Threedimensional shear wave velocity imaging by ambient

Originally published as: Pilz, M., Parolai, S., Picozzi, M., Bindi, D. (2012): Three-dimensional shear wave velocity imaging by ambient seismic noise tomography. - Geophysical Journal International, 189, 1, pp. 501—512. DOI: http://doi.org/10.1111/j.1365-246X.2011.05340.x

Geophys. J. Int. (2012) 189, 501–512 doi: 10.1111/j.1365-246X.2011.05340.x

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Three-dimensional shear wave velocity imaging by ambient seismicnoise tomography

Marco Pilz, Stefano Parolai, Matteo Picozzi and Dino BindiHelmholtzzentrum Potsdam, German Research Center for Geosciences, Helmholtzstr.7, 14467 Potsdam, Germany. E-mail: [email protected]

Accepted 2011 December 16. Received 2011 December 15; in original form 2011 October 5

S U M M A R Y3-D shear wave velocity images are of particular interest for engineering seismology. To obtaininformation about the local subsoil structure, we present a one-step inversion procedure basedon the computation of high-frequency correlation functions between stations of a small-scalearray deployed for recording ambient seismic noise. The calculation of Rayleigh wave phasevelocities is based on the frequency-domain SPatial AutoCorrelation technique. Constitutively,a tomographic inversion of the traveltimes estimated for each frequency is performed, allowingthe laterally varying 3-D surface wave velocity structure below the array to be retrieved. Wetest our technique by using simulations of seismic noise for a simple realistic site and by usingreal-world recordings from a small-scale array performed at the Nauen test site (Germany).The results imply that the cross-sections from passive seismic interferometry provide a clearimage of the local structural heterogeneities and the shear wave velocities are satisfactorilyreproduced. The velocity structure is also found to be in good agreement with the results ofgeoelectrical measurements, indicating the potential of the method to be easily applied forderiving the shallow 3-D velocity structure in urban areas and for monitoring purposes.

Key words: Interferometry; Surface waves and free oscillations; Seismic tomography.

1 I N T RO D U C T I O N

The shear wave velocity, vs, is a key parameter related to the assess-ment of the local amplification of ground motion and is most com-monly used in engineering seismology. Examples include empiricalpredictions of strong ground motion (e.g. Boore et al. 1997), sitecoefficients for building codes (BSSC 2004), liquefaction potentialcharacterization (e.g. Stokoe & Nazarian 1985; Andrus & Stokoe2000) as well as input for numerical simulations of the response ofsedimentary basins (e.g. Bao et al. 1998; Pitarka et al. 1998; Pilzet al. 2011). Within this context, the shear wave velocity valuesare usually measured in situ by means of various invasive (bore-hole) or non-invasive (shear wave refraction and reflection studies)techniques. Although these techniques provide accurate and well-resolved values of vs they suffer from several drawbacks, such asincreasing costs with required depth or only pointwise estimates.

In recent years, passive seismic techniques have received consid-erable attention. The success of these applications is explained bythe fact that these methods are based on surface waves, which areby far the strongest waves excited by ambient seismic noise. Manyrecent studies on passive seismic processing have focused mainly ontwo applications: The simplest one, the single horizontal-to-vertical(H/V) technique, requires one station’s recordings only to derive lo-cal site conditions and uses the vertical component as a reference. Ifa strong impedance contrast between the sediments and the under-lying bedrock is present, the peak of the H/V spectral ratio appearsat the fundamental resonance frequency of the soil and is correlated

to the shear wave velocity and the thickness of the low-velocitylayers. For a known thickness of sediments, the shear wave velocitystructure of the subsoil can then be obtained (e.g. Fah et al. 2001,2003; Scherbaum et al. 2003; Parolai et al. 2006; Pilz et al. 2010).

Since the beginning of this century, another procedure, seismicinterferometry, has rapidly become popular in a variety of applica-tions (see reviews by Campillo 2006; Curtis et al. 2006; Wapenaaret al. 2008; Schuster 2009; Snieder et al. 2009). One of the mostintriguing applications of the method, shown both theoretically andexperimentally, is that a random wavefield has correlations, which,on average, take the form of the Green’s function of the media(e.g. Rickett & Claerbout 1999; Lobkis & Weaver 2001; Weaver &Lobkis 2001; Snieder 2004; Wapenaar 2004; Sabra et al. 2005a, b).Between pairs of receivers, the Green’s function can be extractedfrom cross-correlations of ambient noise recorded at both receivers,in turn allowing an estimate of the propagation delay between thestations.

Such traveltime measurements of Rayleigh waves reconstructedfrom seismic noise have been used at lower frequencies to producehigh-resolution images on continental (Yang et al. 2007; Bensenet al. 2008) and regional (Shapiro et al. 2005; Kang & Shin 2006;Yao et al. 2006; Lin et al. 2007; Moschetti et al. 2007) scales.However, traveltime inversion is a non-linear problem since the raypath is velocity-dependent. Consequently, small changes to the in-terface position might consequently also produce changes in raytrajectories. In contrast to global tomographic studies we are notseeking small perturbations to an established reference model, but

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we rather attempt to locate clear velocity differences on a smallerlocal scale. Within this context, the applicability of the method hasalso been shown for higher frequencies at local scales (Chavez-Garcıa & Luzon 2005; Brenguier et al. 2007; Picozzi et al. 2009).Since, keeping in mind the level of errors of the input data for shal-low seismic surveys, small deviations of the paths from a straightline can be tolerated. However, the use of high-frequency seismicnoise interferometry requires a better understanding of the originof seismic noise and of the spatial and temporal distribution of itssources (e.g. Pederson & Kruger 2007; Halliday & Curtis 2008,2009). In particular, it remains important to establish reliable con-ditions under which noise can be considered to be well randomized.

Even if the origin of seismic noise is not fully known, Aki (1957)observed that cross-correlations between stations aligned in differ-ent directions did not differ substantially, concluding that ‘we mayregard the microtremor as being propagated in every direction, eachwith almost uniform power’. Subsequently, Asten (2006), Bensenet al. (2007) and Harmon et al. (2010) stated that the effect of inho-mogeneous source distributions on isotropic velocity estimates willbe likely to be within the error bars of isotropic velocity studies.

Among several studies, there are two approaches in the interpre-tation of the correlation of seismic noise between pairs of stations.The first one, based on diffusion theory, has shown that it is pos-sible to estimate the time domain Green’s function in a mediumfrom measurements of ambient seismic noise (Snieder 2006). Thesecond approach is based on the SPatial AutoCorrelation (SPAC)method (e.g. Aki 1957, 1965, Cox 1973).

Several authors (e.g. Chavez-Garcıa & Luzon 2005; Sanchez-Sesma & Campillo 2006; Chavez-Garcıa & Rodriguez 2007; Yokoi& Margaryan 2008) have demonstrated the equivalence between theresults obtained by both techniques for a small-scale experiment ata site with a homogeneous subsoil structure. Further, observationalstudies of ground motion (Prieto & Beroza 2008) and attenuation(Prieto et al. 2009; Lawrence & Prieto 2011) have presented anoptimistic picture of this ability to exploit real-world measurementsof the amplitude of ambient noise. However, for non-homogeneoussubsoil conditions (2- or 3-D structure), the SPAC method mightonly provide an average estimate of the S-wave velocity structure.On the other hand, one can expect that, similarly to what is obtainedover regional scales, local heterogeneities will definitively affect thenoise propagation between sensors, and hence, can be retrieved byanalysing the Green’s function estimated by the cross-correlationof the signals recorded at these stations.

Nowadays, retrieving the local 3-D S-wave velocity structurequantitatively by the analysis of ambient vibrations has gained con-siderable attention as a low-cost tool (e.g. Picozzi et al. 2009;Renalier et al. 2010). In these studies, however, the 3-D S-wavevelocity structure is obtained in a two-step approach by first cal-culating fundamental group or phase velocity dispersion curves.The second step involves an inversion of the dispersion curves forobtaining local 1-D S-wave velocity-depth profiles, with the lowerfrequency component constraining deeper structures than the high-frequency parts of the waves. Using this approach the final 2- and3-D S-wave velocity models can only be obtained by interpolationbetween the individual 1-D S-wave velocity profiles.

In this paper, we propose a new method for a one-step tomo-graphic inversion scheme based on seismic noise that allows oneto obtain an approximate 3-D S-wave velocity model that is use-ful for seismic engineering, as well as microzonation and mon-itoring purposes. This is feasible because the S-wave velocity isthe dominant property for the fundamental mode of high-frequencyRayleigh waves. Since traditional sensitivity kernels are rather prob-

lematic in 3-D (Marquering et al. 1999), we solve this problem by aweighted 3-D inversion procedure. Hereunto, we will first describehow Rayleigh wave phase velocities are calculated. To validate theprocedure, a synthetic example is presented, including a discussionon some of the problems that can be encountered in tomographicinversion. We further test the performance of the technique usingreal data collected at Nauen test site in northern Germany.

2 M O D E L L I N G O F S E I S M I C N O I S E

The expression ‘ambient seismic noise’ is a generic term for bothlow- and high-frequency signals. Cultural activities are the originof the high-frequency signals, also called microtremors, whereasoceanic and atmospheric disturbances result in the low-frequencysignals or microseisms (Longuet-Higgins 1950). For the followinganalyses, we only consider cultural noise higher than a few hertz.Such synthetic wavefields of the cultural noise can be modelled asa distribution of impulsive point forces located at the surface orsubsurface with random force orientation and amplitude (Lachet &Bard 1994). Following Coutel & Mora (1998), the force term in thewave equation can be written as,

fx (x, t) = a (x, t) sin [2πθ (x, t)]fz (x, t) = a (x, t) cos [2πθ (x, t)]

, (1)

with θ being the orientation of the force relative to the vertical and abeing the amplitude of the force. To simulate a random distributionof forces (both in direction and amplitude), θ and a are taken asuniformly distributed random numbers between 0 and 1.

For our first test, noise synthetics were computed using the spec-tral element code GeoELSE, developed by the Center for Ad-vanced Studies, Research and Development, in Sardinia and theDepartment of Structural Engineering of the Politecnico di Milano(Faccioli et al. 1997). The code was developed for the analysis ofvarious wave propagation phenomena, including man-made vibra-tions and allows for sources and receivers at shallow depths.

For the calculation of noise synthetics, we consider a simplephysical model representative of a soft-material site. It consistsof a homogeneous layer, in which a second layer dipping at 20◦

and characterized by a much higher S-wave velocity is embedded(Fig. 1). For the wavefield generation, a random force is applied ateach time step at 168 different locations, arbitrarily distributed overthe entire modelling surface (Fig. 2). The final data set is composedof 30 min of noise synthetics; such a duration is similar to thatusually considered in real-world experiments. Examples of noisesynthetics, bandpass filtered between 2 and 25 Hz and recorded atone receiver, are shown in Fig. 3(a). Fig. 3(b) presents the verticalGreen’s function and its envelope calculated for a central frequencyof 8 Hz. It can be seen that, in general, the Green’s functions arefound to be rather symmetric, pointing out an almost uniform noisesource distribution surrounding the two stations, which is requiredby the theory.

3 C A L C U L AT I O N O F R AY L E I G H WAV EP H A S E V E L O C I T Y

We first determined the Rayleigh wave phase velocities, given thatthe uncertainty in the phase velocity measurement is much less thanthat of the measurement of group velocity (Bensen et al. 2008). Thecalculation of the phase velocities is based on the frequency-domainSPAC technique of Aki (1957), which also allows the inclusion ofvery long wavelengths for a given station spacing. If the microtremor

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3-D seismic noise tomography 503

Figure 1. The simple 3-D structure used for calculation of noise synthetics. The black dots represent the spatial distribution of the receivers. The red dotindicates the receiver for which seismic noise time-series are shown in Fig. 3.

Figure 2. Spatial distribution of noise sources (triangles) and receivers (squares). The black rectangle indicates the part shown in detail in Fig. 1.

wavefield is stochastic and stationary in both space and time, the realportion of the azimuthally averaged correlation function betweentwo stations ρ (r, ω0) is related to the phase velocity by

ρ(r, ω0) = J0

(ω0

c(ω0)r

), (2)

where r is the interstation distance, c(ω0) is the phase velocity foran angular frequency, ω0, and J 0 is the zeroth-order Bessel functionof the first kind. Eq. (2) clearly indicates that if one measures ρ(r,ω0) for a fixed r and for various ω0, the phase velocity dispersioncurve can be obtained by equating the zero crossings of the realpart of the spectrum with the zero crossings of the Bessel function.

In particular, if the distribution of noise sources is azimuthallyuniform, the imaginary part of the spectrum is zero and close tozero for modest azimuthal noise variations (Cox 1973; Webb 1986).

Recently, it was shown (Weaver & Lobkis 2004; Chavez-Garcıaet al. 2005) that it is possible to substitute an average over timeof the correlation in the frequency domain between two stationsfor a fixed interstation distance instead of the azimuthal average ofthe correlation required by SPAC, a concept already mentioned byAki (1957). The basic idea of this hypothesis is that the average ofthe cross-correlation between two stations for a long-enough timewindow averages all the different directions of the waves composingthe microtremors, that is, a single station pair can be seen as being

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504 M . Pilz et al.

Figure 3. (a) Simulation of seismic noise time-series computed using thesource distribution shown in Fig. 2. (b) Vertical component of the Green’sfunction for a frequency of 8 Hz. The envelope of the Green’s functionshown in red.

equivalent for an azimuthal average of many station pairs with thesame interstation distance. Although this theory is valid only if thesources of ambient vibrations or the subsoil structure do not imposea predominant direction of propagation, Derode et al. (2003) showedexperimentally that an inhomogeneous source distribution will havea lower effects on the calculated traveltimes of the waves than ontheir signal-to-noise ratios. Even if the majority of the noise sourcesare clustered in a narrow azimuthal range, Picozzi et al. (2009)demonstrated that for frequencies of some hertz, the calculatedtraveltimes are stable if there is still sufficient noise signal fromall other directions. Therefore, even if the strongest noise emergesonly from a few directions, we are sure that the correlation functioncan still be calculated reliably. As an example, Fig. 4(a) shows theresulting correlation function plotted against frequency taking twostations of the synthetic data set with an interstation distance of37 m.

Although the amplitude of the real part of the spectrum dependson the background noise spectrum, the locations of the zero cross-ings in the spectrum should be insensitive to variations in the spectralpower of the background noise, unless there are no strong transientsignals. We choose to use the locations of the zero crossings toderive dispersion values using a series expansion to calculate thezero crossings of the Bessel function (Elbert 2001). If ωn denotesthe frequency of the nth observed zero crossing and zn representsthe nth zero of J 0, the corresponding phase velocities (blue dots in

Figure 4. Example of dispersion curve estimation. (a) Real part of the noisecorrelation function along with zero crossings (black starts). (b) Blue dotsrepresent the phase velocity dispersion curve derived from the zero crossingsusing eq. (1).

Fig. 4b) are obtained by,

c(ωn) = wnr

zn. (3)

However, in real-world data the association of a given zero witha particular zero crossing of J 0 can be difficult since the noisemay cause missed or additional zero crossings. Following Ekstromet al. (2009), one may distinguish the positive-to-negative and thenegative-to-positive zero crossings by generating two dispersioncurves that can be assessed for consistency. The difference betweenthe phase velocities of the two curves at each sampled frequencymay be used as a quality criterion.

For the lowest analysed frequency, the calculated velocities aremore scattered than for higher frequencies (Fig. 4b). When thewavelength becomes too large with respect to the interstation dis-tance, the phase difference between the stations becomes negligi-ble. This trend is particularly evident when the geophone intervalsare small. Increasing the frequency shows a decrease in the esti-mated velocities, demonstrating the dispersive character of Rayleighwaves.

C© 2012 The Authors, GJI, 189, 501–512



As a general constraint, the SPAC technique establishes a formalrelationship between the correlation coefficients and the phase ve-locity dispersion curve which needs to obey the limits of Henstridge(1979), where it was shown that correlation functions can only becalculated correctly when 2 < λ/r < 15.7 (Henstridge criterion),where, λ = 2πc(ωn)/ω0, is the wavelength of the wave considered.However, although in our case the left inequality is not always ful-filled, Tsai & Moschetti (2010) pointed out that the errors are mostlyless than a few per cent, even for relatively small λ/r, if the distri-bution of noise is uniform enough, which is likely to be the casein reality. In the following, since the S-wave velocity is the domi-nant property of the fundamental mode of high-frequency Rayleighwaves, we perform inversions for a S-wave velocity model estima-tion while considering the observed velocities in the 3-D frequency-dependent tomography as being representative of Rayleigh wavephase velocities.

4 T H E A L G O R I T H M F O R 3 - D S U R FA C EWAV E T O M O G R A P H Y

To determine the traveltimes between the sensors, the first step ofdata processing consists of preparing waveform data from each in-dividual station. Since our interest lies in detecting local variationsin S-wave velocity for the shallowest part of the subsurface andsince the distances of the array only span a range of some metres toaround 100 m, we are strongly interested in high-frequency noise(higher than a few hertz). As demonstrated by Chavez-Garcıa &Rodrıguez (2007), the local velocity structure and the interstationdistance strongly affect the frequency range for which the phase ve-locities can be determined reliably. Chavez-Garcıa & Luzon (2005)have shown that Love waves dominate the records in the lower fre-quency range (lower than a few hertz), whereas Rayleigh wavesare prevalent in the higher frequency band of their recordings. Forour analyses, we consider only the vertical component of the cross-correlation functions, which are strongly dominated by Rayleighwaves. Therefore, a second-order Butterworth high-pass filter wasapplied to these data using a corner frequency of 0.9 Hz, with theaim of removing high-amplitude low-frequency irregularities thattend to obscure noise signals. From the continuous data sets, 60noise time windows, each 30 s long, were extracted. Picozzi et al.(2009) have shown that this duration is sufficient for obtaining areliable estimate of the Green’s function between two sensors fordistances of some tens of metres. After removing the linear trendfrom each window, a 5 per cent cosine taper was applied at bothends.

After preparing the time-series, one-bit normalized data wereused to calculate the cross-correlations, following Campillo & Paul(2003) and Bensen et al. (2007). Although some interstation dis-tances may not be suitable to obtain reliable results, we performcross-correlations between all possible station pairs. This yields atotal of n(n − 1)/2 possible station pairs, where n is the numberof stations. For each of the chosen frequencies and for each pairof receivers, all calculated cross-correlations were stacked togetherwhich, on average, improves the signal-to-noise ratio (Bensen et al.2007). Finally, the traveltimes between all stations were calculatedfrom the estimated phase velocities using the known distances be-tween the sensors.

In the second step, the estimated traveltimes were inverted bya tomographic approach to calculate a 3-D velocity model. In ageneral form, the traveltime, t, between the source and a receiveralong a ray path L along a line element dl for a continuous slowness,

s, (inverse velocity) is given in its integral form as,

t =∫

Lsdl. (4)

This means that the traveltime is based on a linear combinationof slowness. However, since the ray path is velocity-dependent,traveltime inversion is a non-linear problem, although, in media withslight velocity anomalies the deviation of the paths from a straightline will either be of the same order as the dimension of the blocksor will be less than a quarter of the wavelength. On the one hand,deviations smaller than the block size will not introduce significanterrors and, on the other, deviations smaller than a quarter wavelengthare within the limits of image resolution. Hence, following Kugleret al. (2007) and keeping in mind the level of errors of the input datafor shallow seismic surveys, we are sure that a bias of a few percentcan be tolerated keeping the solution linear, which will be discussedin more detailed later. This means that eq. (4) can be expressed in asimple discrete matrix form to be used in practical applications,

t = L1s, (5)

where L1 is an RF × MN matrix with R being the number of rays forF different frequencies crossing the medium that is subdivided intoa reasonable number of M cells in each of the N horizontal layers.Since deviations smaller than the block size will not introduce sig-nificant errors and since differences less than a quarter wavelengthwill be within the limits of image resolution, the phase velocityis assumed to be constant within each of the cells. Therefore, thepropagation paths can be considered to be straight rays within thecells.

However, to ensure the accuracy of the inversion, the setting ofthe cell size and the number of rays travelling through every cell isnot a negligible issue. On the one hand, too few rays in each cellwill cause the inversion not to converge or to converge too slowlyto reflect the real velocities. By contrast, a too-large number of rayswill lead to a waste of data, resulting in a lower resolution. Moreover,since the spatial resolution will also depend on the frequency rangeused, the quarter-wavelength criterion suggests that for a frequencyrange higher than a few hertz and low-velocity sediments, a cellsize of several metres will provide balanced ray coverage.

The basic problem of tomography is to solve eq. (5) for the a pri-ori unknown slowness. In general, uncertainties in measured trav-eltimes can degrade the solution and can produce spurious velocityanomalies. Since traditional sensitivity kernels are rather problem-atic in 3-D (Marquering et al. 1999), we adopted an iterative pro-cedure for solving eq. (5) using damped last squares or singularvalue decomposition, following Long & Kocaoglu (2001). For thefirst iteration, we start with a homogeneous 3-D model s(k) (i.e. thesame velocity for all the cells) and calculate the new solution s(k+1)

by solving

W�t(k)= L2�s(k), (6)

in which �t is the vector of the normalized misfit between theobserved and theoretical traveltimes, [t0 − t(k)]/t0. Accordingly,�s is the vector of the normalized slowness modification [s(k) −s(k−1)]/s(k). To reduce the risk of divergence and to stabilize theiteration process, the adaptive bi-weight estimation (Tukey 1974),which is a kind of the maximum likelihood estimation, is appliedby introducing a diagonal weighting matrix, W, with RF × RFelements. The design matrix, L2, in eq. (6) consists of two blocks,

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506 M . Pilz et al.

namely,

L2 =[

WL1

K(ε2

)M

], (7)

in which the upper-block elements represent the ray path segmentmatrix L1 weighted by W. In particular, the measured slownesscontains information about the underlying structure to depths cor-responding to approximately one-third to one-half the wavelengthof each frequency.

The lower elements K and M in eq. (7) describe constraints onthe solution, making use of the damping coefficient ε. As pointedout by Marquardt (1963) and by Ammon & Vidale (1993), thedamping factor significantly controls the speed of convergence andfurther acts as a constraint on the model space (Tarantola 1987).After some trial and error testing of different values, we fixed ε to0.7 to give the finest resolution without causing instabilities in theinversion.

The MNF × MNF matrix, K, weights the data depending onthe number, length, orientation and vertical penetration depth ofeach ray path segment crossing each cell. According to Yanovskaya& Ditmar (1990), for the 3-D problem the weights turn out tobe representable as a product of two functions, one depending onthe horizontal coordinates and one depending on the depth, thatis, the frequency. For the first function, the singular values (a1,a2) of the ray density matrix were used to calculate its ellipticity√

(a21−a2

2 )/a21 , based on a proposal of Kissling (1988). If several rays

with different azimuths cross the cell, the ellipticity is close to 1and a good resolution is achieved. Therefore, the horizontal weightswere computed by multiplying the ellipticity for the number of rayscrossing each cell.

The second function calculates the vertical weights to account fordifferent penetration depths of the different frequencies. Using a ho-mogeneous starting model for the first iteration, the vertical weightsare based on the analytical solution of displacement components, u,for Rayleigh waves in a half-space, following Hill (2010):

u3(z) = lelkz − sepkz, (8)

with z < 0 being the depth below the surface; k = ω/cR is theRayleigh wavenumber, in which cR is the Rayleigh wave phasevelocity as given in eq. (3) and ω the angular frequency. Since wecan assume the Rayleigh wave velocity to be around 0.92 vs, whichis realistic for typical values of Poisson’s ratio (0.2 < ν < 0.4), thisleads to l = 0.8475, p = 0.3933 and s = 1.4679 (see Bullen 1963;Lay & Wallace 1995). The corresponding vertical weights, w, arecalculated as the normalized average values of u3 within each layer,

w =∫ d2

d1 u3(z)dz∫ ∞0 u3(z)dz

, (9)

with d1 and d2 being the lower and upper bound of each layer,respectively.

Using these equations will provide a first approximation of thebehaviour of surface wave penetration at depth. Since eqs (8) and (9)strongly depend on the underlying Rayleigh wave phase velocities,the vertical weights for all the cells and accordingly the matrix Kare updated after each inversion. Eq. (9) further implies that thevertical weights vary smoothly, from which it follows that there isno practical limitation of the vertical resolution.

Matrix M in eq. (7) constrains the solution to vary smoothly in thehorizontal domain, that is, the slowness of each cell is related alsoto the slowness of all the surrounding cells with weights dependingon the cell’s location. The smoothness constraints are implementedby adding a system of equations to the original traveltime inversion

problem, following Ammon & Vidale (1993). Note, however, thatthese constraints do not act as a low-pass filter in the classicalsense; therefore, not all sharp contrasts in the model are necessarilysmoothed.

Finally, we solve eq. (6) using the singular value decompositiontechnique (e.g. Golub & Reinsch 1970; Arai & Tokimatsu 2004).The decomposition of the matrix, L2, leads to a product of threematrices, UVT, in which U and V are (RF + MNF) × (RF + MNF)and MN × MN matrices, respectively and is a (RF + MNF) ×MN matrix that has the singular values of L2, aii, in the diagonalelements of the matrix. Decomposing L2 in eq. (6) and utilizing theorthogonal property of the matrices U and V, we obtain,

�s(k) = V†UTW�t(k). (10)

If some of the eigenvalues of † are small, errors in the datacould cause strong fluctuations in the solution. To suppress thoseundesirable effects in solving eq. (10), the diagonal componentsof the matrix †, aii

−1, are replaced by (aii + ε/aii)−1 (Levenberg1944; Marquardt 1963) with the damping factor, ε (Fletcher 1971).

From eq. (10), the slowness vectors are updated after each inver-sion step until a reasonable compromise between the reduction ofthe rms error between the observations and the predictions and thenorm of the solution is reached.

The final slowness s is used to calculate the 3-D shear wavevelocity, vs. Slowness is related to the shear wave velocity by,

s = 1

cvs, (11)

that is, c < 1. For typical values of Poisson’s ratio (0.2 < ν <

0.4) and a homogeneous half-space, c takes values 0.9 < c < 0.95.As already mentioned, we used c = 0.92 for all calculations. Wewill discuss later that this approximation will hold for engineeringpurposes.

5 VA L I DAT I O N O F T H E P RO P O S E DI N V E R S I O N S C H E M E

5.1 Synthetic example

For evaluating to what degree the initial model can be restored andto check the resolution of the results, we used the synthetic dataset with a known subsurface structure as described in Section 2.Since the spatial resolution depends on both the frequency and theinversion technique and following Nolet (1987) and Jensen et al.(2000), a horizontal resolution of 10 m can be achieved for theselected frequency range (4.5 Hz ≤ f ≤ 13 Hz). For the verticalresolution we chose a value of 5 m to calculate the average velocitiesfor each layer.

Fig. 5 shows the inversion results and the average S-wave veloc-ities for the indicated depths using five different frequencies (4.5,6, 8.3, 10.1 and 13 Hz) to calculate the interstation traveltimes es-timated by the correlation function. The comparison between theinput structure and the calculated velocities shows that the struc-ture is well reproduced. For the uppermost layer the velocity is al-most constant, whereas for deeper layers the widening high-velocityblock can clearly be identified. Also, in terms of absolute values, theS-wave velocities are fairly well reproduced, whereas, the velocitycontrasts are sometimes less sharp. This is based on the fact that weused a smoothing parameter in the solution (matrices K and M ineq. 7).

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Figure 5. Left-hand panel: input model for validation test. Right-hand panel: inversion results. The dots represent locations of the sensors. The black linesshow the path coverage. The images were obtained after 200 iterations. For each layer, the average S-wave velocities at depths indicated in the upper-rightcorner are shown.

Due to the difficulty in calculating meaningful uncertainty esti-mates of the weighted and smoothed inversion procedure, modernstatistical techniques, such as bootstrap, (Efron 1979; Koch 1992)have been proposed over the last few decades. Here, we propose theapplication of the bootstrap technique to the tomographic algorithmpresented in this work. Bootstrap methods work by repeated inver-sions of the bootstrap data set obtained by randomly resamplingthe original data set. The resampling is carried out by replacinga row of the matrix with another one, with both rows randomlychosen. Each bootstrap sample has the same size as the originaldata set and the bootstrap estimate of the standard error is givenby the standard deviation of the distribution of the reconstructedvelocity models (called bootstrap replications) obtained for eachbootstrap sample. Finally, the standard deviation for the velocitymodel obtained by the inversion of the original data set is estimatedfrom the values of the bootstrap standard error and the number ofdata.

We performed 500 bootstrap inversions to estimate the stabilityof the solution. Fig. 6 shows the standard deviation estimated for thetomographic models for two different depths. The standard devia-tion is often significantly smaller than 30 m/s and often, especiallyin parts with high ray coverage, below 10 m/s. We also observeslightly higher uncertainties for deeper parts of the model due tolarger variations in the short frequency data. A similar observationwas reported by Yang et al. (2008). Nevertheless, the results confirmthe reliability of the applied inversion algorithm to obtain accurateS-wave velocity distributions.

5.2 A real-world example: the Nauen test site

We now apply the proposed procedure to a real data set,obtained from an array consisting of 21 three componentseismological stations deployed at the test site of Nauen (Ger-many) for one week in 2007 May (http://www.geophysik.tu-berlin.de/menue/forschung/testfeld_nauen/uebersicht). Further de-tails about this experiment can be found in Picozzi et al. (2009). Thegeology of the Nauen site is representative for large areas of north-ern Germany with Quaternary sediments overlying Tertiary clays.In the actual area of investigation, these sediments mainly consistof fluvial sands bordered by glacial till. The topography is charac-terized by flat hills underlain by till and plains comprised of glacialfluvial sands and gravels. Therefore, there was no need to considerthe shape of the topography in the inversion. In the studied area,there is an unconfined shallow aquifer consisting of fine to mediumsands underlain by an aquiclude of marly and clayey glacial till.North of the site, the glacial till approaches the surface in a nearlyE–W strike. Electrical resistivity cross-sections indicating the 2-D subsurface structure and the distribution of the seismic stationsare shown in Fig. 7. It is obvious that the array covers structuralvariations at the site and a good azimuthal coverage exists.

After correcting the noise recordings for the instrumental re-sponse, the recorded data were processed using the same techniqueas described above. Inversion results for four horizontal slices ob-tained after 200 iterations using four different frequencies (5, 7, 10and 14 Hz) are presented in Fig. 8. The calculated velocities areshown only for cells which are crossed by rays and a few adjacent

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Figure 6. Standard deviations of S-wave velocities for different depths computed after 500 repeated bootstrap inversions.

Figure 7. Left-hand panel: satellite photography of the Nauen test site showing the location of sensors. Red lines indicate the location of the geoelectricprofiles. Right-hand panel: geoelectric profiles (Yaramanci et al. 2002).

ones uncrossed by rays but influenced by the smoothing constraintsimposed on the solution.

The occurrence of strong variations in the site conditions is vis-ible at a first glance. The topmost layer shows an almost constantvelocity distribution, suggesting that the medium is quite homoge-neous down to 5 m. For the subjacent layer between 5 and 10 m,higher velocities are found in the northeastern part of the study area.Below 10 m, the area of high velocities extends towards the south-west. This is a clear indication of a two-dimensional structure thatwas also found in two 2-D geoelectric cross-sections (see Fig. 7).Profile A clearly indicates a layer with low electrical resistivity be-low 5 m in the northeastern part, which is identified to be glacialtill. This layer extends towards the southwest and covers the entirestudy area below ∼17 m (Yaramanci et al. 2002).

As can be seen when comparing Figs 7 and 9, this area corre-sponds well with the absolute depth and the impedance contrast level

in our cross-sections. Also, the absolute velocities agree well withGoldbeck (2002) who reported S-wave velocities between 200 and450 m/s for profile A. Fig. 9 further shows a fair agreement of our in-terpolated results with the findings of Picozzi et al. (2009). However,the latter authors calculated the 1-D S-wave velocity-depth profiles.Their 2-D S-wave velocity cross-sections could only be obtainedafter the interpolation of the inversion results of the Rayleigh wavevelocity dispersion curves along a profile of 10 cells. In contrast,our proposed inversion method directly allows the calculation of afirst-order 3-D S-wave velocity model.

6 D I S C U S S I O N

Our proposed technique aims at estimating a 3-D model of thelocal S-wave velocity structure at shallow depths. However, the

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Figure 8. Inversion results using data collected at the Nauen test site. The images were obtained after 200 iterations. Yellow lines mark the locations of thegeoelectric cross-sections shown in Fig. 7.

results strongly depend on the site specific conditions, in particularthe numbers of receivers and the distances between them. If theinterstation distance relative to the wavelength is too large, thewavelengths may be smaller than required by the basic samplingtheorem. However, in this case, it is still possible to measure groupvelocity, as has been shown previously by Campillo & Paul (2003)and Shapiro & Campillo (2004). On the other hand, a higher numberof receivers and ray paths will significantly scale down the level ofuncertainty because the number of cells with sparse ray coverage isreduced.

The calculation of the final S-wave velocity model (eq. 11) isan approximation and in a strict sense only valid for a homoge-

neous half-space. However, it is still reasonable because the S-wavevelocity is the dominant property for the fundamental mode ofhigh-frequency Rayleigh waves. Of course, for a layered structurethe Rayleigh wave disperses when the wavelengths are in the rangeof 1–30 times the layer thickness (Stokoe et al. 1994), but Xia et al.(1999) never found variations of more than 25 per cent in shearwave velocity between this approximation and their true multilayermodel. On the other hand, using an alternative procedure by in-verting our phase velocity results to obtain vs-depth profiles wouldinvolve additional uncertainties, since it would further require in-formation about the densities and P-wave velocities introduced bythe inversion process. Furthermore, also the inversion of dispersion

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510 M . Pilz et al.

Figure 9. Comparison of S-wave velocity cross-sections along profile Aobtained by Picozzi et al. (2009) after interpolation between 2-D velocity-depth profiles from dispersion curves (top panel) and by our 3-D inversiontechnique (bottom panel).

curves calls also for homogeneity. If the wave path is horizontallyheterogeneous, Kennett & Yoshizawa (2002) and Strobbia & Foti(2006) show that this can cause significant perturbations in the ob-served velocity. As a consequence, Lin & Lin (2007) point out thatartefacts may be introduced in spatially more-dimensional shearwave velocity imaging if lateral heterogeneity is not accounted for.Hereunto, our 3-D model provides a fair estimate of the prevailingvelocity structure that can be used for engineering and monitoringpurposes.

To estimate the exact resolution limits of our inversion technique,one has to take into consideration that the horizontal resolution willdepend on many site specific factors, such as the velocity structure,the used frequencies and the wave propagation paths. Therefore,no strict formula can be provided. Keeping in mind the quarter-wavelength criterion and the selected frequency range, a horizontalresolution of 10 m is fairly reachable for low-velocity sediments.However, when the size of the cells is too large, our methodologytends to smear sharp discontinuities in velocity and is, therefore,also likely to obscure fine scale velocity structures. Nonetheless, formost of the engineering applications, such small-scale peculiaritiesare of minor interest.

Since we introduced a smoothing parameter in the solution (ma-trices K and M in eq. 7) the formulation above assumes that theproperties of the medium are only changing smoothly at every pointof the medium. Of course, it is easy to generalize the equations tothe case where there are discontinuities as explicit parameters ofthe inverse problem. However, if such an approach is followed,non-existent discontinuities may also be introduced as the iterativeinversion proceeds as steep gradients, but, obviously, this is notideal.

In contrast, since the weights of the individual cells vary smoothlyvertically, there will also be only a steady change in the resulting ve-locities, that is, vertical discontinuities will be resolved less sharply.For deeper parts of the investigated area, only smaller frequenciesshow non-zero weights, reducing the resolution for these parts.Subsequently, since at each frequency information about the under-lying velocity structure to a depth of about one-third to one-halfwavelength is provided, the penetration depth only depends on thelowest frequency used. The use of very low frequencies might beproblematic due to difficulties in the correct determination of the

traveltimes, averaging out smaller scale S-wave velocity contrasts.On the other hand, the absolute number of data sets (frequencies)will not have a significant influence on the inversion results as longas a wide enough range of frequencies uniformly encounters thedepths of interest. In particular, there appears to be no decreasein accuracy due to reductions in the number of used frequencies,consistent with Rix et al. (1991). Analyses on the number of neces-sary frequencies have indicated that five or six frequencies equallydistributed over the frequency range of interest are sufficient, sinceneighbouring frequencies provide similar information (not shownhere). Of course, more frequencies can be used at the expense ofreduced inversion speed.

In addition, to estimate the effect of the initial value selectionon the final results, we ran both inversions using many differenthomogeneous starting models. Within the region of highest raycoverage, the standard deviation is much less than 10 per cent. Onlya few isolated spots show deviations up to 20 per cent in areas ofsparse ray coverage. The choice of the starting value was found tobe critical to the convergence of the inversion procedure only if anunrealistic initial velocity (i.e. the difference to the final velocitywas larger than a factor 4) is chosen.

7 C O N C LU S I O N S

We have used synthetic and real-world data sets based on the verticalcomponent of seismic noise recordings to perform a tomographic3-D inversion for obtaining images of the S-wave velocity structure.Based on the correlation of seismic noise recordings, Rayleigh wavephase velocities and corresponding traveltimes have been calculatedbetween each pair of seismic sensors. With only a limited numberof seismic stations and recording times of several tens of minutes,detailed images of the local subsoil structure can be obtained. Thereliability of the proposed technique was validated using syntheticdata sets and bootstrap showing that the results are well constrained.

Using real recordings of seismic noise and comparing the resultswith independently calculated geophysical results, a good agree-ment for the position of the main geophysical boundaries is found,highlighting the potential of the method to be used where othergeophysical methods might fail. Since reliable traveltime estimatesfor the frequency range investigated can be obtained in almost realtime, the results imply the use of the proposed procedure as an ex-ploration and monitoring tool. In future work, a joint inversion ofRayleigh and Love wave data sets, the latter estimated from hori-zontal components, should help to further constrain the inversionand to improve the resolution of the model.

A C K N OW L E D G M E N T S

We would like to acknowledge the editor J. Trampert and two anony-mous reviewers for providing helpful comments that allowed us tosignificantly improve the manuscript. We are grateful to B. Liss, A.Manconi, R. Milkereit and R. Bauz for the fieldwork. K. Flemingkindly revised our English. Instruments were provided by the Geo-physical Instrumental Pool of the Helmholtz Center, Potsdam. Wedeeply thank CRS4 and Politecnico di Milano for the cooperationusing GeoELSE.

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Threedimensional shear wave velocity imaging by ambient