the usage of data from distant stations. [58] A plausible and consistent explanation is that the reverberations are caused by the 410- and the 660-km discontinuities beneath Japan. The signals are better adjusted by placing the discontinuities at about 390 and 670 km depth which can be explained by the vicinity of a cold region in the mantle due to the subducting Pacific plate. The 410and 660-km discontinuities have a positive and a negative Clapeyron slope, respectively, and occur at smaller and higher pressures in the presence of cold material. Following Kirby et al. [1996], the 410-km discontinuity may be elevated to 350 km while the 660-km discontinuity can warp down to about 720 km due to subducting slabs. A systematic observation of these phases can provide temperature estimates and limits the fate of subducted material at the base of the upper mantle [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 interpretation. Other plausible explanations may exist and a detailed study is required to reduce possible ambiguities. Nevertheless, our examples served to show the abilities of the filter and the importance of using tools which can reveal important 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 the time series need not be stationary and instantaneous attributes can be designed. This is an attractive property which is used in different data-adaptive filters [e.g., Carrozzo et al.,

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

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2002; Pinnegar and Eaton, 2003; Schimmel and Gallart, 2004; Diallo et al., 2005]. The latter two filters use the S-transform and the Continuous Wavelet transform, respectively, to enhance signals by their instantaneous polarization. Pinnegar and Eaton [2003] utilize ratios of amplitude S-spectra while Carrozzo et al. [2002] perform wavelet decompositions to correlate portions of details of adjacent traces. [61] Other powerful strategies are based on eigenanalyses and are mostly applied to large data volumes to extract energetic coherent signals. In contrast to expansions with harmonic functions (for example, Fourier series), the basis functions are data-adaptive. However, traveltime variations, such as signals arriving on curved trajectories, require a larger number 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 signals which enable to constrain fine structure. This differs from methods which favor the extraction of main features, i.e., the energetic signals. Weak signals are more sensitive to noise than the large amplitude signals and can only be detected by their repeated appearance due to the multitude of other signals and noise. Small traveltime perturbations and small interferences already make the weak signal detection difficult. Our filter strategy permits to detect signals independent of their amplitude as long as there is a mean coherence at the different frequency components. The filtered waveforms are reconstructed using the coherenceweighted signal components.

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

6. Conclusions

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

[63] We present a data-adaptive lateral-coherence filter for array data. This filter attenuates incoherent noise as a function of frequency and slowness. The lateral-coherent signals are enhanced through the incoherent signal attenuation. We show that weak but coherent signals can be detected with our processing tool. The quality of the signal reconstruction depends on the corruption of the frequency-dependent coherence by noise. If noise and signals share the same frequency components, then depending on the degree of corruption, coherent signals may still be detected due to the randomness of noise. If there is no mean coherence over different frequency 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 different data types in many different scenarios. The concept of the method is simple, and adaptations for special data or applications remain easy. In our examples from passive and active seismology, the filter successfully attenuates incoherent noise and reveals signals originally hidden in noise. To improve the detection of coda signals, the data should be aligned with respect to a reference phase to increase the lateral coherence. [65] Finally, there exist many signal detection tools due to the variability of signal and noise characteristics, different data configurations, and tasks. There exists no best method, and the selection of processing tools depends much on the problem at hand. Our method is not an all-round method, and, such as with any other method, one should be aware about its limitations and functioning for an adequate use. In

Appendix A: The New Inverse S-Transform [66] Here we show that the application of the new inverse S-transform [Schimmel and Gallart, 2005] on the S-spectrum of a time series u(t) yields u(t). For further discussions on the inverse S-transform and its discretization, see work by Simon et al., [2007]. The inverse S-transform is given by using the weight function F(t,f) = 1 in equation (11): 1 pffiffiffiffiffiffi Z S ð; f Þ i2f ~ df : uð Þ ¼ k 2 e jf j

ðA1Þ

1

[67] Substituting S(t, f) (equations (3) and (5)) into equation (A1), canceling out the normalization factors pffiffiffiffiffiffi k 2p/jfj, and rearranging the order of integration gives Z1 ~ uð Þ ¼

2

Z1

uðt Þ4

1

3 e

f 2 ðt Þ2 2k 2

ei2f ðtÞ df 5dt:

ðA2Þ

1

Equation (A2) can be written as the convolution ~ u(t) = u * I(t) with Z1 I ð Þ ¼

e

f 22 2k 2

þ i2f

df :

ðA3Þ

1

Z1 I ð Þ ¼

e

f 22 2k 2

½cosð2f Þ þ i sinð2f Þ df ;

ðA4Þ

1

to simplify its integration. The imaginary part of the integral is odd and becomes zero. Therefore equation (A4) reduces to Z1 I ð Þ ¼

e

f 22 2k 2

cosð2f Þdf ;

ðA5Þ

1

[69] For t = 0, we obtain Z1 I ðt ¼ 0Þ ¼

df ¼ 1;

ðA6Þ

1

and for t 6¼ 0 we get I ðt 6¼ 0Þ ¼

pffiffiffiffiffiffi 2k 2 p2 1 2pke ; jtj

ðA7Þ

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

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pffiffiffiffiffiffi 2k2p2 . It can be seen that I(t 6¼ 0) becomes quickly 2pke zero for k 1 (for example, for realistic values of k: c(k = 1) ’ 108, c(k = 1.5) ’ 1019, c(k = 2) ’ 1034). This means that with I(0) = 1, I(t) approaches a Dirac delta function d(t) and we obtain ~u(t) ’ u(t) using equation (A2). [70] Acknowledgments. We thank S. Figueras from the Cartographic Institute of Catalonia (ICC), R. Antoń from the National Geographic Institute (IGN), and J. Diaz and M. Ruiz from the Institute of Earth Sciences Jaume Almera for providing us with their data. We are grateful to C. Simon and S. Ventosa for fruitful and valuable discussions on the inverse S-transform. The Associate Editor D. Toomey, F. Simons, D. Eaton, and an anonymous reviewer provided constructive reviews which improved our manuscript. Plots were made using Generic Mapping Tools (GMT) by Wessel and Smith [1991] and Seismic Unix by Cohen and Stockwell [1999]. This research has been enabled by the Spanish Ministry of Education and Science through a Ramon and Cajal Fellowship and the Consolider-Ingenio 2010 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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