Mapping the Moho with seismic surface waves: A review...

Tectonophysics xxx (2013) xxx–xxx

TECTO-125734; No of Pages 18

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Tectonophysics

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Mapping the Moho with seismic surface waves: A review, resolution analysis, andrecommended inversion strategies

Sergei Lebedev a,⁎, Joanne M.-C. Adam a,b, Thomas Meier c

a Dublin Institute for Advanced Studies, School of Cosmic Physics, Geophysics Section, 5 Merrion Square, Dublin 2, Irelandb Trinity College Dublin, Department of Geology, Dublin 2, Irelandc Christian-Albrechts University Kiel, Institute of Geophysics, Kiel, Germany

E-mail address: [email protected] (S. Lebedev).

0040-1951/$ – see front matter © 2013 Elsevier B.V. Allhttp://dx.doi.org/10.1016/j.tecto.2012.12.030

Please cite this article as: Lebedev, S., et al.,inversion strategies, Tectonophysics (2013)

a b s t r a c t
a r t i c l e i n f o
Article history:Received 30 June 2012Received in revised form 21 December 2012Accepted 28 December 2012Available online xxxx

Keywords:Rayleigh waveLove waveMohorovičić discontinuityModel spaceInversionTomography

The strong sensitivity of seismic surface waves to the Moho is evident from a mere visual inspection of their dis-persion curves or waveforms. Rayleigh and Love waves have been used to study the Earth's crust since the earlydays of modern seismology. Yet, strong trade-offs between the Moho depth and crustal and mantle structure insurface-wave inversions prompted doubts regarding their capacity to resolve the Moho. Here, we reviewsurface-wave studies of the Moho, with a focus on early work, and then use model-space mapping to establishthe waves' sensitivity to the Moho depth and the resolution of their inversion for it. If seismic wavespeeds withinthe crust and upper mantle are known, thenMoho-depth variations of a few kilometres produce large (>1%) per-turbations in phase velocities. However, in inversions of surface-wave data with no a priori information(wavespeeds not known), strong Moho-depth/shear-speed trade-offs will mask ~90% of the Moho-depth signal,with remaining phase-velocity perturbations ~0.1% only. In order to resolve the Moho with surface waves alone,errors in the data must thus be small (up to ~0.2% for resolving continental Moho). With larger errors, Moho-depth resolution is not warranted and depends on error distribution with period. An effective strategy for the in-version of surface-wave data alone for theMoho depth is to,first, constrain the crustal and upper-mantle structureby inversion in a broad period range and then determine the Moho depth in inversion in a narrow period rangemost sensitive to it, with the first-step results used as reference. Prior information on crustal andmantle structurereduces the trade-offs and thus enables resolving the Moho depth with noisier data; such information should beused whenever available. Joint analysis or inversion of surface-wave and other data (receiver functions, topogra-phy, gravity) can reduce uncertainties further and facilitate Moho mapping.

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Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 02. Surface-wave studies of the crust and the Moho . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 03. Sensitivity of surface waves to the Moho. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 04. Trade-offs between the Moho depth and other model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 05. Inversion of surface-wave measurements for the Moho depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

5.1. Mapping the model space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 05.2. Resolution and trade-offs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 05.3. Inversion of measured data: Northern Kaapvaal Craton. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

6. Noise in the data: how much is too much for the Moho to be resolved? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 07. Recommended inversion strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

7.1. Inversion of surface-wave data only, with no a priori information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 07.2. A priori information: include whenever available! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 07.3. Joint analysis and inversion of surface-wave and other data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

8. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 09. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

⁎ Corresponding author. Tel.: +353 1 653 5147x240.

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Mapping the Moho with seismic surface waves: A review, resolution analysis, and recommended, http://dx.doi.org/10.1016/j.tecto.2012.12.030

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1. Introduction

The Mohorovičić discontinuity, often referred to as the Moho, sep-arates the Earth's crust from the underlying mantle. Compositionaldifferences between the lighter crust and the denser upper mantlegive rise to an increase in seismic velocities across the Moho, fromthe crust to the mantle. The discontinuity can thus be identified seis-mically as the location of the seismic-velocity increase (Mohorovičić,1910).

During the century since the discovery of the Moho (Mohorovičić,1910), the discontinuity, which can be either sharp or gradational,has been detected and imaged in numerous locations around theworld, at various length-scales and with different seismic techniques.Controlled-source seismic surveys yield high resolution of the entirecrust and the Moho by sampling them densely with rays of reflectedor refracted seismic body waves, propagating between local sourcesand receivers (Prodehl and Mooney, 2011, and references therein).Relatively expensive and labour-intensive, controlled-source experi-ments can be complemented by “passive” seismic studies that usenatural seismic sources (local or teleseismic earthquakes or ambientseismic noise). The passive imaging approaches include the analysisof P-to-S wave conversions at the Moho (e.g., Bostock et al., 2002;Kind et al., 2002; Nabelek et al., 2009; Stankiewicz et al., 2002; Zhuand Kanamori, 2000), surface-wave imaging, including inversionsof surface-wave dispersion curves or waveforms and surface-wavetomography (e.g., Das and Nolet, 1995; Endrun et al., 2004; Yanget al., 2008), joint inversions of the P-to-S conversions (receiver func-tions) and surface-wave data (e.g., Julià et al., 2000; Tkalčić et al.,2012), local body-wave tomography (e.g., Koulakov and Sobolev,2006), and even SS waveform stacking (Rychert and Shearer, 2010).Regional crustal models and Moho maps have also been constructedusing combinations of both active-source and passive seismic data,as well as other geophysical and geological data (e.g., Grad et al.,2009; Kissling, 1993; Molinari and Morelli, 2011; Tesauro et al.,2008; Thybo, 2001).

Seismic surface waves are particularly sensitive to the structure ofthe crust and uppermost mantle — and, thus, to the depth of theMoho. Because these waves propagate along the Earth's surface, mea-surements of their speeds characterise average elastic properties ofthe crust and upper mantle between seismic sources and stations orbetween different stations. The Moho can thus be imaged even in lo-cations with no stations or sources.

The two main types of surface waves are Rayleigh waves and Lovewaves (Aki and Richards, 1980; Dahlen and Tromp, 1998; Kennett,1983, 2001; Levshin et al., 1989; Nolet, 2008). The speeds of Rayleighwaves depend primarily on the speeds of the vertically polarised Swaves in the crust and mantle and, also, on P-wave speeds and densi-ty; the particle motion associated with Rayleigh waves in an isotropic,laterally homogeneous Earth model is within the great circle planecontaining the source and the receiver. The speeds of Love waves de-pend primarily on the speeds of the horizontally polarised S wavesand, also, on density; the associated particle motion is approximatelyperpendicular to the great circle plane.

The depth sensitivity of surface waves depends on their period:the longer the period, the deeper within the Earth the waves sample(Fig. 1). This makes surface waves strongly dispersive.

Dispersion curves of surface waves (their phase or group velocitiesplotted as a function of period or frequency) show a characteristicsharp increase with period associated with the Moho (Figs. 2, 3).This increase reflects the S-wave velocity increase across the discon-tinuity, and its period range depends on the depth of the Moho: it oc-curs at longer periods if the Moho is deeper. The depth of the Mohocan thus be estimated roughly by a mere visual inspection of asurface-wave dispersion curve (Figs. 2, 3).

Inferences on the crustal structure and thickness have beendrawn from surface-wave observations since the early days of modern

Please cite this article as: Lebedev, S., et al., Mapping the Moho with seisinversion strategies, Tectonophysics (2013), http://dx.doi.org/10.1016/j

seismology. It also became apparent early that the crustal models in-ferred from the dispersion data can be highly non-unique. Althoughthe Moho depth has been an inversion parameter in numerous surface-wave studies, the data's sensitivity to the Moho and, in particular, theresolution of the Moho properties given by inversions of surface-wavedata with measurement errors are still uncertain and not agreed upon.

In this paper we overview the classic surface-wave studies since thelate 19th–early 20th century, as well as some of the more recent workfocussing on the Moho. We then investigate in detail the sensitivity ofsurface-wave phase velocities to theMoho depth and the trade-offs be-tween Moho-depth and crustal and mantle shear-velocity parametersin inversions of surface-wave dispersion. Exploring the model spacesin inversions of synthetic and real data, we examine the resolution ofthe Moho by surface-wave measurements as a data type. Finally, wediscuss strategies for an accurate estimation of the Moho depth usingsurface-wave data and illustrate some of them with applications tophase-velocity measurements from southern Africa.

2. Surface-wave studies of the crust and the Moho

Rayleigh waves were identified on seismic recordings by Oldham(1899), and already at that time Wiechert (1899) speculated that

mic surface waves: A review, resolution analysis, and recommended.tecto.2012.12.030

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Fig. 2. The signature of the Moho in phase-velocity curves of surface waves. The phasevelocities of the fundamental-mode Rayleigh and Love waves were computed for anoceanic model with a 5-km water layer and a 6-km thick crust (top), a continentalmodel with a 37-km thick crust (middle), and a model with a 65-km thick crust thatfits surface-wave data from NE Tibet (Agius and Lebedev, 2010). The period rangeswith the characteristic phase-velocity increase with period due to the S-velocity in-creases at the Moho are marked with grey shading.

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Fig. 3. The signature of the Moho in group-velocity curves of surface waves. The groupvelocities of the fundamental-mode Rayleigh and Love waves were computed for thesame oceanic and continental models as in Fig. 2.

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the velocities of surface waves – which he called “main waves” –

could be used to study the properties of the outer shells of theEarth, by means of measuring phase differences between signalsrecorded at nearby stations. In the early 20th century, velocities ofsurface waves have been estimated, at first, without taking their dis-persion into account. Angenheister (1906) gave a velocity estimate of3.1 km/s for “long waves”, also citing similar, earlier estimates byOmori. Reid (1910) called surface waves “regular waves”, while alsoestimating their velocities.

Golitsyn (cited here from his selected-works compilation: Golitsyn,1960) used minor and major arc recordings of the 1908 Messina earth-quake made at Pulkovo observatory and computed a global-average,surface-wave velocity of 3.53 km/s; dispersion, again, was not consid-ered. This value, interestingly, is very similar to the group velocities ofLove waves in a typical continent at periods below 25 s, well knowntoday (Fig. 3). Golitsyn also argued that the velocity of surface wavesshould depend on the physical properties of the upper layers of theEarth and be different beneath continents and oceans.

Love (1911) demonstrated the existence of transversely polarisedand dispersive surface waves in layered media (the Love waves). Theobservation of Love waves was a direct indication for the layeringwithin the Earth.

Tams (1921) compared Rayleigh waves propagating along oceanicand continental paths and proposed that they had different velocitiesbecause the crust beneath oceans, unlike the crust beneath conti-nents, did not comprise a granitic layer with relatively low seismicvelocities within it. He deliberately did not account for dispersion,considering the accuracy of available measurements insufficient.Angenheister (1921) argued that surface waves are well suited to


study the properties of the Earth's crust and attempted to measureboth the velocities and amplitudes of Love and Rayleigh waves. Healso pointed out the differences of surface-wave propagation alongoceanic and continental paths and reported, correctly, that recordingsat shorter epicentral distances are dominated by shorter periodwaves compared to those at longer distances. The relation betweendominant periods and crustal thickness, however, was not handledaccurately, leading to erroneous estimates of crustal thicknesses.

In order to describe Love wave propagation in realistic models ofthe crust, Meissner (1921) gave an expression for Love waves ina crust with a linear increase of seismic velocities with depth.Stoneley (1925) clarified the differences between group and phasevelocities. He then gave a quantitative expression for Rayleigh-wavedispersion in an Earth model with a compressible fluid over an elastichalf-space (Stoneley, 1926). This was particularly useful because themajority of early surface wave observations were performed forpaths that traversed oceans. Furthermore, the expressions were im-mediately applicable at the time because the problem could be solvedanalytically.

Surface-wave dispersion in an arbitrarily layered elastic half-spacewas determined by Meissner (1926) for Love waves and by Jeffreys(1935) for Rayleigh waves. Meissner (1926) also noted the non-uniqueness of dispersion-curve inversions and gave examples ofdifferent one-dimensional (1-D) Earth models that produced verysimilar dispersion curves. He concluded that highly accurate mea-surements using dense networks would be required in order to deter-mine the structure of the outer layers of the Earth.

In the early 1920s Gutenberg undertook the first systematicstudies of surface-wave dispersion for both Love and Rayleigh waves(e.g., Gutenberg, 1924), also including measurements by Macelwane(1923). He identified the nowwell known normal surface-wave disper-sion, characterised by a general increase of surface-wave speeds with


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period and indicative of the increase of elastic velocities with depth.He also inferred different crustal thicknesses for Eurasia, Americaand the Atlantic and Pacific Oceans. Testing crustal thicknesses of 30,60 and 120 km, he estimated the crustal thickness for Eurasia to bearound 50 km. This was a remarkable result, even though he com-pared, incorrectly, measured group velocities with theoretical phasevelocities.

Neumann (1929) found evidence for lateral heterogeneity of thePacific plate by analysing Love and Rayleigh waves. Carder (1934)summarised Love and Rayleigh wave characteristics known at thattime; the theoretical understanding of surface waves and the inver-sion tools that were available, however, were not sufficient to drawaccurate conclusions on crustal structure.

Ewing and Press (1950) gave a remarkable synthesis of surface-wave observations, using their own as well as previous measure-ments (Bullen, 1939; DeLisle, 1941; Wilson and Baykal, 1948). Thevery title of their paper, “Crustal structure and surface-wave disper-sion,” emphasised the inherent link of the early surface observationsto the properties of crust and the Moho. The analysis was based onsophisticated manual readings of group-velocity dispersion, usingtime-domain measurements of the arrival times of the dominantperiods (about 15–30 s) in the dispersed waveform. The Airy phasedescribed by Pekeris (1948) was identified correctly, based onStoneley's equation, and theoretical group-velocity curves were fittedto the observations. The strong influence of water and sediments onRayleigh-wave velocities was clearly established. Interpreting the re-sults, Ewing and Press (1950, 1952) implicitly applied ray theory andderived estimates for path-average, sub-crustal velocities and theMoho depths by estimating the continental portion of the paths.The limited bandwidth of their observations, however, and their useof a simplified Earth model with one layer (water and sediments)overlying a half-space, implied that their results were most meaning-ful for sub-crustal velocities, and less so for the properties of the crust.Citing Love-wave, group-velocity observations by Wilson (1940),Ewing and Press (1950) also noted that Love waves show highergroup velocities for oceanic paths compared to continental paths inthe period range of 20–100 s and confirmed that, in contrast to Rayleighwaves, Love waves are insensitive to the water layer.

Brilliant and Ewing (1954) measured, in the time domain,Rayleigh-wave phase differences between stations in the US, elimi-nating the phase shifts due to the source and the oceanic portionsof the paths. They determined the first phase-velocity curve forNorth America between 18 s and 32 s, a period range where phasevelocities are sensitive mainly to the crust and the Moho.

Evernden (1954) analysed group velocities of Love waves be-tween 7 and 45 s for the Pacific Basin. Using Stoneley's analytical ex-pressions for the dispersion of Love waves in a three-layered model,he concluded that a sedimentary layer, a high-velocity crust and aseismic-velocity increase from the crust to the mantle were all neces-sary to explain the measurements. He also noted that the results ofthe surface-wave analysis were compatible with those of seismic re-fraction surveys. These general conclusions still stand today, althoughthe crustal and mantle models have since been improved substantial-ly in their details.

The solution for Rayleigh-wave phase velocities in a model with awater layer overlying two solid layers made it possible to interpretRayleigh-wave group velocity curves measured along oceanic paths.Oliver et al. (1955) discussed available dispersion measurementsfor Rayleigh and Love waves. They concluded that a high velocitycrust is present under the oceans and were able to rule out thehypothesised existence of a large continent submerged beneath thePacific Ocean. They also showed that at periods longer than about25 s the dispersion curves for the Atlantic and Pacific basins weresimilar.

Press et al. (1956) made the first single-station measurementsover a 10–70 s period range for a pure continental path, between


Algeria and South Africa. They noted the similarity of their measure-ments to those by Brilliant and Ewing (1954) for North Americaand, also, to a theoretical curve corresponding to a 35-km-thick, ho-mogeneous crust overlying the mantle. They concluded, as well,that a gradual velocity increase in the crust and the mantle mightbe needed to explain the measured dispersion curves.

Press (1956) measured phase velocities by examining phase dif-ferences across arrays of stations, each array comprising only threestations. The reading algorithm he applied can be described as a visu-al f-k analysis in the time domain. Based on the measurements, hepresented a two-dimensional (2-D) cross-section for the crustalstructure in California, with (over-estimated) Moho-depth variationsfrom ~15 km near the coast to ~50 km beneath the Sierra Nevada.

In the 1960s, the emergence of computer programs for surface-wave analysis presented unprecedented new opportunities for accu-rate analysis and inversion of surface-wave data. Brune et al. (1960)analysed the phase of a dispersed waveform in the time domain andproposed improved reading schemes for phase-velocity determina-tion. Alterman et al. (1961) calculated Rayleigh-wave, phase andgroup velocities in a 10–700 s period range and discussed the effectsof gravity and the Earth's sphericity on the waves' propagation, aswell as the relation between surface waves and the Earth's free oscil-lations. Using the measurements of Ewing and Press (1956) and Nafeand Brune (1960), Alterman et al. (1961) also presented evidencesupporting the Gutenberg's model of the Earth with a low velocityasthenosphere.

Dorman and Ewing (1962) developed a linearised scheme for theinversion of surface-wave measurements and computed a Mohodepth of about 39 km for the New York–Pennsylvania area. Bruneand Dorman (1963) compared Love and Rayleigh waveforms at sta-tions within the Canadian Shield in the time domain. They deter-mined phase velocities in a 5–40 s period range and inverted themfor a 1-D, S-wave velocity model with a multilayered crust andupper mantle, detecting high S-wave velocities within the mantlelithosphere of the craton. They also calculated synthetic dispersedwaveforms for their model.

Toksöz and Ben-Menahem (1963) followed an earlier suggestionby Sato (1955) and measured phase velocities in the frequency do-main, using successive passages of surface waves at a single station.McEvilly (1964) measured the phase difference between two stationsin the frequency domain, for both Love and Rayleigh waves. Invertingthe resulting dispersion curves, he established that different 1-Dmodels were needed for the horizontally and vertically polarised Swaves (Vsh and Vsv, respectively). This observation became knownas the Love–Rayleigh discrepancy.

Santo and Sato (1966) developed a regionalization technique thatmay be seen as the first attempted group-velocity tomography.Knopoff et al. (1967) described filter and triangulation techniques forthe determination of phase velocities. The determination of group ve-locities using spectrograms was then suggested by Landisman et al.(1969).

Since the 1970s, the number of surface-wave studies has grownsteadily. Compilations and reviews of surface-wave analyses in thebeginning of this period are given by Dziewonski (1970), Knopoff(1972), Seidl and Müller (1977), Kovach (1978) and Levshin et al.(1989). With long-period surface-wave measurements increasinglyaccurate and abundant, surface waves were now used extensivelyfor the study of the upper mantle. In the course of inversions of thelong-period data, the substantial sensitivity of surface-wave speedsto crustal structure was often accounted for by means of “crustalcorrections”: the effect of the crustal structure on surface-wavemeasurements was evaluated using a priori crustal models, usuallyconstrained by other seismic methods (e.g., Bassin et al., 2000;Boschi and Ekstrom, 2002; Bozdag and Trampert, 2008; Ferreiraet al., 2010; Kustowski et al., 2007; Lekic et al., 2010; Marone andRomanowicz, 2007; Montagner and Jobert, 1988; Mooney et al.,




1998; Nataf and Ricard, 1996; Nataf et al., 1986; Nolet, 1990; Panninget al., 2010; Woodhouse and Dziewonski, 1984).

Thanks to the rapid growth of broadband seismic networks sincethe 1990s, increasingly large surface-wave datasets were used in re-gional and global imaging. Many tomographic inversions includedthe crustal structure and thickness as inversion parameters (e.g., Dasand Nolet, 1995; Lebedev and Nolet, 2003; Lebedev et al., 1997; Liand Romanowicz, 1996; Pasyanos and Walter, 2002; Shapiro andRitzwoller, 2002; Van der Lee and Nolet, 1997) (Fig. 4), and somesurface-wave studies targeted primarily the Moho itself (Das andNolet, 1995, 1998; Marone et al., 2003; Meier et al., 2007a, 2007b).The main difficulty in resolving theMohowith surfacewaves remainedthe non-uniqueness of seismic-velocity andMoho depthmodels consis-tent with surface-wave observations. Resolving the trade-offs betweenthe Moho depth and seismic velocities required highly accurate mea-surements at intermediate and relatively short periods (Fig. 2). Phasevelocities of surface waves, however, were difficult to measure at shortperiods, with the waveforms of teleseismic surface waves distorted bydiffraction at periods below 15–20 s, and with regional source-stationmeasurements biased substantially even by small errors in earthquakelocations.

The surface-wave crustal imaging has been rejuvenated in the2000s by the emergence of new, array techniques for surface-wavemeasurements. Phase velocities of short-period surface waves arenow measured routinely using pairs or arrays of broadband stations.The measurements are mainly by means of cross-correlation of eitherdiffracted surface waves from teleseismic earthquakes (Meier et al.,2004) or of surface waves within the ambient seismic noise (Shapiroand Campillo, 2004).

The cross-correlation of surface-wave recordings from nearby sta-tions is, essentially, the classical “two-station method” (Brilliant andEwing, 1954; McEvilly, 1964; Press, 1956; Sato, 1955; Toksöz andBen-Menahem, 1963). The difference of the modern and traditionalapplications is in the types of the signal they use. The classicaltwo-station method was applied only to teleseismic surface wavesthat obeyed surface-wave ray theory, i.e., were not distorted by diffrac-tion. Of the new techniques, teleseismic cross-correlations (Meier et al.,2004) can extract inter-station phase-velocity measurements evenfrom wave fields diffracted at teleseismic distances, and the ambientnoise cross-correlations (Shapiro and Campillo, 2004) make use of the

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Fig. 4. Resolving the Moho in East Asia-Western Pacific with surface-wave, waveform tomogic inversion of surface-wave forms for crustal and mantle shear-speed structure. The 1D backthe inversion was assembled from results of multi-mode waveform inversions of around 40model, the 3D inversion reproduces reasonably correct Moho depths, demonstrating the se


constructive interference of surface waves within the ambient noisewave field that arrive to a pair of stations at and near the station-station azimuth. Interestingly, the wave fields used for these measure-ments cannot at all be described by ray theory, but the phase andgroup velocities extracted from the cross-correlation functions candefine surface-wave propagation along inter-station paths in a ray-theoretical framework.

Over the last few years, the new methods have been applied tobroadband array data from around the world. The newly abundantshort-period and broad-band surface-wave measurements are, onceagain, bringing the crust into the focus of surface-wave seismology(e.g., Adam and Lebedev, 2012; Bensen et al., 2007; Deschamps etal., 2008b; Endrun et al., 2008, 2011; Lin et al., 2011; Moschetti etal., 2010; Pawlak et al., 2012; Polat et al., 2012; Shapiro et al., 2005;Yang et al., 2008, 2011, 2012; Yao et al., 2008; Zhang et al., 2007,2009) (Fig. 5). It is thus particularly appropriate at this time to exam-ine in detail the sensitivity of surface waves to the Moho and the res-olution of the Moho properties that they can provide.

3. Sensitivity of surface waves to the Moho

Characteristic signatures of the crustal thickness are clearly seenin various surface-wave observables, including phase-velocity curves(Fig. 2), group-velocity curves (Fig. 3), and waveforms of surface-wave trains on broad-band seismograms. The wave forms are closelyrelated to the frequency-dependent phase velocities. In a weaklyheterogeneous Earth, a complete seismogram can be computed as asuperposition of the fundamental and higher surface-wave modesusing the JWKB (Jeffreys–Wentzel–Kramers–Brillouin) approxima-tion as:

s ωð Þ ¼ ∑m

Am ωð Þ exp iωΔCm ωð Þ� �

; ð1Þ

where the sum is over modes m, ω is the circular frequency, Δ is thesource–station distance, Cm ωð Þ are the average phase velocities ofthe modes along the source-station path, and Am(ω) are the complexamplitudes of the modes, depending on the source mechanism andthe Earth structure in the source region, as well as on geometricalspreading and attenuation (Dahlen and Tromp, 1998).

140˚E

0˚

20˚N

40˚N

140˚E

0˚

20˚N

40˚N

15 18 21 30 40 50 60 75

Moho depth, km

100˚E 120˚E 140˚E100˚E 120˚E 140˚E

form tomography Moho, from CRUST2. 0

raphy (Lebedev and Nolet, 2003). Centre: Moho depths resulting from a 3D tomograph-ground model had a 25-kmMoho depth. The large system of linear equations solved in00 seismograms with source-station paths within the region. Even with a 1D referencensitivity of surface-wave waveforms to the Moho.


image of Fig.�4


32 34 36 37 38 39 40 41 42 43 45 50

crustal thickness km

-32

-28

-24

-20

20 24 28 32 20 24 28 3220 24 28 32

Receiver functions Ambient noise Ambient noise& teleseismic tomograpy

-32

-28

-24

-20A B C

Fig. 5. Results of the Moho mapping in southern Africa using receiver functions (left, Nair et al., 2006) and surface-wave phase velocities, measured with ambient noise (centre) andboth ambient noise and teleseismic signals (right) (Yang et al., 2008). (Figure courtesy of Yingjie Yang.)


Group velocity U is the velocity of propagation of the surfacewave's energy; it depends on the phase velocity and its frequency de-rivative as

U ¼ C1− ω=Cð Þ dC=dωð Þ : ð2Þ

While exploring the surface waves' sensitivity to the properties ofthe Moho, we shall focus on the phase and group velocities only,while noting that different surface-wave observables may have differ-ent useful properties (for example, local minima in the group velocitycurve, causing an Airy phase, have some sensitivity to the sharpnessof the Moho).

Fig. 6 illustrates the sensitivity of the Rayleigh and Love phase-velocity curves to the Moho depth in a typical continental modelwith a 37-km thick crust. If seismic velocities in the crust and uppermantle can be fixed (i.e., assumed to be known), then small Moho-

0

50

100

150

Dep

th, k

m

3.5 4.0 4.5S wave velocity, km/s

NormalContinent

3.5

4.0

4.5

5.0

C, k

m/s

Rayleigh

3.5

4.0

4.5

5.0

C, k

m/s

5 10 20 50Period, s

Love

A

Fig. 6. Sensitivity of surface-wave phase velocities to the depth of the Moho in a typical contthe Moho is shifted up and down, at 1 km increments, from its 37-km reference depth. GLove-wave phase velocity curves computed for these models (B and C, respectively), and thmodel that has a 37-km thick crust (A). Black lines correspond to the models with the Moh


depth variations of only a few kilometres will correspond to easily de-tectable (>1%) perturbations in phase velocities.

Group velocities (Fig. 7) show an even stronger sensitivity to theMoho depth. For a typical continental crustal thickness (37 km), aMoho-depth change of only 1 km for Rayleigh or 2 km for Lovewaves results in a group-velocity perturbations up to almost 1%.

The sensitivity of surface waves to the Moho beneath oceans(Figs. 8 and 9) is different from that beneath continents, both becauseof the small thickness of the oceanic crust and because of the pres-ence of the water layer, which has a strong effect on the propagationof Rayleigh waves (Figs. 1–3). Until recently, it has been difficult tomeasure phase or group velocities in oceans at periods sufficientlyshort to resolve the shallow oceanic Moho. In the last few years,deployments of arrays of Ocean-Bottom Seismometers (OBS) have fi-nally provided the data for such measurements (e.g., Harmon et al.,2012; Yao et al., 2011). Although measurement errors in thesurface-wave data from OBS arrays are relatively large, the signal of

Phase velocity

100 200

-2

-1

0

1

2δC

, %

-2

-1

0

1

2

δC, %

5 10 20 50 100 200Period, s

E

DB

C

inental model. Vs and other model parameters are fixed in the crust and the mantle, andrey and black lines show the 1-D models tested (A), the corresponding Rayleigh- ande relative changes in phase velocities (D, E), with respect to the curve for the referenceo depth within 3 km of the reference value.


image of Fig.�5

image of Fig.�6


0

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Dep

th, k

m

3.5 4.0 4.5S wave velocity, km/s

NormalContinent

3.0

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U, k

m/s

Rayleigh

Group velocity

3.5

4.0

4.5

U, k

m/s

5 10 20 50 100 200Period, s

Love

-6

-4

-2

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2

4

6

δU, %

-6

-4

-2

0

2

4

6

δU, %

5 10 20 50 100 200Period, s

E

DB

C

A

Fig. 7. Sensitivity of surface-wave group velocities to the depth of the Moho in a typical continental model. Definitions of the profiles and curves are as in Fig. 6.


the crustal structure and thickness in these data is also large. A 1-kmperturbation in the depth of an oceanic Moho corresponds to a per-turbation of 0.75% in phase velocity and over 2% in group velocity ofRayleigh waves (Figs. 8 and 9).

Fig. 10 summarises the sensitivity of phase and group velocities tothe Moho depth in different tectonic settings. The cumulative misfitsbetween the perturbed and reference phase- and group-velocitycurves (top row) are computed over the entire length of the broad-band curves, with sample spacing increasing logarithmically with in-creasing period so as to equalize, roughly, the weight of the structuralinformation given by different parts of the phase-velocity curve, sen-sitive to different depth intervals within the Earth (Bartzsch et al.,2011). The misfits do not have a physical meaning; comparisons ofthe misfits in the analysis and inversion of the same phase-velocitycurves, however, are consistent and meaningful. The misfits showsteep valleys with clear minima at the correct (reference) Mohodepth values. For continents with either normal or thickened crust,Rayleigh and Love waves show a similar sensitivity to the Moho,with the periods of maximum sensitivity increasing with an increas-ing Moho depth, and with the period ranges of sensitivity broaderfor Love waves compared to Rayleigh waves. Generally, perturbationsin the depth of a deeper Moho can be expected to translate into

0

10

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30

40

50

Dep

th, k

m

0 1 2 3 4S wave velocity, km/s

Ocean

2

3

4

5

C, k

m/s

Rayleigh

3.5

4.0

4.5

5.0

C, k

m/s

5 10 20 50 1Period, s

Love

A

Fig. 8. Sensitivity of surface-wave phase velocities to the depth of the Moho in a


smaller phase-velocity changes compared to those in the depth of ashallower Moho, because the sensitivity kernels of surface wavessampling the deeper Moho will be broader (Fig. 1). The thickestcrust beneath high plateaux, however, is also characterised by lowseismic velocities within it (e.g., Agius and Lebedev, 2010; Yang etal., 2012), which enhances the crust-mantle, seismic-velocity contrastand, thus, the visibility of the Moho.

4. Trade-offs between the Moho depth and othermodel parameters

The effect of the Moho on surface wave speeds reflects primarilythe shear-wave speed increase from the crust to the mantle. Varia-tions in shear speeds in the lower crust or uppermost mantle giverise to perturbations in surface-wave speeds similar to those due toMoho-depth variations. If seismic velocities in the crust and mantleare not known a priori – as is the case most often – then an inversionof surface-wave data will suffer from a trade-off between the param-eters for the Moho depth and the crustal and mantle shear speeds.The resulting model non-uniqueness translates into uncertainty inthe Moho depth.

Phase velocity

00 200

-2

-1

0

1

2

δC, %

-2

-1

0

1

2

δC, %

5 10 20 50 100 200Period, s

E

DB

C

typical oceanic model. Definitions of the profiles and curves are as in Fig. 6.


image of Fig.�7

image of Fig.�8


0

10

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40

50

Dep

th, k

m

0 1 2 3 4S wave velocity, km/s

Ocean

1

2

3

4

5

U, k

m/s

Rayleigh

Group velocity

3.0

3.5

4.0

4.5

5.0

U, k

m/s

5 10 20 50 100 200Period, s

Love

-6

-4

-2

0

2

4

6

δU, %

-6

-4

-2

0

2

4

6

δU, %

5 10 20 50 100 200Period, s

E

DB

C

A

Fig. 9. Sensitivity of surface-wave group velocities to the depth of the Moho in a typical oceanic model. Definitions of the profiles and curves are as in Fig. 6.


The trade-offs are quantified in Fig. 11. In each of the two-parameterplanes, the parameter on the horizontal axis is for the Moho depth, andthe parameters on the vertical axes are for shear-speed perturbations inthe lower crust and in theuppermostmantle and for the thickness of theMoho. For both Rayleigh and Love waves, the Moho depth and shearspeeds above and below theMoho show the expected trade-offs: amis-fit due to an increase (decrease) of the Moho depth can be compensat-ed, to a large extent, by an increase (decrease) of the wavespeeds aboveor below theMoho. This is fundamentally due to the broad depth rangeof surface-wave depth sensitivity functions (Fig. 1).

The trade-off between the Moho depth and its thickness is weak,and the sensitivity of surface waves to the Moho thickness in generalis low. Surface waves alone are thus insufficient to determine wheth-er the crust–mantle transition is a sharp discontinuity or a gradientover a depth range. The fine structure of a discontinuity can, however,be investigated by means of joint analysis of surface-wave data ormodels and other data, such as receiver functions (e.g., Endrunet al., 2004; Julià et al., 2000; Lebedev et al., 2002a, 2002b; Shen etal., 2013; Tkalčić et al., 2012). The incorporation of such additionaldata can also reduce the trade-offs between the Moho depth andshear speeds (Fig. 11).

5. Inversion of surface-wave measurements for the Moho depth

We now set up an inversion procedure that will help us to not onlydetermine the best-fitting Moho-depth values but also explore theproperties of the multi-parameter model space that are most relevantto the Moho depth and its uncertainty. The procedure is similar tothat described by Bartzsch et al. (2011), who projected thesmallest-misfit surface in a multi-dimensional parameter space onto atwo-parameter plane (the two parameters in that study being thedepth and the thickness of the lithosphere-asthenosphere boundary).Here, we use a one-parameter axis instead of a two-parameter planeand focus on the Moho depth only (the Moho thickness being difficultto constrain with surface waves with useful accuracy (Fig. 11)). Ourgoal is to investigate the general properties of the inversion ofsurface-wave data for the Moho depth.

5.1. Mapping the model space

For every point along the Moho-depth axis, we perform anon-linear gradient search inversion in which the Moho depth isfixed and the crustal and mantle structure is varied, so as to minimise


the misfit between the synthetic andmeasured phase-velocity curves.Perturbations to the background shear-speed profiles (Fig. 12A) areparameterised using 15–20 boxcar (crust) and triangle (mantle)basis functions, with the width of the basis functions increasingwith depth (see Bartzsch et al., 2011, for details). It is importantthat the crustal and mantle structure is over-parameterised, i.e. thatthe number of basis functions is large enough so that the choice of aparticular number does not affect the minimum misfit achievablewith various Moho depths. At the same time, the shear-speed profilesare constrained to be relatively smooth, both implicitly, by the finitewidths of the basis functions (10 km or greater depth ranges inthe crust; a few tens of km in the mantle) and by the slight normdamping applied to the inversion parameters. Given reasonablyaccurate phase-velocity measurements, small damping is sufficientto rule out exotic models with unrealistic shear-speed values.Compressional-wave speed perturbations are coupled to shear-wavespeed ones (δVP (m/s)=δVS (m/s)). (This assumption is reasonablefor the upper mantle but not always for the crust, particularly insedimentary layers. In the examples below, sedimentary layers areabsent or insignificant, but in general the variations in crustalPoisson's ratios will add to the uncertainties of the inversion for theMoho depth; a priori information on the structure of the sedimentsis thus particularly valuable (Section 7.2).) The non-linear gradientsearch is performed with the Levenberg–Marquardt algorithm. Syn-thetic phase velocities are computed directly from one-dimensional(1-D) Earth models at every step during the gradient search, using afast version of the MINEOS modes code (Masters, http://igppweb.ucsd.edu/∼gabi/rem.dir/surface/minos.html), which we modifiedfrom the version of Nolet (1990). The gradient search is not linearisedand converges to true best-fitting solutions (Erduran et al., 2008).

The inversion procedure is thus a grid search (the grid, in this case,being one-dimensional) that comprises numerous non-linear gradi-ent searches, one at each knot along the Moho axis. The gradientsearches determine best-fitting shear-speed profiles that minimisethe misfit as much as possible with the Moho depth fixed at thevalue that defines the point on the axis. Any trade-offs between theMoho depth and shear speeds above and below it will contribute tominimizing the impact of the Moho on the misfit function (that is,the gradient-search inversion will compensate, as much as possible,the impact of changes in the Moho depth with changes in shearspeeds above and below it). If the Moho depth, however, is not con-sistent with the data, then the best possible fit will still be relativelypoor.


http://igppweb.ucsd.edu/~gabi/rem.dir/surface/minos.html

http://igppweb.ucsd.edu/~gabi/rem.dir/surface/minos.html

image of Fig.�9


0.0

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2.5

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fit x

103

10 20 30 40 50 60 70 80Depth, km

Phase velocity

Tibet

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Ocean : Rayleigh: Love

Rayleigh waves

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iod,

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iod,

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Tibet

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Rayleigh waves

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Ocean

30 40

Normalcontinent

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iod,

s

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Per

iod,

s

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Tibet

Depth, km

Phase and group velocity perturbation, % -3 -2 -1 0 1 2 3

Fig. 10. Sensitivity of the phase-velocity (left) and group-velocity (right) curves of fundamental-mode surface waves to the depth of the Moho in different tectonic settings. Top:misfits computed over the length of the broad-band curves. The misfits are due to deviations of the Moho depths from their reference values (“ocean”: 11 km relative to the seasurface; “normal continent”: 37 km from the surface; “Tibet”: 65 km from the surface). Seismic velocities in the crust and mantle are fixed. Middle and bottom: Phase- andgroup-velocity changes at each period due to changes of the Moho depth from its reference value. The 1-D models and corresponding phase- and group-velocity perturbationsfor a “normal continent” (37-km Moho depth) are the same as in Figs. 6 and 7; for an ocean — same as in Figs. 8 and 9. The “Tibet” reference model has a 65-km thick crust anda relatively low, 3.6 km/s S-wave velocity in the lower crust (Agius and Lebedev, 2010).


5.2. Resolution and trade-offs

We first apply the model space mapping procedure to syntheticphase-velocity curves, computed for a reference model with a37-km Moho depth (Fig. 12A). The results show that both the Ray-leigh and Love wave data have the capacity to resolve the Mohodepth accurately: the V-shaped misfit curves have a clear minimumat the correct Moho depth (Fig. 12B).


Although the V shapes of the misfit curves (Fig. 12B) look similarto those in the sensitivity tests where only the Moho depth was var-ied (Fig. 10, top), the curves are, in fact, quite different: the misfits arenow around 10 times smaller. The best-fitting, phase-velocity curvesfor all the Moho depths tested in the inversion are much closer to thereference curve (synthetic data) and to each other (Fig. 12C, D) thanthe different curves in the sensitivity tests (Fig. 6B, C). Thisorder-of-magnitude reduction in the misfits is due to the trade-offs


image of Fig.�10


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Moh

o th

ickn

ess,

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Misfit x1040 2 4 6 8 10

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er-m

antle

velo

city

var

iatio

n, k

m/s

Low

er-c

rust

velo

city

var

iatio

n, k

m/s

Fig. 11. Trade-offs between the Moho depth and other seismic model parameters. The reference model (a cross in each frame) is as in Fig. 6(A), with a 37-km deep Moho. In each ofthe tests, the Moho depth and one other parameter were perturbed within the ranges shown, with the rest of the model unchanged, and the misfit was computed at each pointwithin the 2-parameter planes. The misfit is the average relative difference between the perturbed and reference broad-band phase-velocity curves computed over their entirelength (5–250 s). Top: the trade-off between the Moho depth and S-wave velocities in the lower crust (between 15-km depth and the Moho). Middle: the trade-off betweenthe Moho depth and S-wave velocities in the uppermost mantle (between the Moho and a 100-km depth). Bottom: the (weak) trade-offs between the depth and thickness ofthe Moho. Variations in the Moho thickness were parameterised using a layer with a linear seismic-velocity increase within it, centred at the value of the Moho depth.


between the Moho depth and the crustal and mantle structure. Theadjustments in the crustal and mantle structure determined in thecourse of an inversion can compensate for an incorrect Moho wellenough to mask around 90% of the signal.

The effects of the trade-offs can be clarified further by a comparisonof the relative differences of phase-velocity curves in the inversion(where both the Moho depth and the crustal and mantle structurewere varied, Fig. 12E-H) and in the sensitivity tests (where only theMoho depth was varied, Figs. 6D, E and 10, middle and bottom left). Ifonly the Moho depth is perturbed, then its change by a few kilometresresults in phase-velocity perturbations that vary gradually with periodand reach a maximum on the order of 1% (Figs. 5, 6). In the inversion,where the effect of the Moho-depth perturbations is partly compensat-ed byperturbations in crustal andmantle seismic-velocity structure, thesame Moho-depth changes result in oscillatory phase-velocity pertur-bations up to a maximum on the order of 0.1% only.


5.3. Inversion of measured data: Northern Kaapvaal Craton

Applying the inversion to real data, we now invert phase-velocitycurves measured in northern Kaapvaal Craton (24-26S, 26-32E),southern Africa (see the map in Fig. 5). Adam and Lebedev (2012)computed the average curves for this region by averaging thousandsof inter-station measurements, obtained by both cross-correlationand multimode waveform inversion (Lebedev et al., 2006, 2009;Meier et al., 2004). (The region-average measurements and inver-sions are meaningful because the Moho depth shows variations ofonly a few kilometres across the northern Kaapvaal Craton, andshear-velocity heterogeneity is also limited, according to publishedreceiver-function studies and tomography (e.g., Kgaswane et al.,2009; Nair et al., 2006; Yang et al., 2008).) The highly accuratephase-velocity curves span very broad period ranges, particularlyfor Rayleigh waves (up to 5–400 s).


image of Fig.�11


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th, k

m3.5 4.0 4.5

S wave velocity, km/s

Normalcontinent

A

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fit x

104

30 36 42 48Moho depth, km

B

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, km

/s

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Love

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δC, %

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F


5

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iod,

s

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10

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Per

iod,

s

Phase velocity perturbation δC, %-0.2 -0.1 0.0 0.1 0.2

G

H

Fig. 12. Resolution tests: model-space-map inversions of synthetic phase-velocity curves for the Moho depth. The inversions of Rayleigh (C, E, G) and Love (D, F, H) waves wereperformed separately. The model spaces were explored by means of a uniform sampling of the Moho-depth axis, with a non-linear, gradient-search inversion at each point. Ineach of the gradient searches, the Moho depth is fixed but the crustal and mantle shear-speed structure is allowed to vary, so that the trade-offs between the Moho depth andshear velocities are taken into account. A: the best-fitting shear-speed profiles for each Moho depth in the 29–49 km range. B: the minimum data-synthetic misfits given by thegradient-search inversions at each of the 1-km-spaced points on the Moho-depth axis (crosses). C, D: Rayleigh and Love phase-velocity curves, coloured for the reference modeland grey and black (very close to or behind the coloured curves) for all the profiles in (A). E, F: differences between the best-fitting phase-velocity curves determined for variousMoho depths and the reference curves computed for the reference model with a 37-km Moho depth. Black lines in A, E, F indicate globally best-fitting models and curves, withcumulative misfits below the threshold indicated by the dashed line in (B). G, H: Period-dependent differences between the best-fitting phase-velocity curves at each (fixed)Moho depth and the reference curves. The characteristic patterns of alternating positive and negative differences in the Moho depth-period plane reflect the trade-offs of theMoho depth and shear-speed structure in the crust and the mantle.


In Fig. 13 we show the results of three different inversions of theRayleigh-wave phase velocities. In the first (Fig. 13, top row: C, F, I),we inverted the measured dispersion curve in a relatively broad peri-od range (5–70 s), in which it had substantial sensitivity to the upperand lower crust, to the Moho, and to the lithospheric mantle. Themeasured curve can be fit with synthetic curves closely (within aline thickness in Fig. 13C). The misfits can be seen more clearlywhen relative phase-velocity differences are plotted (Fig. 13F); theysuggest that the noise in the measurements is up to 0.1–0.2%, varyingwith period. All the shear-velocity profiles corresponding to the syn-thetic phase-velocity curves in Fig. 13C and F (as well as in Fig. 13D, E,G and H) are plotted in Fig. 13A. The pattern of frequency-dependentphase-velocity perturbations due to Moho-depth changes (Fig. 13I) issimilar to that in inversions of noise-free, synthetic data (Fig. 12G),but with distortions due to the errors in the measurements.

In the second inversion (Fig. 13, middle row: D, G, J), we attemptto remove the effects of the noise and invert the dispersion data inthe same period range as in the first inversion but “smoothed” before-hand. The smoothing was by means of an over-parameterised andunder-damped, gradient-search inversion of the phase-velocitycurve for a 1-D shear-velocity profile (the profile itself being of no im-portance). While this inversion can fit structural information in thedata, it cannot fit random errors with a strong period dependence(“high-frequency noise”), inconsistent with any plausible Earthmodels. Random errors thus get smoothed out to a large extent. Com-pared to the original-data inversion, the inversion of the smootheddispersion curve reaches smaller misfits for best-fitting Moho depths(Fig. 13G). It also shows a more regular pattern of perturbations ofbest-fitting phase velocities as a function of the Moho depth (Fig. 13J).

The smoothed-curve inversion (Fig. 13D, G, J) confirms that whenrandom errors in the data are reduced, frequency-dependent misfitsdisplay patterns that are more similar to those in synthetic-data in-versions, and the Moho depth can probably be resolved. The smooth-ing, however, may by itself introduce new biases into the data. For


this reason, the inversions of smoothed data are best used for testingand validation, and not as the primary way to determine the Mohodepth. Ideally, the results of the smoothed-data and original-datainversions should be consistent, indicating their robustness (e.g.,Deschamps et al., 2008a; Endrun et al., 2011). The cumulative misfitcurves in our inversions for the Moho depth, however, do not showsuch consistency and are substantially different for the two inver-sions (Fig. 13B): the larger misfits given by original-data inversionsform a broader smallest-misfit valley, centred at Moho depths thatare 7–8 km greater, compared to the misfits in the smoothed-curveinversion. This implies that the accuracy of the original-data inver-sion has suffered from the errors in the data (which we estimatedto be up to ~0.2%).

In order to reduce the effect of measurement errors, we set up athird inversion (Fig. 13, bottom row: E, H, K). We now invert anarrow-band curve, in a period range most sensitive to the Moho(15–32 s). Because this narrow-band curve has limited sensitivity tothe crustal and mantle structure, an accurate reference profile ofcrustal and mantle shear-velocity must be used. Such profile is pro-vided by the results of the original, broad-band inversion.

The narrow-band inversion shows a steep misfit valley, withbest-fitting Moho depths in the 37–41 km range (Fig. 13B). Thesevalues are roughly consistentwith theMohodepths of 40–45 kmdeter-mined in the region using receiver functions (Kgaswane et al., 2009;Nair et al., 2006) and the Moho depths of 40–43 km constrained byRayleigh-wave measurements in a 6–40 s period range, made withcross-correlations of ambient seismic noise and inverted with startingmodels similar to those given by receiver-function analysis (Yang etal., 2008) (Fig. 5). Receiver-function measurements have their own un-certainties due to trade-offs of theMohodepth and the crustal Vp/Vs ra-tios; in surface-wave inversions, uncertainties result from trade-offs ofthe Moho depth and crustal and uppermost-mantle shear-velocitystructure. These uncertainties are the most likely reason for the appar-ent small discrepancy between the differentmeasurements in northern


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Fig. 13. Inversion of the Rayleigh-wave phase-velocity curve from the northern Kaapvaal Craton. A: best-fitting, shear-speed profiles computed for Moho depths fixed at valuesbetween 28 and 60 km. The black profiles correspond to the black phase-velocity curves in (H). B: minimum misfits given by the gradient searches with the Moho depth fixedat various values (crosses) and the crustal and mantle structure allowed to vary. Blue, red and green curves correspond to the inversion of the measured broad-band curve (toprow: C, F, I), smoothed broad-band curve (middle row: D, G, J), and a narrow-band curve over periods most sensitive to the Moho (bottom row: E, H, K), respectively. Whenthe measured, broad-band curve is inverted, the modest noise at the shortest and longest periods contributes to misfits sufficiently to make the Moho depth very uncertain. Inver-sion of the narrow-band curve, with an accurate background shear-speed model pre-computed in a preliminary broad-band inversion, yields the most robust results.


Kaapvaal Craton. Joint analysis of surface-wave and receiver-functiondata could help to reduce some of these uncertainties and, also, to con-strain thefine structure of theMoho. Small discrepancies notwithstand-ing, the close agreement between the results of the receiver-functionanalysis and those yielded by the surface-wave inversion with no apriori information (shear speeds in the crust and upper mantle wereallowed to vary in unlimited, very broad ranges, and the trial Mohodepth values spanned a very broad, 28–60 km range) validates the in-version set-up and confirms the resolving power of surface waves.

The inversion procedure that is optimal thus has two steps: in thefirst step, we use a broad-band dispersion curve to determine themantle and crustal structure with a reasonable accuracy (Fig. 13A);in the second step, we use that as a reference model (which can stillbe perturbed) while inverting only the part of the curve in the narrowperiod range where the signal of the Moho depth is the strongest andmost likely to be well above the noise level. (This assumes that theMoho is associated with a seismic-velocity contrast. This contrast isseen, empirically, in the steep increase in the phase velocity at pe-riods sampling primarily the depth range around the Moho. It is thisperiod range that is used in the second-step inversion.)

Love-wave phase-velocity curves show sensitivity to the Mohodepth similar to that of the Rayleigh-wave ones. Unfortunately, thereis usually more noise in Love-wave measurements. For the northernKaapvaal Craton, errors in the Love-wave phase-velocity curve ofAdam and Lebedev (2012) appear to be up to 0.2–0.3% at 5–50 s andup to 0.5% at 60–70 s. Inversions of Love-wave phase velocities,performed in the same way as those for Rayleigh waves (Fig. 13) didnot provide robust solutions for the Moho depth. We also attemptedjoint Love and Rayleigh inversions, allowing for radial anisotropy, but


Love-wave data did not contribute usefully to constraining the Mohodepth, due to the higher levels of noise in them.

6. Noise in the data: how much is too much for the Mohoto be resolved?

As we saw in Section 5.3, errors in surface-wave measurements(noise in the data) can bias the Moho-depth values yielded by the in-version of the data. This will occur regardless of what inversion ap-proach is used. The trade-offs between the Moho depth and crustaland mantle seismic velocities make the signal of the Moho depth inthe data very subtle: ~0.1–0.2% of the phase velocity values. If the in-version of the data accounts for the trade-offs correctly and if no apriori information on seismic velocities is available, then an amountof noise that is similar to or higher in amplitude than the signal ofthe Moho may bias the results of the inversion.

The resolvability of the depth of a continental Moho with surface-wave, phase-velocity data alone (and with no a priori information) isnot warranted if the noise level exceeds ~0.2%. With stronger noise,the Moho may or may not be resolved correctly, depending on thecharacter of the noise.

We illustrate the effects of different noise patterns in Fig. 14. (Forcompleteness, misfits as a function of periods are presented inSupplementary Fig. 1). Five different synthetic phase-velocity curveswere inverted in different model-space-map inversion tests. One ofthe five curves was computed for a cratonic seismic-velocity profile(Fig. 14A, dashed line), and the other four were obtained by an addi-tion of different patterns of noise to this curve.


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Fig. 14. The effect of measurement errors on the results of surface-wave inversions for the Moho depth. A: synthetic phase-velocity curves were computed for the “true referencemodel” (dashed line); in the different tests, both this model and AK135 (solid line) were used as the reference. B–E: different patterns of noise added to the synthetic curves beforetheir inversion. F, G: results of model-space-map inversions for the Moho depth. The minimum of every misfit curve shows the best fitting Moho value. While the effect of the ref-erence model is small, errors in the data of only 0.3–0.6% can cause large errors (up to 10 km) in the retrieved Moho depth, depending on their distribution with period. Completepresentation of misfits as a function of period for each of the tests is given in Supplementary Data (SFig. 1).


In order to isolate the effect of the assumed reference model on themodel-space-map inversion (this effect is due to the damping appliedin the gradient searches), each of the five dispersion curves wasinverted twice, first with the correct, “true reference model”, and thenwith a substantially different, global reference model (Fig. 14A, solidline). These tests confirmed that the influence of the reference modelis limited, much smaller than that of the noise in each case (Fig. 14F, G).

The first two noise patterns were estimated from real data, mea-sured across the Limpopo Belt (Adam and Lebedev, 2012). The “com-plete estimated noise” (Fig. 14C) is the difference between themeasured and synthetic phase velocities, the latter computed for a“preferred,” smooth 1-D profile obtained in a damped, gradient-search inversion of the data. The “high-frequency noise” is estimatedin the same way but with the gradient-search inversion under-damped, and the 1-D profile showing some unrealistic roughness,most likely introduced by the inversion so as to fit the smoothercomponents of the noise. Regardless of how accurately these noisepatterns represent the actual errors in the Limpopo measurements,


they are reasonable estimates and are well suited for the purposesof our tests as examples of noise distribution with period.

The “high-frequency noise” pattern is characterised by random er-rors oscillating and changing sign every few seconds along the periodaxis. This noise pattern increases the misfits but has little effect on theMoho-depth values yielded by the inversions (misfit-curve minima inFig. 14G). The influence of such random-noise patterns was investi-gated previously by Bartzsch et al. (2011) in their surface-wave inver-sions for the depth of the lithosphere–asthenosphere boundary. If theerrors were random (did not persist over broad period ranges), thentheir effect on the best-fitting parameter values was small, even ifthe noise amplitude was as high as 1.0%. Here, we use noise estimatesnot from a random-number generator, as Bartzsch et al. (2011), butfrom real data. We arrive, however, to the same conclusion: random,“high-frequency” noise will be unlikely to bias the results of the in-version, in our case for the Moho depth.

In contrast, the complete estimated noise (Fig. 14C) contains errorsthat persist over relatively broad period ranges. This noise pattern


image of Fig.�14



causes large errors in theMoho depth values. Although the noise is onlyup to 0.3% in the 11–80 s period range and up to 0.6% in the 5–11 s pe-riod range, it causes a 10-km error in the Moho depth (Fig. 14G). Onecomponent of this noise pattern is a sharp phase-velocity increase at10–11 s. Approximating this component using “ramp” functions with0.3% and 0.5% amplitudes (Fig. 14D, E), we find that it alone gives riseto Moho-depth errors of a few kilometres (Fig. 14F).

It is clear from these tests that errors in phase-velocity data of only0.3–0.5% can prevent the determination of the Moho depth with use-ful accuracy, particularly if the errors vary slowly along the periodaxis. This result is general and not specific to an inversion method.

Given that there is always noise in the data, the question now is:how can we reduce the effect of noise, so as to resolve the Moho? Un-fortunately, the most damaging errors are the ones that vary smooth-ly with period, and such errors can map, to a large extent, intoartificial structure in seismic-velocity models. If period ranges withrelatively large suspected errors can be identified – for example, byan observed increase in data-synthetic, dispersion-curve misfit, byan increase of standard deviations or standard errors, or by an in-crease in frequency-dependent waveform misfit – excluding such pe-riod ranges from the inversion would be most effective (Fig. 13).

Another way to improve the resolvability of the Moho is to includea priori information in the inversions, as is done often. The errors test-ed in this section are relatively small. Most surface-wave studies ofthe Moho to date have used data with errors larger than these. Doesthis mean that the results of all these studies should be in doubt?Not if they incorporated accurate additional, a priori information, ex-plicitly or implicitly. For example, in our inversions shown in Fig. 13we included no a priori information and allowed the lower-crustaland uppermost-mantle sheer speeds to vary in very broad ranges of3.5–4.2 km/s and 4.2–4.6 km/s, respectively. Often, geological or geo-physical constraints will be available to make these ranges muchnarrower. This will steepen the misfit valleys and allow the Moho tobe resolved even with noise higher than a few tenths of a percent.

The Moho can thus be resolved even with relatively noisy data ifaccurate a priori information on the crustal and mantle seismic veloc-ities (or on the difference between the two) is available. Such con-straints have been used extensively in surface-wave crustal studies,either explicitly, with a clear analysis of the ranges of values consis-tent with existing data, or implicitly, through the choice of a referenceor starting model. It is important to keep in mind that the accuracy ofthe Moho depth yielded by such a constrained inversion will dependdirectly on the accuracy of the a priori constraints.

7. Recommended inversion strategies

Basic strategies for the surface-wave inversion for the Moho depthare the same for the different types of surface-wave observables (phasevelocities, group velocities or waveforms) and are dictated by the sensi-tivity of surface waves and the trade-offs between the discontinuitydepth and seismic-velocity structure. Resolving the Moho depth withsurface waves alone is possible but is guaranteed to work only if the er-rors of themeasurements are very small (e.g., up to ~0.2% of phase veloc-ities for mapping the Moho beneath continents). Data with larger errorscan still be used effectively if accurate a priori information is availablethat can reduce the possible ranges of seismic velocities in the crustand uppermost mantle or of the velocity contrast across the Moho.(More information translates into better resolution in the presence ofthe same errors.) Similarly, relatively noisy surface-wave data can besuccessfully inverted for the Moho jointly with data of other types thatprovide complementary sensitivity, such as receiver functions.

7.1. Inversion of surface-wave data only, with no a priori information

Contrary to the conventional wisdom, inverting surface-wavemeasurements in a period range as broad as possible directly for the


Moho depth is not optimal. Because the Moho depth is constrainedby very subtle signal in surface-wave data, even small errors atshort periods (sensitive primarily to the upper crust) or longer pe-riods (sensitive primarily to the deep mantle lithosphere) can biasthe inversion results, because of the trade-offs between seismic veloc-ities at different depths. In order to determine the Moho depth usingsurface-wave data only, with no a priori information on crustal ormantle structure, the most effective inversion strategy is to first con-strain the crustal and mantle structure by an inversion of the data in abroad period range and then, as a second step, to find best-fittingMoho depths in an inversion in a narrow period range with themost sensitivity to the Moho, using the results of the first step as areference model (see Section 5.3 for details).

7.2. A priori information: include whenever available!

A successful inversion of surfacewaves alone for theMoho depth re-quires high accuracy of the measurements, because the strong sensitiv-ity of surface waves to the Moho is reduced, in inversions, by severetrade-offs of the depth of the Moho and the shear-speed structureabove and below it. Accurate a priori constraints on crustal and mantlestructure will reduce the trade-offs and should be sought when possi-ble. Such constraints come, for example, from controlled-source crustalimaging in the region, from receiver-function studies, or from crustalxenoliths (e.g., Christensen and Mooney, 1995).

7.3. Joint analysis and inversion of surface-wave and other data

Joint analysis or joint inversion of surface-wave and other data canreduce model uncertainties, making Moho mapping possible evenwhen the accuracy of surface-wave measurements alone is insuffi-cient for the purpose. Surface-wave and receiver-function data com-plement each other especially well: both types of data are yieldedby passive imaging methods and can be obtained from a small arrayof broadband stations situated virtually anywhere in the world.And, whereas surfacewaveswith their broad depth sensitivity kernels(Fig. 1) provide strong constraints on shear speeds within depthranges (i.e., on smooth variations of shear speed with depth), receiverfunctions have particular sensitivity to sharp discontinuities (e.g.,Endrun et al., 2004; Julià et al., 2000; Shen et al., 2013; Tkalčić et al.,2012).

While the joint inversion of seismic data of different types is wellestablished, recent developments in computational petrological andgeophysical modelling now also make increasingly feasible joint in-versions of surface-wave and other geophysical and geological data.Fullea et al. (2012) developed a joint inversion of surface-wave dis-persion curves and topography and applied it to data from centralMongolia, also incorporating constraints from surface heat flow mea-surements, controlled-source seismic data, and crustal xenolith anal-ysis. Designed for determination of the thermal structure of thelithosphere with a high vertical resolution, such petro-physical inver-sion also constrains the depth of the Moho, using the complementarysensitivities of the surface-wave and other data.

8. Discussion

In our investigation of the resolution of surface-wave inversionsfor the Moho depth, we considered only one type of surface-wave ob-servables, their phase velocities. Our results, however, characterisegeneral data-model relationships and apply, with obvious adjust-ments, to inversions of other surface-wave observables, includinggroup velocities and waveforms.

Group velocities have a higher sensitivity to the Moho depth, com-pared to phase velocities, in the absence of errors in the measure-ments or with the same error levels (Figs. 6–11). This advantage ofgroup velocities is offset, normally, by the larger actual errors in




their measurements. In group-velocity tomography, further errorsmay come from the group delays' off-ray sensitivity to local phase-velocity perturbations and their dispersion (Eq. (2)), in additionto the expected sensitivity to local group velocity perturbations(Dahlen and Zhou, 2006). In spite of this, the high sensitivity ofgroup velocities to the Moho depth has been exploited successfullyin Moho mapping, including in joint inversions of group and phasevelocities (e.g., Shapiro and Ritzwoller, 2002).

Although we have focussed so far on the fundamental-mode sur-face waves only, higher surface-wave modes also sample the crustand the Moho. The growing global data sets of higher-mode,phase-velocity measurements in increasingly broad frequency ranges(Schaeffer and Lebedev, submitted for publication; Van Heijst andWoodhouse, 1999; Visser et al., 2007) will offer new types of con-straints on the global crustal structure. At a regional scale, the utilityof higher modes has already been clearly established: Levshin et al.(2005) measured group velocities of Rayleigh- and Love-wave firstcrustal overtones in the 7 – 18 s period range and showed that thejoint inversion of the fundamental and higher mode measurementsincreased the resolution of crustal structure.

Lateral variations in the Moho depth (Moho topography) can beresolved using tomography based on surface-wave ray theory ifthey are smooth enough so that the validity of surface-wave ray the-ory is warranted (Dahlen and Tromp, 1998) or, at least, so that suffi-cient amount of surface-wave signals in the time-frequency planescan be selected empirically, such that they can be modelled usingray theory (Das and Nolet, 1995; Lebedev et al., 2005; Schaeffer andLebedev, submitted for publication). Strong lateral changes in theMoho depth result in strong lateral seismic-velocity changes andgive rise to scattering and multipathing (e.g., Levshin et al., 1992;Meier and Malischewsky, 2000; Meier et al., 1997). The Moho topog-raphy itself, if not accounted for accurately, can also cause biases inthe interpretation of surface-wave measurements that average overpaths or areas (Levshin and Ratnikova, 1984). In order to resolvesmall-scale Moho topography these effects have to be taken into ac-count in the inversion for local phase or shear velocities (Wielandt,1993).

The strong trade-offs between the depth of the Moho and theshear speeds just above and just below it – discussed throughoutthis paper – also have implications for large-scale mantle tomographythat uses intermediate- and long-period surface waves. Here, thetrade-offs play a positive role, reducing the effect of uncertainties inthe Moho depth on mantle models. If the inversion set-up includesa few parameters for wavespeeds within the crust and a reasonablydense parameterisation in the lithospheric mantle, and if the refer-ence model for the tomography is 3-D and includes a realistic crust,such as CRUST2.0 (Bassin et al., 2000) or a more detailed regionalmodel (e.g., Grad et al., 2009; Molinari and Morelli, 2011; Tesauroet al., 2008), then the Moho depth may not need to be perturbed inthe inversion at all (e.g., Lebedev and van der Hilst, 2008; Legendreet al., 2012; Schaeffer and Lebedev, submitted for publication). The ef-fects of differences between the true and 3-D-reference Moho depthswill be compensated by perturbations within the crust almost entirely,with an accuracy most likely better than that of the source-station,surface-wave measurements themselves, affected by uncertainties insource parameters.

9. Conclusions

Surface waves have been used to study the outer layers of theEarth since the early days of modern seismology. Their sensitivity tothe crustal thickness (the depth to the Moho) has been establishedand applied in structural studies already in the first half of the 20thcentury. The beginning of the 21st century has seen the crustalsurface-wave seismology re-energised by the emergence of newtechniques for broad-band surface-wave measurements, using the


increasingly abundant data from arrays of seismic stations. Thanksto the progress in the precision and bandwidth of the measurements,surface-wave imaging of the crust and the Moho is now reaching anew level of accuracy.

Both Rayleigh and Love waves have strong sensitivity to the Mohodepth. Tests with synthetic data show that if seismic wavespeedswithin the crust and upper mantle can be fixed (assumed to beknown), then Moho-depth variations of a few kilometres producelarge (>1%) perturbations in phase velocities, varying graduallywith period.

In inversions of surface-wave data with no a priori information,the Moho depth shows strong trade-offs with shear-wave speeds inthe lower crust and uppermost mantle. Adjustments in the crustaland mantle structure can compensate for as much as 90% of theMoho-depth signal in surface-wave data. In the inversion, changesof a few kilometres in the depth of a continental Moho result in oscil-latory phase-velocity perturbations that reach a maximum on theorder of 0.1% only. For the inversion to be guaranteed to resolve theMoho depth with useful precision, very high accuracy of the measure-ments is thus required, with errors up to 0.1–0.2% at most. Errors thatpersist over broad period ranges are particularly harmful; random er-rors that vary rapidly with period have a smaller effect.

An effective strategy for the inversion of surface-wave data alonefor the Moho depth, with no a priori information, is to first constrainthe crustal and upper-mantle structure by an inversion in a broad pe-riod range, and then, as a second step, to find best-fitting Mohodepths in an inversion of the data in a narrow period range that hasthe most sensitivity to the Moho, with the results of the first stepused as a reference model.

A priori constraints on crustal andmantle structure (from controlled-source imaging, receiver-functions, xenoliths or regional geology) willreduce the trade-offs. Joint analysis or inversion of surface-wave andother data (receiver functions, topography, gravity) can reduce modeluncertainties andmakeMohomapping possible evenwhen the accuracyof surface-wavemeasurements alonewould be insufficient. Alone or as apart of multi-disciplinary datasets, surface-wave data offer unique sensi-tivity to the crustal and upper-mantle structure and are becoming in-creasingly important in the seismic imaging of the crust and the Moho.

Supplementary data to this article can be found online at http://dx.doi.org/10.1016/j.tecto.2012.12.030.

Acknowledgements

We thank the editors, I. Artemieva, L. Brown, B.L.N. Kennett andH. Thybo for their work on this volume. Insightful comments and sug-gestions by Anatoli Levshin, an anonymous reviewer and one of theeditors have helped us to improve the clarity of the paper. Mostfigures were created with the Generic Mapping Tools (Wessel andSmith, 1998). This work was funded by the Dublin Institute forAdvanced Studies and Science Foundation Ireland (grants 08/RFP/GEO1704 and 09/RFP/GEO2550).

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Mapping the Moho with seismic surface waves: A review...

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