characteristics of seismic noise: fundamental and higher...

Characteristics of Seismic Noise: Fundamental and Higher Mode

Energy Observed in the Northeast of the Netherlands

by W. P. Kimman,* X. Campman, and J. Trampert

Abstract We study seismic noise recorded in the northeast of the Netherlands bybeamforming and by using empirical Green’s functions obtained by seismic interfero-metry. From beamforming we found differences in noise directions in different fre-quency bands. The main source region for primary microseisms (0.05–0.08 Hz) is inthe west-northwest direction, while the secondary microseisms (0.1–0.14 Hz) have awest-southwest back azimuth. Furthermore, we observed a fast (∼4 km=s) arrivalcorresponding to the Rayleigh wave overtone. This arrival is also in the secondarymicroseism band (between 0.15 and 0.2 Hz), but has a west-northwest back azimuth.Both arrivals in the secondary microseism band gain in strength during winter, as doesthe average wave height in the North Atlantic. We measured phase velocity dispersioncurves from both beamforming and noise cross-correlations, as well as group velocityfrom the latter. These are then jointly inverted for an average 1D S-wave model. Theresults show how the combination of different methods leads to a more complete char-acterization of the propagation modes and an improved knowledge of the subsurface,especially as the group velocity measurements increase the upper frequency limit ofanalysis, providing valuable information of the shallowest subsurface.

Introduction

Array studies of microseismic noise have gained in-creasing attention in recent years. This is partly becauseland-based measurements can be used for monitoring oceanwaves in areas where no buoy data are available (Bromirskiet al., 1999). More importantly, by cross-correlating seismicnoise measured at two locations, one can study the wave pro-pagation between the two stations. Many useful applicationsof this principle have been proposed since its first demonstra-tion in seismology (e.g., Gouédard, Stehly, et al., 2008; Snie-der et al., 2009; Wapenaar, Draganov, et al., 2010; Wapenaar,Slob, et al., 2010). Tomography using surface waves mea-sured from empirical Green’s functions has become a valu-able tool for determining high-resolution crustal structures.An important assumption in Green’s function retrieval is thatof an isotropic noise field, which is generally not valid. Agood understanding of the characteristics and distributionof seismic noise is therefore necessary for ambient noisetomography. Microseisms, the continuously excited back-ground oscillations present in seismogram recordings, aremainly caused by ocean disturbances and dominate the noisecontent in the frequency band that was considered in thisstudy (0.05 to 1 Hz). Microseisms mainly consist of funda-mental mode surface-wave energy (Bonnefoy et al., 2006);

therefore, the majority of noise correlation studies onlyrecover the fundamental mode surface wave. Recently, sev-eral studies reported the measurement of overtones (Harmonet al., 2007; Nishida et al., 2008; Brooks et al., 2009;Yao et al., 2011) or body waves (Roux et al., 2005; Draga-nov et al., 2007; Gerstoft et al., 2008; Draganov et al., 2009).The dominant spectral amplitude of ambient seismic noise isbetween 0.05 and 0.2 Hz. A distinction is often made be-tween the primary microseisms (0.05–0.08 Hz) and the oftenstronger, secondary microseisms (0.1–0.2 Hz). Primarymicroseisms have frequencies similar to the dominant oceanwaves, while secondary microseisms have a dominantfrequency twice as high. Both are related to ocean and atmo-spheric disturbances, but can have different origins becauseof differences in source mechanisms. Longuet-Higgins(1950) showed that nonlinear ocean-wave interaction cancause pressure oscillations at twice the frequency of theocean waves, therefore likely to be the cause of secondarymicroseisms. A requirement for this interaction is that oceanwaves travel in (nearly) opposite directions, for example, dueto coastal reflection of ocean waves (Haubrich and McCamy,1969). Previous studies demonstrated that the power andback azimuth of microseisms are indeed correlated withocean-wave heights (e.g., Essen et al., 2003; Chevrot et al.,2007), and even to individual storms (Bromirski, 2001) orhurricanes (Gerstoft et al., 2006). Kedar et al. (2008)

*Now at WesternGeco, Schlumberger House, Buckingham Gate,Gatwick, RH6 0NZ, West Sussex, United Kingdom.

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Bulletin of the Seismological Society of America, Vol. 102, No. 4, pp. 1388–1399, August 2012, doi: 10.1785/0120110069

combined information from wave interactions with bathyme-try to model the most likely source areas for secondarymicroseisms. In general, while the secondary microseismsseem to be generated by both coastal and deep ocean sources(Haubrich et al., 1963; Bromirsky and Dunnebier, 2002;Tanimoto et al., 2006; Chevrot et al., 2007), most studiesindicate a shallow origin for the primary microseisms(Hasselman, 1963; Cessaro, 1994; Gerstoft and Tanimoto,2007). Stehly et al. (2006), however, found primary micro-seisms generated in midocean, by backprojecting the cross-correlation functions.

Even though the original assumption of an isotropicnoise field is generally not met, strongly directional noise isnot an insurmountable problem for Green’s function extrac-tion. Convergence to the directly travelling part of theGreen’s function is faster, and weak arrivals are more likelyto be retrieved if the noise is directional (Kimman and Tram-pert, 2010), as long as the interstation paths are approxi-mately in line with the dominant source direction. If theazimuthal source coverage is very narrow but in line, a smallphase error (up to π=4) can occur for interstation paths smal-ler than three wavelengths (Tsai, 2009). Even interstationpaths slightly off the dominant source direction can stillproduce remarkably accurate cross-correlation functions(figs. 5, 6 in Kimman and Trampert, 2010; fig. 10 in Peder-sen et al., 2007).

We used a combination of beamforming and the cross-correlation technique to determine the nature of the micro-seisms and obtain a 1D shear wave speed structure beneaththe studied area. To identify the noise content, we installed atemporary array in the northeast of the Netherlands (Fig. 1).Noise can be expected to originate from both the nearbyNorth Sea, as well as the Atlantic Ocean. The North AtlanticOcean is well-known for its capacity to generate microseismsthat can be measured in Europe and North America (Essenet al., 2003; Kedar et al., 2008), especially during wintertimes (Stehly et al., 2006). Schmalfeldt (1978) located pri-mary microseisms acting near the German coast, while thesecondary microseisms originated near the Norwegian coast.This area was also found to be a strong generator for micro-seisms by Friedrich et al. (1998), Essen et al. (2003), andothers. Darbyshire (1991) furthermore described secondarymicroseism sources in the North Channel and the BristolChannel with arrays located in Wales.

Data and Preprocessing

We aim at a higher resolution compared with standard(crustal) noise studies, having wavelengths down to less than1 km, which means analyzing higher frequencies and shorterinterstation distances. However, for dispersion analysis thefrequency band should not be too narrow. To compromise

Figure 1. (a) Location of the array, and (b) station configuration. (c) The resulting distribution of distances and (1-way) azimuths for allinterstation paths. (d) A typical array response. In this example, we computed the response to a plane wave coming in at 315°, with a slownessof 0:4 s=m and a frequency of 0:175 Hz.

Fundamental and Higher Mode Energy Observed in the Northeast of the Netherlands 1389

between these constraints, the 15 broadband (Trillium 120P)seismometers were placed in a circular configuration, so thatthe interstation distances varied between 3.1 and 46.6 km.The stations were roughly positioned in four radii (2.5, 8,15, and 23 km) centered in the Annerveen region in theNetherlands (53.085° N, 6.765° E; see Fig. 1). This config-uration provides a wide range of interstation distances, aswell as a reasonably evenly distributed azimuthal coverage(Fig. 1c). The recording was continuous from September2007 until June 2008.

Figure 1d shows the resulting array response for anincoming plane wave with back azimuth of 315° and a fre-quency of 0.15 Hz, travelling at 2:5 km=s. By analyzing theresponse over a range of slownesses and frequencies, weestimate that beamforming measurements from the array arereliable in the frequency band between 0.05 and 0.35 Hz.Slownesses and azimuths retrieved for frequencies lowerthan 0.05 Hz suffer from low resolution, while at higherfrequencies the response suffers from spatial aliasing.

The vertical component displacement data are prepro-cessed in 12-hr segments. These are corrected for trend andmean, after which the instrument response is removed, and a4-point zero-phase band-pass Butterworth filter is applied(corner frequencies 0.025 and 4 Hz). The data are subse-quently downsampled (to 8 Hz sampling rate) and frequency-equalized by inversely weighting the complex signal with asmoothed version of the amplitude spectrum.

Beamforming Output

To interpret the retrieved Green’s functions, it is essen-tial to have a good understanding of the distribution of thenoise sources (Marzorati and Bindi, 2008; Koper et al., 2009;Yao et al., 2009). We analyzed the noise source distributionwith beamforming (e.g., Rost and Thomas, 2002; Tanimotoand Prindle, 2007; Gerstoft and Tanimoto, 2007). We use afrequency-domain beamforming (FDB) algorithm based onLacoss et al. (1969).

This produces an estimate of the frequency-wavenumber(f-k) power spectrum of the noise field by calculating for eachfrequency, the power of the noise field along azimuths of syn-thetic plane waves with a range of wavenumber vectors k.

This is done under the assumption that the heterogeneityacross the array can be ignored. Instead of calculating thepower as a function of the horizontal wavenumbers, we usethe fact that the horizontal slowness is proportional to themagnitude of thewavenumber vector, while the back azimuthgives its direction. The FDB output is then a 3D volume offrequency, horizontal slowness, and back azimuth. We showtwo types of displays from this volume: (1) cross-sectionslices with constant frequency-bins, and (2) slowness-binslices. The former is the standard beamforming output, whilethe latter emphasizes the difference in directionality of themicroseism sources at different frequencies. In this case,we consider the sum over all slownesses.

In the process of Green’s function retrieval, the cross-correlations are averaged over long periods of time. Wetherefore investigate long time averages of the source distri-bution. Figure 2 shows the beamforming output as a functionof slowness and back azimuth for three different frequenciesfor the month of January. The data were averaged over allearthquake-free days. The selected frequencies correspondto the primary microseism (0.07 Hz) and two peaks in thesecondary microseism band (0.12 and 0.16 Hz). We note thatthe primary microseism signal (Fig. 2a) has a maximum inthe western direction, but is present over a large back-azimuth range. The secondary microseism signal (Fig. 2b)source distribution is more localized. The source is locatedto the west-southwest, with its associated wave field travel-ling at around 2:5 km=s. At 0.16 Hz, we see a shift in backazimuth. A clear maximum can be seen, corresponding totravelling waves with a velocity of nearly 4 km=s and origi-nating from a source with a back azimuth of 290°. Such a fastvelocity could be an indication of higher-mode Rayleighwaves. The difference in characteristics between 0.12 Hzand 0.16 Hz is better visualized if we plot the radial axisof the beamforming output as a function of frequency insteadof slowness. This is shown in Figure 3 for each of theavailable nine months of data. The figure shows the sumof the FDB output power over slownesses in the range of0:5–0:2 s=km as a function of frequency and back azimuth.Distinct variations of directionality with frequency areclearly visible. Three peaks can be recognized, correspond-ing to the primary and secondary microseisms, the latter

Figure 2. Beamforming output for the month of January. On the radial axis is slowness (0:2 − 0:5 s=km), plotted for (a) 0.07, (b) 0.12,and (c) 0.16 Hz. The color-code represents energy in dB. The first secondary microseisms (b) have a more west-southwest back azimuthcompared with the primary microseism (a). At 0.16 Hz, still in the secondary microseism bandwidth, a fast (3:5–4 km=s) arrival can bedistinguished with a west-northwest directionality.

1390 W. P. Kimman, X. Campman, and J. Trampert

composed of two distinct peaks. The strength of the primarymicroseisms stays roughly constant. The relative strength ofthe secondary microseisms builds up during winter.

Figure 4 shows the backprojected dominant sourcedirections for January 2008. The solid lines indicate the direc-tions associated with the maximum average power for the

primary microseismic source (P) and two secondary micro-seismic sources (S1 and S2). While the array has the closestproximity to the coast to the north, the primary microseismback azimuth is dominated by a west-northwest direction.Excitation of the primary microseisms is in the broad rangebetween 270° and 360°. Stehly et al. (2006) report a similarwest-northwest directionality (from cross-correlation ampli-tudes) from an array covering western Europe. Under thegenerally accepted notion that the primary microseism oftenhave a shallow origin, we consider the nearby North Sea coastas the most likely candidate for the generation of primarymicroseisms.

In Figure 5 we show the average wave height for thesame months corresponding to Figure 3 (Swail et al., 2006).Both the average pattern and the strength with time seem tocorrespond roughly to the secondary microseisms. Essenet al. (2003) find a predominant source region off the coastof Norway for the secondary microseisms recorded inGermany. We observe, however, a more western directional-ity (or even west-southwest for some months) for secondarymicroseisms travelling through our array. Storms near theBritish coast (Fig. 5) are possible sources for this secondarymicroseism, with the likely mechanism to be opposing

Figure 3. The beamforming output as a function of back azimuth and frequency, for nine different months. The radial axis representsfrequency from 0.05 to 0.3 Hz. The figure is compiled from stacks of earthquake-free days for each month, and all slownesses between 0.2and 0:5 s=km. The months of September and May are represented by six days only.

75°W 60°W

45°W 30°W 15°W 0°

15°E

30°N

45°N

60°N

Figure 4. Backprojection of the azimuths of the beamformingmaxima for the month of January. We considered the primary (P)and two secondary (S1, S2) microseism maxima, backprojectedalong the great-circle path.


waves, which generally cause (1) a higher amplitude peakand (2) a peak close to twice the primary frequency (whichis mostly true for S1). The English Channel has a geometrythat lends itself well to the necessary interference of oppo-sitely travelling ocean waves, described by Friedrichet al. (1998).

The relative high-frequency content of the fast arrivalsuggests it is generated by a mechanism in which the naturalspread of wave energy generates interactions between thespectral components of a single active sea; hence, no oppos-ing waves are necessary for this interaction (Kibblewhite andEwans, 1985). The transition from a deep-water approach toa narrow continental margin was suggested to be the expla-nation for the occurrence of higher modes in the study ofBrooks et al. (2009). The frequency range where we observea possible higher mode corresponds to theirs. Also, other pre-vious studies (Harmon et al., 2007; Yao et al., 2011) reportedthe observation of higher modes between 0.14 and 0.28 Hz.Kedar et al. (2008) described how microseisms can be gen-erated from nonlinear ocean-wave interaction in the AtlanticOcean, where bathymetry is an important factor. More workis needed, however, to explain the excitation of the highermodes predominantly in this frequency band. Kedar et al.(2008) furthermore show an area south of Greenland to be

a main contributor to (secondary) microseisms in the NorthAtlantic. The back azimuth to this region from our array is290°–300°, corresponding with the third peak (S2) inFigure 3. However, the largest wave heights during the win-ter period have occurred in the Rockall Plateau/RockallThrough region (the northwest of Ireland) (same back azi-muth, Fig. 4). Furthermore, the latter region also containsa narrow continental margin with steep differences in bathy-metry. Therefore, this area is likely to be the source for thegeneration of higher mode microseisms. Further studiesusing more than one array should pinpoint the noise sourcemore precisely.

Empirical Green’s Functions

As already mentioned, the data are preprocessed into478 half-daily traces. To obtain empirical Green’s functions,the traces are 1-bit cross-correlated in short time segments(156 s) with overlapping windows (of 62.5 s). These arelinearly stacked to form 478 cross-correlation functions. Ifthe mean amplitude of the short time segment is larger thana specific threshold (six times the root mean square [rms]amplitude, averaged over the half-day), the segment isrejected. This is done (additionally to taking the 1-bit

Figure 5. The average wave height in the North Atlantic per month, over the recording period (Swail et al., 2006).


correlation) to prevent bias from local noise, teleseismicevents, and induced microseismicity that is commonly pre-sent in the region due to hydrocarbon exploration (van Ecket al., 2006). In the frequency domain, the empirical Green’sfunction is given by

Gim�xA; xB;ω� − G�im�xA; xB;ω�

≈ 2iωA�ω�hui�xA;ω�u�m�xB;ω�i; (1)

where � is complex conjugation, and h i stands for averagingof the spatial ensemble of sources (Wapenaar and Fokkema,2006), because we correlate noise displacements (u)resulting from sources that act simultaneously in time.Gim�xA; xB;ω� is the displacement Green’s function in thei-direction at location xA, due to an impulsive point forcein the m-direction at location xB. Because the amplitude can-not be expected to be retrieved accurately, we neglect thesource spectrum and the frequency dependent amplitudefactor A�ω� that should theoretically be considered (Hallidayand Curtis, 2008; Kimman and Trampert, 2010). Asequation (1) implies, a negative time derivative (iω in thefrequency domain) is required to obtain the empiricalGreen’s function from the noise cross-correlation function.

We treat the causal and anticausal part of the cross-correlation functions separately (they are not averaged). Theresulting 478 half-days are stacked with a (time-frequency)phase-weighted stacking technique (Kimman, 2011; Schim-mel et al., 2011). A phase-error up to π=4 can occur, depend-ing on the source distribution (Lin et al., 2008; Tsai, 2009). Ifthe average interstation distance D is larger than roughlythree wavelengths, a coverage of �25° around the stationaryphase azimuth is sufficient for the resulting phase velocityerrors to be small (Kimman and Trampert, 2010). AlthoughLin et al. (2008) show this not to be a problem for themajority of measurements, to be on the safe side, we filteredout the early arriving energy in the time-frequency domain,for which 2.5 wavelengths are less than the interstation dis-tance D. [This corresponds to anything arriving before2:5=�t · f� in the time (t)–frequency (f) domain.] Therefore,we assume that the π=4 term occurring in the Green’s func-tion has been retrieved correctly, and we apply this phaseshift to the seismograms to maintain only the phase differ-ence between two stations due to propagation. The retrievedempirical Green’s functions are plotted as a function of dis-tance in Figure 6.

The correlation gather shows a fast low-frequency dis-persive arrival. At higher frequencies (>0:5 Hz), a dominantphase of approximately 550–600 m=s can be traced acrossthe array. Also, an even slower (very weak) arrival appearsacross the shortest interstation distances, with an apparentspeed of 320–350 m=s. Our region of interest is character-ized by soft sediments in the uppermost kilometers(TNO-NITG, 2004). For surface waves this leads to a largevelocity drop between low-frequency waves sensitive to thecrust (and deeper) and frequencies confined, and sensitive

only, to the first few hundreds of meters depth. Estimatingthe structural characteristics below the array in terms of sur-face wave modes can give insight in determining the modalcomposition of the noise wave field. We will estimate the 1Daveraged velocity profile by an inversion of the averagedispersion curves.

Dispersion Measurements

Phase Velocity from Beamforming

The beamforming output is a 3D array A�c; f;ϕ�,describing the average phase velocity c as a function of fre-quency f, corresponding to different back-azimuths ϕ.Hence, we can pick a dispersion curve from this volume.In Figure 7, we plot the Rayleigh wave phase velocities fromDecember and January, for the dominant back azimuths ofboth the 275°–305° (west-northwest) and 235°–265° (west-southwest) range. Based on the power of the beamformingestimate (Fig. 3), the green curves indicate the phase velocityof the first secondary microseism S1, between 0.1 and0.14 Hz; the red curves show the dispersion curves below(P), and above (S2) in this frequency range. There is goodconsistency at the lower frequencies, and the transitionbetween P and S1 (∼0:1 Hz) is rather smooth. At 0.14–0.23 Hz, the S2 microseism signal is characterized by highvelocities. (A slower, secondary maximum is not observed.)

0 20 40 60 80 1000

10

20

30

40

50

Time (s)D

ista

nce

(km

)

Empirical Green’s functions, 0.05 to 1 Hz

Figure 6. The retrieved Green’s functions. Shown is the causalpart retrieved �90° around the main noise direction (290°). Thephase velocity is analyzed from the low frequency, fast travellingarrivals (green triangle). The group velocity is estimated fromthe higher frequency slow arrivals (red triangle).


We will show that the jump at 0.15 Hz is the transition of thefundamental mode to the first overtone.

Phase Velocity from Empirical Green’s Functions

We measure the average phase velocity as a function offrequency over the complete gather of retrieved interstationGreen’s functions. Therefore, we again make the approxima-tion of a plane wave travelling in a 1D-layered medium, andtransform the gather to the f-k domain. First, the Green’sfunctions are (zero-phase) band-pass filtered (between 0.05and 0.4 Hz), and a cosine taper is applied to select the time-window between velocities travelling with an upper andlower limit (here 1.2 and 4:5 km=s; green triangle in Fig. 6).

We include the interstation paths that are in the main noisedirection, 290° � 45°. The maximum coherence in the f-kdomain is analyzed to estimate the velocity as a function offrequency. In Figure 8 we show the obtained dispersioncurves. We experimented with excluding paths with differentorientations with respect to, and smaller coverage around, themain source back azimuth, but this yielded relative minordeviations from the result shown. For comparison to thebeamforming result, the phase velocity extracted from theGreen’s functions (the maximum in Fig. 8) is shown in blackin Figure 7. It is in agreement with the beamforming esti-mates, but provides additional information at high frequen-cies. We interpret both the segments at low (<0:14 Hz) andat higher (>0:2 Hz) frequencies to be a fundamental mode,and the band in between to be a higher mode.

Uncertainties in Phase Velocity Measurements

All measurements are shown in Figures 9–11, togetherwith their error bars. For beamforming, picks have beenmade for three-day averages of phase velocity from thepower estimate, and the error bars represent the 95% confi-dence (�2σ) interval from their variation during the monthsof December and January (during which the directionalitywas reasonably stable). For the Green’s function estimate,the accuracy, defined by a drop to half the maximum (a dropof 6 dB) in Figure 8, has been taken as uncertainty. Given thestability of the beamforming estimates this is most likelyconservative compared with computing the variance frommultiple measurements of the maximum over time. Com-pared with the beamforming result, the dispersion from theGreen’s functions is less accurate at the lowest frequencies,but it can, however, be traced to higher frequencies.

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.41.5

2

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5

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Green’s functions235−265 degrees275−305 degrees

P

S2

S1

Figure 7. Phase velocity estimation from beamforming andGreen’s function analysis. We estimated the maximum of the beam-form output for each back azimuth. The green curves represent theback azimuths 235°–265° (with an increment of 1° per curve), andthe red curves represent the back azimuths 275°–295°. The phasevelocity estimated from the Green’s functions is shown in black.

Figure 8. The left panel shows the causal part of the estimated Green’s functions�45° around the main noise direction, band-pass filteredbetween 0.05 and 0.45 Hz. The estimated f-k velocity estimation is shown on the right. The maximum value for each frequency (black) istaken as the dispersion curve measurement.


Group Velocity from Empirical Green’s Functions

To make use of all available information, we also in-cluded measurements of the interstation group velocity. Forthe lowest frequencies, phase velocity was the most suitablemeasurement, because the average velocity across the arraycould be considered. However, above 0.4 Hz the phasevelocity estimation will be aliased. Therefore, we turn tomeasurements of group velocities, because the empiricalGreen’s functions contain information above this frequency.We use the well-established frequency–time analysis methodFTAN (Levshin, 1972) to measure the maximum energy ofthe individual empirical Green’s function in the time-frequency domain, which gives additional information aboutthe higher frequencies (red triangle in Fig. 6). Again, we only

measure the empirical Green’s function between 290° � 45°back azimuth. FTAN employs a floating filtering technique toseparate the fundamental mode from the rest in the signal,and enables the user to pick the dispersion curve onthe smoothed time-frequency representation of the empiricalGreen’s function. We averaged the dispersion curves over thenumber of interstation paths we considered and calculatedthe standard deviation from the variation over the differentpaths. The resulting group velocity curve (Fig. 11) is rela-tively smooth and ranges between 0.7 to 0:5 km=s and 0.4to 1 Hz. The error bars represent the 95% confidence intervalcalculated from the variation over the different paths used foreach frequency. We will compare jointly inverting all disper-sion curves with the case where we invert for phase veloc-ity only.

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m/s

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m)

Best 1000 models, M0 constrained

Figure 9. (a) The measured phase velocities obtained from both beamforming (the two low-frequency segments) and empirical Green’sfunctions (higher frequencies). The best-fitting 25 dispersion curves are plotted in red. Dashed red, the first higher mode dispersion curves areplotted. (b) The S-wave profiles belonging to the best 1000 models. The misfit is shown as shades of gray; the darker the gray, the smaller themisfit. The profile corresponding to the smallest misfit is shown in green.

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Best 1000, M0 and M1 constrained

Figure 10. (a) The joint misfit for both the fundamental and the higher mode Rayleigh waves. (b) The resulting S-wave profiles for thebest 1000 models. Details as in the caption for Figure 9.


1D Model Estimation

To gain further understanding of the prevailing noisewave field in terms of propagating surface wave modes, astructural inversion is performed. Similar to Brooks et al.(2009), we aim to confirm the presence of fundamental andhigher modes by a known background model.

Our measured dispersion profiles are characterized by asharp velocity drop. This means that the higher frequenciesare confined to the low-velocity layers. The resulting inver-sion is highly nonlinear and nonunique; a large range ofpossible models could produce such dispersion curves. Wetherefore perform a Monte Carlo search to explore the para-meter space, minimizing the rms misfit between the observedand computed dispersion curves.

We base a prior starting model on well data (the upperfew hundred meters), and the 3D structural model ofTNO-NITG (2004). Because for the latter only P-wave struc-tural information is available, we use empirically obtainedrelations derived for sand and mudstones for an estimate of

shear wave speed (Castagna’s relation; Castagna et al., 1985)and density (Gardner et al., 1974). The average thickness ofthe Moho in the region is taken to be 28.5 km (Remmelts andDuin, 1990); below this depth we use values from the pre-liminary reference Earth model (PREM; Dziewonski andAnderson, 1981).

To retrieve the S-velocity structure at shallow andintermediate depth, we parameterize our model as shown inTable 1. The sampling is performed according to a Gaussiandistribution around the prior model. In total, 106 models werecreated and 6 parameters were varied. In addition, a require-ment is implemented for the models to have only an increas-ing velocity with depth, and the Moho must be within 15% ofthe average Moho depth. We compute the misfit of our mea-sured dispersion curves with the theoretical phase velocitiespredicted from these models, computed with Herrmann’sprograms in seismology (Herrmann, 1978).

The best-fitting 25 dispersion curves are shown inFigure 9. In this example, only the two fundamentalmode phase velocity curves have been used to constrain

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U (

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z (k

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Best 1000, phase & group

Figure 11. The joint misfit for (a) all phase velocity segments and (b) group velocity measurements, and (c) the resulting 1000 bestS-wave profiles. Details as in the caption for Figure 9

Table 1Summary of the Explored Models*

Thickness (km) α (km=s) β (km=s) ρ (kg=m3)

1. Shallow subsurface 0.03 1.7 0.34 0.72. Upper North Sea 0.47(1� σ · p1) 1.8 0.7(1� σ · p2) 0.31α1=4

3. Lower North Sea 0.65(1� σ · p1) 2.4 0.9(1� σ · p2) 0.31α1=4

4. Chalk 1.4(1� σ · p3) 3.8 2.10(1� σ · p4) 0.31α1=4

5. Rhineland-Limburg 5(1� σ · p5) 4.1 2.36(1� σ · p6) 0.31α1=4

9. Lower crust 20.9(1� σ · p5) 6.2 3.5(1� σ · p6) 2.610. Mantle 1 185.6 8.0 4.4 3.611. Halfspace ∞ 10.0 5.2 3.9

*The layer thicknesses and P-wave speeds are representative (but simplified) of theformations in the area (TNO-NITG, 2004). The six parameters p1 to p6 are random values,Gaussian distributed around 0. σ denotes the standard deviation of these parameters andis 0.15 always. The standard deviation of the sampled models is therefore always 15%around the mean value. The relations between P-wave speed and S-wave speed(β � �α − 1:36�=1:16) and density (0:31α1=4) are Castagna’s relation and Gardner’s rule,respectively.


the inversion. The layer thicknesses and shear wave speedsof the best 1000 models are plotted in Figure 9b. At the low-est frequencies the beamforming measurements become un-reliable (considering the large wavelengths compared withthe size of the array) and, if included, the inversion resultsshow overestimation in velocity for all models. We thereforerestricted the inversion to include only frequencies above0.07 Hz. The inversion of the fundamental mode phase ve-locity produces models that explain the overtone measure-ment well. Inverting for both the first higher mode and thefundamental mode, as shown in Figure 10, produces similarmodels but with smaller misfits for the first higher mode,leading to reduced uncertainty in the posterior distributionof S-wave velocities and depths.

Further improvement is achieved if we also include thegroup velocity measurements at higher frequencies (Fig. 11).We focused on the high frequencies because these provideadditional information and are measured more reliably thanlow frequencies, given the larger interstation distance towavelength. The additional information does not deterioratethe phase velocity misfit, but indeed improves our knowl-edge of the shallowest (the upper 1.5 km) subsurface,reflected in small variations in the models (Fig. 11c).

Discussion and Conclusions

By means of beamforming, analysis of retrieved Green’sfunctions, and synthesizing this information into a 1D inver-sion, wewere able to determine the noise characteristics of themost dominant propagation modes measured over theAnnerveen array between 1 and 15 s. Even though analyzingnoise cross-correlation functions in a 1D fashion is intrinsi-cally similar to beamforming, the two methods are comple-mentary in the frequency range for extraction of the phasevelocity. In both methods the maximum coherence is soughtfor a trial slowness between each station pair, under a planewave assumption. In the Green’s function method, the illumi-nating wave field is already assumed to be isotropic, while inthe beamforming, the actual (apparent) velocity is obtainedfor all directions. Beamforming covers the low-frequencyregime (<3wavelengths), because only a phase-shift of planewaves arriving at different stations are analyzed. In contrast,the notion of far-field surface wave Green’s functions is inap-propriate in this regime. In contrast, the Green’s functionmethod is more successful at higher frequencies. In itsassumption of an isotropic wave field, the 2D array is effec-tively projected on a 1D problem. The spatial aliasing limit isincreased because the Shannon criterion is based on thesmallest difference between interstation distances and not oninterstation distances themselves. This was discussed byGouédard, Cornou, and Roux (2008) and confirmed by nu-merical simulations. The introduced error from directionalnoise can be corrected for, but as we see, even by a simpleselection along the dominant source back azimuth, we are ableto reach higher frequencies for the analysis of phase velocity.The 1-bit cross-correlation and the time-frequency phase-

weighted stacking are two important processing steps thatcontribute to the improvement of the signal to noise ratioof the empirical Green’s functions. Furthermore, the long-term averaging process that favors weak sources (coherentonly over long time periods) might have played a role in in-creasing the high-frequency limit for the phase velocity mea-surement. Including the higher mode information reduces thevariation in acceptable velocity models, thereby increasingvertical resolution of the inversion.

The soft sediments characterizing the studied area causea steep drop in the dispersion profile, but as a result, the high-amplitude, high-frequency noise confined to the shallowestlayers enabled retrieval of empirical Green’s functions overrelative long distances; arrivals with wavelengths of 500–700 m can be observed over distances up to 30 km. As aresult, including the average of group velocity measurementsenables us to analyze frequencies higher than the aliasingfrequency for phase velocity measurements. Furthermore,our results suggest that higher mode energy may be moredominantly present in microseismic noise than previouslythought. The observed overtone has a distinctly different ori-gin compared with the fundamental mode in the secondarymicroseism. A possible source region, based on the maxi-mum wave height in winter and the existence of large varia-tions in bathymetry, could be the Rockall Plateau/RockallThrough (the northwest of Ireland). In the analysis of theGreen’s functions it is essential to carefully account forovertones, because they could bias the dispersion analysis offundamental modes. In our case, they overlap the fundamen-tal mode over a significant frequency band. It remains to beexplained why they seem to be present mainly in the specificfrequency band between 0.15 and 0.2 Hz. The observation ofthe first overtone in the secondary microseism may provideadditional constraints on the mechanism of excitation ofmicroseismic noise.

Data and Resources

The data used in this study were acquired by the authorsand are available upon request.

Acknowledgments

We thank Peter Gerstoft for the initial version of his beamformingcode. This work is part of the research programme of the Stichting voorFundamenteel Onderzoek der Materie (FOM), which is financially sup-ported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek(NWO).

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Department of Earth SciencesUtrecht UniversityP.O. Box 800213584 CD UtrechtThe Netherlands

(W.P.K., J.T.)

Shell Global Solutions International B.V.Kessler Park 12288 GS RijswijkThe Netherlands

(X.C.)

Manuscript received 3 March 2011


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