TECTO-125734; No of Pages 18 Tectonophysics xxx (2013) xxx–xxx Contents lists available at SciVerse ScienceDirect Tectonophysics journal homepage: www.elsevier.com/locate/tecto Mapping the Moho with seismic surface waves: A review, resolution analysis, and recommended inversion strategies Sergei Lebedev a,⁎, Joanne M.-C. Adam a, b, Thomas Meier c a b c Dublin Institute for Advanced Studies, School of Cosmic Physics, Geophysics Section, 5 Merrion Square, Dublin 2, Ireland Trinity College Dublin, Department of Geology, Dublin 2, Ireland Christian-Albrechts University Kiel, Institute of Geophysics, Kiel, Germany a r t i c l e i n f o Article history: Received 30 June 2012 Received in revised form 21 December 2012 Accepted 28 December 2012 Available online xxxx Keywords: Rayleigh wave Love wave Mohorovičić discontinuity Model space Inversion Tomography a b s t r a c t The strong sensitivity of seismic surface waves to the Moho is evident from a mere visual inspection of their dispersion curves or waveforms. Rayleigh and Love waves have been used to study the Earth's crust since the early days of modern seismology. Yet, strong trade-offs between the Moho depth and crustal and mantle structure in surface-wave inversions prompted doubts regarding their capacity to resolve the Moho. Here, we review surface-wave studies of the Moho, with a focus on early work, and then use model-space mapping to establish the waves' sensitivity to the Moho depth and the resolution of their inversion for it. If seismic wavespeeds within the crust and upper mantle are known, then Moho-depth variations of a few kilometres produce large (>1%) perturbations 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, Mohodepth resolution is not warranted and depends on error distribution with period. An effective strategy for the inversion of surface-wave data alone for the Moho depth is to, first, constrain the crustal and upper-mantle structure by inversion in a broad period range and then determine the Moho depth in inversion in a narrow period range most sensitive to it, with the first-step results used as reference. Prior information on crustal and mantle structure reduces the trade-offs and thus enables resolving the Moho depth with noisier data; such information should be used whenever available. Joint analysis or inversion of surface-wave and other data (receiver functions, topography, gravity) can reduce uncertainties further and facilitate Moho mapping. © 2013 Elsevier B.V. All rights reserved. Contents 1. 2. 3. 4. 5. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface-wave studies of the crust and the Moho . . . . . . . . . . . . Sensitivity of surface waves to the Moho. . . . . . . . . . . . . . . . Trade-offs between the Moho depth and other model parameters . . . . Inversion of surface-wave measurements for the Moho depth . . . . . . 5.1. Mapping the model space . . . . . . . . . . . . . . . . . . . 5.2. Resolution and trade-offs . . . . . . . . . . . . . . . . . . . 5.3. Inversion of measured data: Northern Kaapvaal Craton. . . . . . 6. Noise in the data: how much is too much for the Moho to be resolved? . 7. Recommended inversion strategies . . . . . . . . . . . . . . . . . . 7.1. Inversion of surface-wave data only, with no a priori information. 7.2. A priori information: include whenever available! . . . . . . . . 7.3. Joint analysis and inversion of surface-wave and other data . . . 8. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ⁎ Corresponding author. Tel.: +353 1 653 5147x240. E-mail address: sergei@cp.dias.ie (S. Lebedev). 0040-1951/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.tecto.2012.12.030 Please cite this article as: Lebedev, S., et al., Mapping the Moho with seismic surface waves: A review, resolution analysis, and recommended inversion strategies, Tectonophysics (2013), http://dx.doi.org/10.1016/j.tecto.2012.12.030 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Phase velocity Group velocity dC/dVs dU/dVs 0 0 10 10 20 20 30 30 40 40 0 0 10 10 20 20 30 30 40 40 50 50 Period: 15s 60 0 Depth, km Depth, km Period: 6s Period: 15s 60 0 50 50 100 100 150 150 200 Depth, km Depth, km Period: 6s 200 Period: 40s 250 0 Depth, km The Mohorovičić discontinuity, often referred to as the Moho, separates the Earth's crust from the underlying mantle. Compositional differences between the lighter crust and the denser upper mantle give rise to an increase in seismic velocities across the Moho, from the crust to the mantle. The discontinuity can thus be identified seismically 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 the world, at various length-scales and with different seismic techniques. Controlled-source seismic surveys yield high resolution of the entire crust and the Moho by sampling them densely with rays of reflected or refracted seismic body waves, propagating between local sources and receivers (Prodehl and Mooney, 2011, and references therein). Relatively expensive and labour-intensive, controlled-source experiments can be complemented by “passive” seismic studies that use natural seismic sources (local or teleseismic earthquakes or ambient seismic noise). The passive imaging approaches include the analysis of 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; Zhu and Kanamori, 2000), surface-wave imaging, including inversions of surface-wave dispersion curves or waveforms and surface-wave tomography (e.g., Das and Nolet, 1995; Endrun et al., 2004; Yang et al., 2008), joint inversions of the P-to-S conversions (receiver functions) 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 constructed using 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 of the crust and uppermost mantle — and, thus, to the depth of the Moho. Because these waves propagate along the Earth's surface, measurements of their speeds characterise average elastic properties of the crust and upper mantle between seismic sources and stations or between different stations. The Moho can thus be imaged even in locations with no stations or sources. The two main types of surface waves are Rayleigh waves and Love waves (Aki and Richards, 1980; Dahlen and Tromp, 1998; Kennett, 1983, 2001; Levshin et al., 1989; Nolet, 2008). The speeds of Rayleigh waves depend primarily on the speeds of the vertically polarised S waves in the crust and mantle and, also, on P-wave speeds and density; the particle motion associated with Rayleigh waves in an isotropic, laterally homogeneous Earth model is within the great circle plane containing the source and the receiver. The speeds of Love waves depend primarily on the speeds of the horizontally polarised S waves and, also, on density; the associated particle motion is approximately perpendicular 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 velocities plotted as a function of period or frequency) show a characteristic sharp increase with period associated with the Moho (Figs. 2, 3). This increase reflects the S-wave velocity increase across the discontinuity, and its period range depends on the depth of the Moho: it occurs at longer periods if the Moho is deeper. The depth of the Moho can thus be estimated roughly by a mere visual inspection of a surface-wave dispersion curve (Figs. 2, 3). Inferences on the crustal structure and thickness have been drawn from surface-wave observations since the early days of modern Depth, km 1. Introduction Depth, km S. Lebedev et al. / Tectonophysics xxx (2013) xxx–xxx Period: 40s 250 0 100 100 200 200 300 300 400 400 500 Period: 100s 500 Period: 100s 0 Depth, km 2 0 dC/dVs : Rayleigh dU/dVs : Love Fig. 1. Depth sensitivity of surface waves. The sensitivity curves are the Fréchet derivatives of the phase and group velocities of the fundamental-mode Rayleigh and Love waves with respect to S-wave velocities at different depths. The derivatives were computed for a continental, 1-D Earth model with a 37-km thick crust, at 4 different periods. Each graph is scaled independently. seismology. It also became apparent early that the crustal models inferred from the dispersion data can be highly non-unique. Although the Moho depth has been an inversion parameter in numerous surfacewave studies, the data's sensitivity to the Moho and, in particular, the resolution of the Moho properties given by inversions of surface-wave data with measurement errors are still uncertain and not agreed upon. In this paper we overview the classic surface-wave studies since the late 19th–early 20th century, as well as some of the more recent work focussing on the Moho. We then investigate in detail the sensitivity of surface-wave phase velocities to the Moho depth and the trade-offs between Moho-depth and crustal and mantle shear-velocity parameters in inversions of surface-wave dispersion. Exploring the model spaces in inversions of synthetic and real data, we examine the resolution of the Moho by surface-wave measurements as a data type. Finally, we discuss strategies for an accurate estimation of the Moho depth using surface-wave data and illustrate some of them with applications to phase-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 Please cite this article as: Lebedev, S., et al., Mapping the Moho with seismic surface waves: A review, resolution analysis, and recommended inversion strategies, Tectonophysics (2013), http://dx.doi.org/10.1016/j.tecto.2012.12.030 S. Lebedev et al. / Tectonophysics xxx (2013) xxx–xxx 100 5 10 20 50 5 4 4 normal continent MOHO 3 3 5 10 20 50 100 5 10 20 50 5 MOHO 4 4 MOHO 3 5 10 20 50 period, s 100 5 10 20 50 3 2 ocean 1 Fig. 2. The signature of the Moho in phase-velocity curves of surface waves. The phase velocities of the fundamental-mode Rayleigh and Love waves were computed for an oceanic model with a 5-km water layer and a 6-km thick crust (top), a continental model with a 37-km thick crust (middle), and a model with a 65-km thick crust that fits surface-wave data from NE Tibet (Agius and Lebedev, 2010). The period ranges with the characteristic phase-velocity increase with period due to the S-velocity increases at the Moho are marked with grey shading. the velocities of surface waves – which he called “main waves” – could be used to study the properties of the outer shells of the Earth, by means of measuring phase differences between signals recorded at nearby stations. In the early 20th century, velocities of surface waves have been estimated, at first, without taking their dispersion into account. Angenheister (1906) gave a velocity estimate of 3.1 km/s for “long waves”, also citing similar, earlier estimates by Omori. Reid (1910) called surface waves “regular waves”, while also estimating their velocities. Golitsyn (cited here from his selected-works compilation: Golitsyn, 1960) used minor and major arc recordings of the 1908 Messina earthquake made at Pulkovo observatory and computed a global-average, surface-wave velocity of 3.53 km/s; dispersion, again, was not considered. This value, interestingly, is very similar to the group velocities of Love waves in a typical continent at periods below 25 s, well known today (Fig. 3). Golitsyn also argued that the velocity of surface waves should depend on the physical properties of the upper layers of the Earth and be different beneath continents and oceans. Love (1911) demonstrated the existence of transversely polarised and dispersive surface waves in layered media (the Love waves). The observation of Love waves was a direct indication for the layering within the Earth. Tams (1921) compared Rayleigh waves propagating along oceanic and continental paths and proposed that they had different velocities because the crust beneath oceans, unlike the crust beneath continents, did not comprise a granitic layer with relatively low seismic velocities 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 20 50 100 5 10 20 100 50 5 5 MOHO 4 4 normal continent 3 5 100 10 3 10 20 50 100 MOHO 5 10 20 50 3 100 5 5 Tibet MOHO 4 4 3 3 MOHO 5 10 20 50 period, s 100 5 10 20 50 group velocity, km/s MOHO 4 5 period, s 5 MOHO 3 100 5 Tibet 4 100 5 Love waves group velocity, km/s 50 5 group velocity, km/s 20 group velocity, km/s 10 group velocity, km/s 3 MOHO phase velocity, km/s MOHO 1 group velocity, km/s ocean phase velocity, km/s 2 phase velocity, km/s 4 3 5 phase velocity, km/s 5 phase velocity, km/s 4 Rayleigh waves Love waves MOHO phase velocity, km/s Rayleigh waves 5 3 100 period, s Fig. 3. The signature of the Moho in group-velocity curves of surface waves. The group velocities of the fundamental-mode Rayleigh and Love waves were computed for the same oceanic and continental models as in Fig. 2. study the properties of the Earth's crust and attempted to measure both the velocities and amplitudes of Love and Rayleigh waves. He also pointed out the differences of surface-wave propagation along oceanic and continental paths and reported, correctly, that recordings at shorter epicentral distances are dominated by shorter period waves compared to those at longer distances. The relation between dominant periods and crustal thickness, however, was not handled accurately, leading to erroneous estimates of crustal thicknesses. In order to describe Love wave propagation in realistic models of the crust, Meissner (1921) gave an expression for Love waves in a crust with a linear increase of seismic velocities with depth. Stoneley (1925) clarified the differences between group and phase velocities. He then gave a quantitative expression for Rayleigh-wave dispersion in an Earth model with a compressible fluid over an elastic half-space (Stoneley, 1926). This was particularly useful because the majority of early surface wave observations were performed for paths that traversed oceans. Furthermore, the expressions were immediately applicable at the time because the problem could be solved analytically. Surface-wave dispersion in an arbitrarily layered elastic half-space was determined by Meissner (1926) for Love waves and by Jeffreys (1935) for Rayleigh waves. Meissner (1926) also noted the nonuniqueness of dispersion-curve inversions and gave examples of different one-dimensional (1-D) Earth models that produced very similar dispersion curves. He concluded that highly accurate measurements using dense networks would be required in order to determine the structure of the outer layers of the Earth. In the early 1920s Gutenberg undertook the first systematic studies of surface-wave dispersion for both Love and Rayleigh waves (e.g., Gutenberg, 1924), also including measurements by Macelwane (1923). He identified the now well known normal surface-wave dispersion, characterised by a general increase of surface-wave speeds with Please cite this article as: Lebedev, S., et al., Mapping the Moho with seismic surface waves: A review, resolution analysis, and recommended inversion strategies, Tectonophysics (2013), http://dx.doi.org/10.1016/j.tecto.2012.12.030 4 S. Lebedev et al. / Tectonophysics xxx (2013) xxx–xxx period and indicative of the increase of elastic velocities with depth. He also inferred different crustal thicknesses for Eurasia, America and the Atlantic and Pacific Oceans. Testing crustal thicknesses of 30, 60 and 120 km, he estimated the crustal thickness for Eurasia to be around 50 km. This was a remarkable result, even though he compared, incorrectly, measured group velocities with theoretical phase velocities. Neumann (1929) found evidence for lateral heterogeneity of the Pacific plate by analysing Love and Rayleigh waves. Carder (1934) summarised Love and Rayleigh wave characteristics known at that time; the theoretical understanding of surface waves and the inversion tools that were available, however, were not sufficient to draw accurate conclusions on crustal structure. Ewing and Press (1950) gave a remarkable synthesis of surfacewave observations, using their own as well as previous measurements (Bullen, 1939; DeLisle, 1941; Wilson and Baykal, 1948). The very title of their paper, “Crustal structure and surface-wave dispersion,” emphasised the inherent link of the early surface observations to the properties of crust and the Moho. The analysis was based on sophisticated manual readings of group-velocity dispersion, using time-domain measurements of the arrival times of the dominant periods (about 15–30 s) in the dispersed waveform. The Airy phase described by Pekeris (1948) was identified correctly, based on Stoneley's equation, and theoretical group-velocity curves were fitted to the observations. The strong influence of water and sediments on Rayleigh-wave velocities was clearly established. Interpreting the results, Ewing and Press (1950, 1952) implicitly applied ray theory and derived estimates for path-average, sub-crustal velocities and the Moho depths by estimating the continental portion of the paths. The limited bandwidth of their observations, however, and their use of a simplified Earth model with one layer (water and sediments) overlying a half-space, implied that their results were most meaningful 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 higher group velocities for oceanic paths compared to continental paths in the period range of 20–100 s and confirmed that, in contrast to Rayleigh waves, 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, eliminating the phase shifts due to the source and the oceanic portions of the paths. They determined the first phase-velocity curve for North America between 18 s and 32 s, a period range where phase velocities are sensitive mainly to the crust and the Moho. Evernden (1954) analysed group velocities of Love waves between 7 and 45 s for the Pacific Basin. Using Stoneley's analytical expressions for the dispersion of Love waves in a three-layered model, he concluded that a sedimentary layer, a high-velocity crust and a seismic-velocity increase from the crust to the mantle were all necessary to explain the measurements. He also noted that the results of the surface-wave analysis were compatible with those of seismic refraction surveys. These general conclusions still stand today, although the crustal and mantle models have since been improved substantially in their details. The solution for Rayleigh-wave phase velocities in a model with a water layer overlying two solid layers made it possible to interpret Rayleigh-wave group velocity curves measured along oceanic paths. Oliver et al. (1955) discussed available dispersion measurements for Rayleigh and Love waves. They concluded that a high velocity crust is present under the oceans and were able to rule out the hypothesised existence of a large continent submerged beneath the Pacific Ocean. They also showed that at periods longer than about 25 s the dispersion curves for the Atlantic and Pacific basins were similar. Press et al. (1956) made the first single-station measurements over a 10–70 s period range for a pure continental path, between Algeria and South Africa. They noted the similarity of their measurements to those by Brilliant and Ewing (1954) for North America and, also, to a theoretical curve corresponding to a 35-km-thick, homogeneous crust overlying the mantle. They concluded, as well, that a gradual velocity increase in the crust and the mantle might be needed to explain the measured dispersion curves. Press (1956) measured phase velocities by examining phase differences across arrays of stations, each array comprising only three stations. The reading algorithm he applied can be described as a visual f-k analysis in the time domain. Based on the measurements, he presented a two-dimensional (2-D) cross-section for the crustal structure in California, with (over-estimated) Moho-depth variations from ~ 15 km near the coast to ~ 50 km beneath the Sierra Nevada. In the 1960s, the emergence of computer programs for surfacewave analysis presented unprecedented new opportunities for accurate analysis and inversion of surface-wave data. Brune et al. (1960) analysed the phase of a dispersed waveform in the time domain and proposed improved reading schemes for phase-velocity determination. Alterman et al. (1961) calculated Rayleigh-wave, phase and group velocities in a 10–700 s period range and discussed the effects of gravity and the Earth's sphericity on the waves' propagation, as well as the relation between surface waves and the Earth's free oscillations. Using the measurements of Ewing and Press (1956) and Nafe and Brune (1960), Alterman et al. (1961) also presented evidence supporting the Gutenberg's model of the Earth with a low velocity asthenosphere. Dorman and Ewing (1962) developed a linearised scheme for the inversion of surface-wave measurements and computed a Moho depth of about 39 km for the New York–Pennsylvania area. Brune and Dorman (1963) compared Love and Rayleigh waveforms at stations within the Canadian Shield in the time domain. They determined phase velocities in a 5–40 s period range and inverted them for a 1-D, S-wave velocity model with a multilayered crust and upper mantle, detecting high S-wave velocities within the mantle lithosphere of the craton. They also calculated synthetic dispersed waveforms for their model. Toksöz and Ben-Menahem (1963) followed an earlier suggestion by Sato (1955) and measured phase velocities in the frequency domain, using successive passages of surface waves at a single station. McEvilly (1964) measured the phase difference between two stations in the frequency domain, for both Love and Rayleigh waves. Inverting the resulting dispersion curves, he established that different 1-D models were needed for the horizontally and vertically polarised S waves (Vsh and Vsv, respectively). This observation became known as the Love–Rayleigh discrepancy. Santo and Sato (1966) developed a regionalization technique that may be seen as the first attempted group-velocity tomography. Knopoff et al. (1967) described filter and triangulation techniques for the determination of phase velocities. The determination of group velocities using spectrograms was then suggested by Landisman et al. (1969). Since the 1970s, the number of surface-wave studies has grown steadily. Compilations and reviews of surface-wave analyses in the beginning 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 increasingly accurate and abundant, surface waves were now used extensively for the study of the upper mantle. In the course of inversions of the long-period data, the substantial sensitivity of surface-wave speeds to crustal structure was often accounted for by means of “crustal corrections”: the effect of the crustal structure on surface-wave measurements was evaluated using a priori crustal models, usually constrained by other seismic methods (e.g., Bassin et al., 2000; Boschi and Ekstrom, 2002; Bozdag and Trampert, 2008; Ferreira et al., 2010; Kustowski et al., 2007; Lekic et al., 2010; Marone and Romanowicz, 2007; Montagner and Jobert, 1988; Mooney et al., Please cite this article as: Lebedev, S., et al., Mapping the Moho with seismic surface waves: A review, resolution analysis, and recommended inversion strategies, Tectonophysics (2013), http://dx.doi.org/10.1016/j.tecto.2012.12.030 S. Lebedev et al. / Tectonophysics xxx (2013) xxx–xxx 1998; Nataf and Ricard, 1996; Nataf et al., 1986; Nolet, 1990; Panning et al., 2010; Woodhouse and Dziewonski, 1984). Thanks to the rapid growth of broadband seismic networks since the 1990s, increasingly large surface-wave datasets were used in regional and global imaging. Many tomographic inversions included the crustal structure and thickness as inversion parameters (e.g., Das and Nolet, 1995; Lebedev and Nolet, 2003; Lebedev et al., 1997; Li and Romanowicz, 1996; Pasyanos and Walter, 2002; Shapiro and Ritzwoller, 2002; Van der Lee and Nolet, 1997) (Fig. 4), and some surface-wave studies targeted primarily the Moho itself (Das and Nolet, 1995, 1998; Marone et al., 2003; Meier et al., 2007a, 2007b). The main difficulty in resolving the Moho with surface waves remained the non-uniqueness of seismic-velocity and Moho depth models consistent with surface-wave observations. Resolving the trade-offs between the Moho depth and seismic velocities required highly accurate measurements at intermediate and relatively short periods (Fig. 2). Phase velocities of surface waves, however, were difficult to measure at short periods, with the waveforms of teleseismic surface waves distorted by diffraction at periods below 15–20 s, and with regional source-station measurements biased substantially even by small errors in earthquake locations. The surface-wave crustal imaging has been rejuvenated in the 2000s by the emergence of new, array techniques for surface-wave measurements. Phase velocities of short-period surface waves are now measured routinely using pairs or arrays of broadband stations. The measurements are mainly by means of cross-correlation of either diffracted surface waves from teleseismic earthquakes (Meier et al., 2004) or of surface waves within the ambient seismic noise (Shapiro and Campillo, 2004). The cross-correlation of surface-wave recordings from nearby stations is, essentially, the classical “two-station method” (Brilliant and Ewing, 1954; McEvilly, 1964; Press, 1956; Sato, 1955; Toksöz and Ben-Menahem, 1963). The difference of the modern and traditional applications is in the types of the signal they use. The classical two-station method was applied only to teleseismic surface waves that obeyed surface-wave ray theory, i.e., were not distorted by diffraction. Of the new techniques, teleseismic cross-correlations (Meier et al., 2004) can extract inter-station phase-velocity measurements even from wave fields diffracted at teleseismic distances, and the ambient noise cross-correlations (Shapiro and Campillo, 2004) make use of the Topography 5 constructive interference of surface waves within the ambient noise wave field that arrive to a pair of stations at and near the stationstation azimuth. Interestingly, the wave fields used for these measurements cannot at all be described by ray theory, but the phase and group velocities extracted from the cross-correlation functions can define surface-wave propagation along inter-station paths in a raytheoretical framework. Over the last few years, the new methods have been applied to broadband array data from around the world. The newly abundant short-period and broad-band surface-wave measurements are, once again, bringing the crust into the focus of surface-wave seismology (e.g., Adam and Lebedev, 2012; Bensen et al., 2007; Deschamps et al., 2008b; Endrun et al., 2008, 2011; Lin et al., 2011; Moschetti et al., 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 examine in detail the sensitivity of surface waves to the Moho and the resolution of the Moho properties that they can provide. 3. Sensitivity of surface waves to the Moho Characteristic signatures of the crustal thickness are clearly seen in various surface-wave observables, including phase-velocity curves (Fig. 2), group-velocity curves (Fig. 3), and waveforms of surfacewave trains on broad-band seismograms. The wave forms are closely related to the frequency-dependent phase velocities. In a weakly heterogeneous Earth, a complete seismogram can be computed as a superposition of the fundamental and higher surface-wave modes using the JWKB (Jeffreys–Wentzel–Kramers–Brillouin) approximation as: sðωÞ ¼ ∑ Am ðωÞ exp iωΔC m ðωÞ ; ð1Þ m where the sum is over modes m, ω is the circular frequency, Δ is the source–station distance, C m ðωÞ are the average phase velocities of the modes along the source-station path, and Am(ω) are the complex amplitudes of the modes, depending on the source mechanism and the Earth structure in the source region, as well as on geometrical spreading and attenuation (Dahlen and Tromp, 1998). Moho, from waveform tomography Moho, from CRUST2. 0 40˚N 40˚N 20˚N 20˚N 0˚ 0˚ 100˚E 120˚E -12 -6 -5 -4 -0 1 140˚E 2 topography, km 3 100˚E 5 8 120˚E 5 140˚E 15 18 100˚E 21 30 40 50 120˚E 60 140˚E 75 Moho depth, km Fig. 4. Resolving the Moho in East Asia-Western Pacific with surface-wave, waveform tomography (Lebedev and Nolet, 2003). Centre: Moho depths resulting from a 3D tomographic inversion of surface-wave forms for crustal and mantle shear-speed structure. The 1D background model had a 25-km Moho depth. The large system of linear equations solved in the inversion was assembled from results of multi-mode waveform inversions of around 4000 seismograms with source-station paths within the region. Even with a 1D reference model, the 3D inversion reproduces reasonably correct Moho depths, demonstrating the sensitivity of surface-wave waveforms to the Moho. Please cite this article as: Lebedev, S., et al., Mapping the Moho with seismic surface waves: A review, resolution analysis, and recommended inversion strategies, Tectonophysics (2013), http://dx.doi.org/10.1016/j.tecto.2012.12.030 6 S. Lebedev et al. / Tectonophysics xxx (2013) xxx–xxx Receiver functions -20 Ambient noise A Ambient noise & teleseismic tomograpy B C -20 -24 -24 -28 -28 -32 -32 20 24 28 32 20 24 28 32 20 24 28 32 32 34 36 37 38 39 40 41 42 43 45 50 crustal thickness km 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) and both 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 surface wave's energy; it depends on the phase velocity and its frequency derivative as U¼ C : 1−ðω=C ÞðdC=dωÞ ð2Þ While exploring the surface waves' sensitivity to the properties of the Moho, we shall focus on the phase and group velocities only, while noting that different surface-wave observables may have different useful properties (for example, local minima in the group velocity curve, causing an Airy phase, have some sensitivity to the sharpness of the Moho). Fig. 6 illustrates the sensitivity of the Rayleigh and Love phasevelocity curves to the Moho depth in a typical continental model with a 37-km thick crust. If seismic velocities in the crust and upper mantle can be fixed (i.e., assumed to be known), then small Moho- depth variations of only a few kilometres will correspond to easily detectable (>1%) perturbations in phase velocities. Group velocities (Fig. 7) show an even stronger sensitivity to the Moho depth. For a typical continental crustal thickness (37 km), a Moho-depth change of only 1 km for Rayleigh or 2 km for Love waves 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 because of the small thickness of the oceanic crust and because of the presence of the water layer, which has a strong effect on the propagation of Rayleigh waves (Figs. 1–3). Until recently, it has been difficult to measure phase or group velocities in oceans at periods sufficiently short to resolve the shallow oceanic Moho. In the last few years, deployments of arrays of Ocean-Bottom Seismometers (OBS) have finally provided the data for such measurements (e.g., Harmon et al., 2012; Yao et al., 2011). Although measurement errors in the surface-wave data from OBS arrays are relatively large, the signal of Phase velocity Rayleigh C, km/s 50 2 5.0 A 1 4.5 0 4.0 δC, % 0 Normal Continent 150 3.5 4.0 4.5 S wave velocity, km/s C, km/s 5.0 100 D -2 2 Love 1 4.5 0 4.0 3.5 δC, % Depth, km -1 B 3.5 -1 C 5 10 20 50 Period, s 100 200 E 5 10 20 50 Period, s 100 200 -2 Fig. 6. Sensitivity of surface-wave phase velocities to the depth of the Moho in a typical continental model. Vs and other model parameters are fixed in the crust and the mantle, and the Moho is shifted up and down, at 1 km increments, from its 37-km reference depth. Grey and black lines show the 1-D models tested (A), the corresponding Rayleigh- and Love-wave phase velocity curves computed for these models (B and C, respectively), and the relative changes in phase velocities (D, E), with respect to the curve for the reference model that has a 37-km thick crust (A). Black lines correspond to the models with the Moho depth within 3 km of the reference value. Please cite this article as: Lebedev, S., et al., Mapping the Moho with seismic surface waves: A review, resolution analysis, and recommended inversion strategies, Tectonophysics (2013), http://dx.doi.org/10.1016/j.tecto.2012.12.030 S. Lebedev et al. / Tectonophysics xxx (2013) xxx–xxx 7 A 4.0 U, km/s 50 Rayleigh 3.5 Depth, km 3.0 4.5 U, km/s 100 Normal Continent B D C E Love 4.0 3.5 150 3.5 4.0 4.5 S wave velocity, km/s 5 10 20 50 Period, s 100 200 5 10 20 50 Period, s 100 200 6 4 2 0 -2 -4 -6 6 4 2 0 -2 -4 δU, % 0 δU, % Group velocity -6 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-km perturbation in the depth of an oceanic Moho corresponds to a perturbation of 0.75% in phase velocity and over 2% in group velocity of Rayleigh waves (Figs. 8 and 9). Fig. 10 summarises the sensitivity of phase and group velocities to the Moho depth in different tectonic settings. The cumulative misfits between the perturbed and reference phase- and group-velocity curves (top row) are computed over the entire length of the broadband curves, with sample spacing increasing logarithmically with increasing period so as to equalize, roughly, the weight of the structural information given by different parts of the phase-velocity curve, sensitive to different depth intervals within the Earth (Bartzsch et al., 2011). The misfits do not have a physical meaning; comparisons of the misfits in the analysis and inversion of the same phase-velocity curves, however, are consistent and meaningful. The misfits show steep valleys with clear minima at the correct (reference) Moho depth 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 increasing Moho depth, and with the period ranges of sensitivity broader for Love waves compared to Rayleigh waves. Generally, perturbations in the depth of a deeper Moho can be expected to translate into smaller phase-velocity changes compared to those in the depth of a shallower Moho, because the sensitivity kernels of surface waves sampling the deeper Moho will be broader (Fig. 1). The thickest crust beneath high plateaux, however, is also characterised by low seismic velocities within it (e.g., Agius and Lebedev, 2010; Yang et al., 2012), which enhances the crust-mantle, seismic-velocity contrast and, thus, the visibility of the Moho. 4. Trade-offs between the Moho depth and other model parameters The effect of the Moho on surface wave speeds reflects primarily the shear-wave speed increase from the crust to the mantle. Variations in shear speeds in the lower crust or uppermost mantle give rise to perturbations in surface-wave speeds similar to those due to Moho-depth variations. If seismic velocities in the crust and mantle are not known a priori – as is the case most often – then an inversion of surface-wave data will suffer from a trade-off between the parameters for the Moho depth and the crustal and mantle shear speeds. The resulting model non-uniqueness translates into uncertainty in the Moho depth. Phase velocity 4 1 3 0 -1 2 20 B 5.0 C, km/s 30 40 Ocean 50 2 Rayleigh 0 1 2 3 4 S wave velocity, km/s δC, % C, km/s 10 Depth, km 5 A D -2 2 Love 1 4.5 0 4.0 3.5 δC, % 0 -1 E C 5 10 20 50 Period, s 100 200 5 10 20 50 Period, s 100 200 -2 Fig. 8. Sensitivity of surface-wave phase velocities to the depth of the Moho in a typical oceanic model. Definitions of the profiles and curves are as in Fig. 6. Please cite this article as: Lebedev, S., et al., Mapping the Moho with seismic surface waves: A review, resolution analysis, and recommended inversion strategies, Tectonophysics (2013), http://dx.doi.org/10.1016/j.tecto.2012.12.030 8 S. Lebedev et al. / Tectonophysics xxx (2013) xxx–xxx Rayleigh 4 U, km/s 10 20 3 2 B 1 D 5.0 30 Love 4.5 U, km/s Depth, km 5 A 40 Ocean 4.0 3.5 C 3.0 50 0 1 2 3 4 S wave velocity, km/s 5 10 20 50 Period, s 100 200 E 5 10 20 50 Period, s 100 200 6 4 2 0 -2 -4 -6 6 4 2 0 -2 -4 δU, % 0 δU, % Group velocity -6 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-parameter planes, the parameter on the horizontal axis is for the Moho depth, and the parameters on the vertical axes are for shear-speed perturbations in the lower crust and in the uppermost mantle and for the thickness of the Moho. For both Rayleigh and Love waves, the Moho depth and shear speeds above and below the Moho show the expected trade-offs: a misfit due to an increase (decrease) of the Moho depth can be compensated, to a large extent, by an increase (decrease) of the wavespeeds above or below the Moho. This is fundamentally due to the broad depth range of 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 general is low. Surface waves alone are thus insufficient to determine whether the crust–mantle transition is a sharp discontinuity or a gradient over a depth range. The fine structure of a discontinuity can, however, be investigated by means of joint analysis of surface-wave data or models and other data, such as receiver functions (e.g., Endrun et al., 2004; Julià et al., 2000; Lebedev et al., 2002a, 2002b; Shen et al., 2013; Tkalčić et al., 2012). The incorporation of such additional data can also reduce the trade-offs between the Moho depth and shear 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 only determine the best-fitting Moho-depth values but also explore the properties of the multi-parameter model space that are most relevant to the Moho depth and its uncertainty. The procedure is similar to that described by Bartzsch et al. (2011), who projected the smallest-misfit surface in a multi-dimensional parameter space onto a two-parameter plane (the two parameters in that study being the depth and the thickness of the lithosphere-asthenosphere boundary). Here, we use a one-parameter axis instead of a two-parameter plane and focus on the Moho depth only (the Moho thickness being difficult to constrain with surface waves with useful accuracy (Fig. 11)). Our goal is to investigate the general properties of the inversion of surface-wave data for the Moho depth. 5.1. Mapping the model space For every point along the Moho-depth axis, we perform a non-linear gradient search inversion in which the Moho depth is fixed and the crustal and mantle structure is varied, so as to minimise the misfit between the synthetic and measured phase-velocity curves. Perturbations to the background shear-speed profiles (Fig. 12A) are parameterised using 15–20 boxcar (crust) and triangle (mantle) basis functions, with the width of the basis functions increasing with depth (see Bartzsch et al., 2011, for details). It is important that the crustal and mantle structure is over-parameterised, i.e. that the number of basis functions is large enough so that the choice of a particular number does not affect the minimum misfit achievable with various Moho depths. At the same time, the shear-speed profiles are constrained to be relatively smooth, both implicitly, by the finite widths of the basis functions (10 km or greater depth ranges in the crust; a few tens of km in the mantle) and by the slight norm damping applied to the inversion parameters. Given reasonably accurate phase-velocity measurements, small damping is sufficient to rule out exotic models with unrealistic shear-speed values. Compressional-wave speed perturbations are coupled to shear-wave speed ones (δVP (m/s) = δVS (m/s)). (This assumption is reasonable for the upper mantle but not always for the crust, particularly in sedimentary layers. In the examples below, sedimentary layers are absent or insignificant, but in general the variations in crustal Poisson's ratios will add to the uncertainties of the inversion for the Moho depth; a priori information on the structure of the sediments is thus particularly valuable (Section 7.2).) The non-linear gradient search is performed with the Levenberg–Marquardt algorithm. Synthetic phase velocities are computed directly from one-dimensional (1-D) Earth models at every step during the gradient search, using a fast version of the MINEOS modes code (Masters, http://igppweb. ucsd.edu/∼gabi/rem.dir/surface/minos.html), which we modified from the version of Nolet (1990). The gradient search is not linearised and 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 gradient searches, one at each knot along the Moho axis. The gradient searches determine best-fitting shear-speed profiles that minimise the misfit as much as possible with the Moho depth fixed at the value that defines the point on the axis. Any trade-offs between the Moho depth and shear speeds above and below it will contribute to minimizing 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 shear speeds above and below it). If the Moho depth, however, is not consistent with the data, then the best possible fit will still be relatively poor. Please cite this article as: Lebedev, S., et al., Mapping the Moho with seismic surface waves: A review, resolution analysis, and recommended inversion strategies, Tectonophysics (2013), http://dx.doi.org/10.1016/j.tecto.2012.12.030 S. Lebedev et al. / Tectonophysics xxx (2013) xxx–xxx Misfit x103 2.0 Normal continent : Rayleigh : Love 2.5 2.0 Tibet 1.5 1.5 1.0 1.0 0.5 0.5 10 20 30 40 50 Depth, km 60 70 80 10 20 Rayleigh waves 200 Period, s : Rayleigh : Love Tibet 0.0 Ocean Normal continent 30 40 50 Depth, km 60 70 0.0 80 Rayleigh waves Tibet Ocean Normal continent Tibet 200 100 100 50 50 20 20 10 10 5 10 20 30 40 50 60 70 80 10 20 Love waves 200 Period, s Normal continent Ocean Ocean Normal continent 30 40 50 60 70 5 80 Love waves Tibet Ocean Normal continent Tibet 200 100 100 50 50 20 20 10 10 5 10 20 30 40 50 Depth, km Period, s Ocean 60 70 80 10 20 30 40 50 Depth, km 60 70 Period, s 2.5 Group velocity Misfit x103 Phase velocity 9 5 80 -3 -2 -1 0 1 2 3 Phase and group velocity perturbation, % 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 sea surface; “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- and group-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 perturbations for 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 and a 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 synthetic phase-velocity curves, computed for a reference model with a 37-km Moho depth (Fig. 12A). The results show that both the Rayleigh and Love wave data have the capacity to resolve the Moho depth accurately: the V-shaped misfit curves have a clear minimum at the correct Moho depth (Fig. 12B). Although the V shapes of the misfit curves (Fig. 12B) look similar to those in the sensitivity tests where only the Moho depth was varied (Fig. 10, top), the curves are, in fact, quite different: the misfits are now around 10 times smaller. The best-fitting, phase-velocity curves for all the Moho depths tested in the inversion are much closer to the reference curve (synthetic data) and to each other (Fig. 12C, D) than the different curves in the sensitivity tests (Fig. 6B, C). This order-of-magnitude reduction in the misfits is due to the trade-offs Please cite this article as: Lebedev, S., et al., Mapping the Moho with seismic surface waves: A review, resolution analysis, and recommended inversion strategies, Tectonophysics (2013), http://dx.doi.org/10.1016/j.tecto.2012.12.030 S. Lebedev et al. / Tectonophysics xxx (2013) xxx–xxx Phase velocity 35 40 45 15 30 35 40 Rayleigh waves 45 30 15 0 15 0.2 0 15 35 40 45 Love waves 30 30 35 40 45 0 30 30 0 30 0.2 0.2 0.1 0.1 0.1 0.1 0.0 0.0 0.0 0.0 -0.1 -0.1 -0.1 -0.1 -0.2 -0.2 -0.2 30 35 40 45 30 35 40 45 15 30 35 40 45 30 35 40 45 15 0 15 0.2 0 15 30 35 40 45 30 35 40 45 30 30 35 40 45 30 35 40 45 0 30 0.2 -0.2 30 0 30 0.2 0.2 0.1 0.1 0.1 0.1 0.0 0.0 0.0 0.0 -0.1 -0.1 -0.1 -0.1 -0.2 -0.2 30 35 40 30 35 40 45 45 15 30 35 40 30 35 40 45 45 15 0 15 30 35 40 45 30 35 40 45 0 15 30 30 35 40 45 30 35 40 45 0 30 0.2 -0.2 30 0 30 30 30 30 30 20 20 20 20 10 10 10 10 0 0 0 30 35 40 45 Moho depth, km 30 35 40 45 Moho depth, km 0 0 30 35 40 45 Moho depth, km 2 4 6 8 30 35 40 Moho thickness, km -0.2 Moho thickness, km Upper-mantle velocity variation, km/s Lower-crust velocity variation, km/s 30 Love waves Lower-crust velocity variation, km/s Rayleigh waves Group velocity Upper-mantle velocity variation, km/s 10 45 Moho depth, km 10 Misfit x10 4 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 of the 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 point within the 2-parameter planes. The misfit is the average relative difference between the perturbed and reference broad-band phase-velocity curves computed over their entire length (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 between the 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 of the 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. The adjustments in the crustal and mantle structure determined in the course of an inversion can compensate for an incorrect Moho well enough to mask around 90% of the signal. The effects of the trade-offs can be clarified further by a comparison of the relative differences of phase-velocity curves in the inversion (where both the Moho depth and the crustal and mantle structure were varied, Fig. 12E-H) and in the sensitivity tests (where only the Moho depth was varied, Figs. 6D, E and 10, middle and bottom left). If only the Moho depth is perturbed, then its change by a few kilometres results in phase-velocity perturbations that vary gradually with period and reach a maximum on the order of 1% (Figs. 5, 6). In the inversion, where the effect of the Moho-depth perturbations is partly compensated by perturbations in crustal and mantle seismic-velocity structure, the same Moho-depth changes result in oscillatory phase-velocity perturbations 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-velocity curves 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 thousands of inter-station measurements, obtained by both cross-correlation and multimode waveform inversion (Lebedev et al., 2006, 2009; Meier et al., 2004). (The region-average measurements and inversions are meaningful because the Moho depth shows variations of only a few kilometres across the northern Kaapvaal Craton, and shear-velocity heterogeneity is also limited, according to published receiver-function studies and tomography (e.g., Kgaswane et al., 2009; Nair et al., 2006; Yang et al., 2008).) The highly accurate phase-velocity curves span very broad period ranges, particularly for Rayleigh waves (up to 5–400 s). Please cite this article as: Lebedev, S., et al., Mapping the Moho with seismic surface waves: A review, resolution analysis, and recommended inversion strategies, Tectonophysics (2013), http://dx.doi.org/10.1016/j.tecto.2012.12.030 S. Lebedev et al. / Tectonophysics xxx (2013) xxx–xxx Normal continent 2 1 0 B 30 36 42 48 Moho depth, km 200 0.0 4.0 C 5.0 C, km/s Misfit x104 150 3 48 0.4 4.5 3.5 Moho depth, km 36 42 E Love 100 -0.4 50 20 G 0.4 4.5 0.0 3.5 D 5 10 20 50 100 200 5 Period, s F 10 20 50 100 200 Period, s -0.4 10 5 200 100 50 20 4.0 H Period, s 100 30 Period, s 50 Period, s 10 20 50 100 200 Rayleigh C, km/s Depth, km A Period, s 10 20 50 100 200 5 δC, % 5.0 5 δC, % S wave velocity, km/s 3.5 4.0 4.5 0 11 10 5 -0.2 -0.1 0.0 0.1 0.2 Phase velocity perturbation δC, % 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 were performed 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. In each 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 and shear 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 the gradient-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 model and 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 various Moho 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, with cumulative 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 the Moho depth and shear-speed structure in the crust and the mantle. In Fig. 13 we show the results of three different inversions of the Rayleigh-wave phase velocities. In the first (Fig. 13, top row: C, F, I), we inverted the measured dispersion curve in a relatively broad period range (5–70 s), in which it had substantial sensitivity to the upper and lower crust, to the Moho, and to the lithospheric mantle. The measured curve can be fit with synthetic curves closely (within a line thickness in Fig. 13C). The misfits can be seen more clearly when relative phase-velocity differences are plotted (Fig. 13F); they suggest that the noise in the measurements is up to 0.1–0.2%, varying with period. All the shear-velocity profiles corresponding to the synthetic 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-dependent phase-velocity perturbations due to Moho-depth changes (Fig. 13I) is similar 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 attempt to remove the effects of the noise and invert the dispersion data in the same period range as in the first inversion but “smoothed” beforehand. The smoothing was by means of an over-parameterised and under-damped, gradient-search inversion of the phase-velocity curve for a 1-D shear-velocity profile (the profile itself being of no importance). While this inversion can fit structural information in the data, it cannot fit random errors with a strong period dependence (“high-frequency noise”), inconsistent with any plausible Earth models. Random errors thus get smoothed out to a large extent. Compared to the original-data inversion, the inversion of the smoothed dispersion curve reaches smaller misfits for best-fitting Moho depths (Fig. 13G). It also shows a more regular pattern of perturbations of best-fitting phase velocities as a function of the Moho depth (Fig. 13J). The smoothed-curve inversion (Fig. 13D, G, J) confirms that when random errors in the data are reduced, frequency-dependent misfits display patterns that are more similar to those in synthetic-data inversions, and the Moho depth can probably be resolved. The smoothing, however, may by itself introduce new biases into the data. For this reason, the inversions of smoothed data are best used for testing and validation, and not as the primary way to determine the Moho depth. Ideally, the results of the smoothed-data and original-data inversions should be consistent, indicating their robustness (e.g., Deschamps et al., 2008a; Endrun et al., 2011). The cumulative misfit curves in our inversions for the Moho depth, however, do not show such consistency and are substantially different for the two inversions (Fig. 13B): the larger misfits given by original-data inversions form a broader smallest-misfit valley, centred at Moho depths that are 7–8 km greater, compared to the misfits in the smoothed-curve inversion. This implies that the accuracy of the original-data inversion has suffered from the errors in the data (which we estimated to be up to ~ 0.2%). In order to reduce the effect of measurement errors, we set up a third inversion (Fig. 13, bottom row: E, H, K). We now invert a narrow-band curve, in a period range most sensitive to the Moho (15–32 s). Because this narrow-band curve has limited sensitivity to the crustal and mantle structure, an accurate reference profile of crustal and mantle shear-velocity must be used. Such profile is provided by the results of the original, broad-band inversion. The narrow-band inversion shows a steep misfit valley, with best-fitting Moho depths in the 37–41 km range (Fig. 13B). These values are roughly consistent with the Moho depths of 40–45 km determined in the region using receiver functions (Kgaswane et al., 2009; Nair et al., 2006) and the Moho depths of 40–43 km constrained by Rayleigh-wave measurements in a 6–40 s period range, made with cross-correlations of ambient seismic noise and inverted with starting models similar to those given by receiver-function analysis (Yang et al., 2008) (Fig. 5). Receiver-function measurements have their own uncertainties due to trade-offs of the Moho depth and the crustal Vp/Vs ratios; in surface-wave inversions, uncertainties result from trade-offs of the Moho depth and crustal and uppermost-mantle shear-velocity structure. These uncertainties are the most likely reason for the apparent small discrepancy between the different measurements in northern Please cite this article as: Lebedev, S., et al., Mapping the Moho with seismic surface waves: A review, resolution analysis, and recommended inversion strategies, Tectonophysics (2013), http://dx.doi.org/10.1016/j.tecto.2012.12.030 S. Lebedev et al. / Tectonophysics xxx (2013) xxx–xxx C, km/s 400 1.5 0.5 0.0 B 30 40 50 60 Moho depth, km 30 F C Smoothed Rayleigh 4.0 0.0 D G Narrow-band Rayleigh 0.0 E 20 Period, s 50 H 5 10 20 Period, s -0.4 Period, s 10 I 5 50 20 10 -0.4 3.6 10 20 J 5 50 0.4 3.8 5 50 -0.4 3.6 3.4 60 0.4 3.8 4.0 Moho depth, km 40 50 0.4 0.0 4.2 1.0 50 3.6 3.4 C, km/s Misfit x104 2.0 Period, s 20 3.8 4.2 N. Kaapvaal Craton 10 Rayleigh 250 350 5 4.0 3.4 300 50 Period, s 200 Period, s 20 20 10 K 50 Period, s 150 10 δC, % A 100 Depth, km 4.2 C, km/s 50 5 δC, % S wave velocity, km/s 3.5 4.0 4.5 5.0 0 δC, % 12 5 -0.2 -0.1 0.0 0.1 0.2 Phase velocity perturbation δC, 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 values between 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 fixed at 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 (top row: 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. When the 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. Inversion 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-function data could help to reduce some of these uncertainties and, also, to constrain the fine structure of the Moho. Small discrepancies notwithstanding, the close agreement between the results of the receiver-function analysis and those yielded by the surface-wave inversion with no a priori information (shear speeds in the crust and upper mantle were allowed to vary in unlimited, very broad ranges, and the trial Moho depth values spanned a very broad, 28–60 km range) validates the inversion set-up and confirms the resolving power of surface waves. The inversion procedure that is optimal thus has two steps: in the first step, we use a broad-band dispersion curve to determine the mantle and crustal structure with a reasonable accuracy (Fig. 13A); in the second step, we use that as a reference model (which can still be perturbed) while inverting only the part of the curve in the narrow period range where the signal of the Moho depth is the strongest and most likely to be well above the noise level. (This assumes that the Moho is associated with a seismic-velocity contrast. This contrast is seen, empirically, in the steep increase in the phase velocity at periods sampling primarily the depth range around the Moho. It is this period range that is used in the second-step inversion.) Love-wave phase-velocity curves show sensitivity to the Moho depth similar to that of the Rayleigh-wave ones. Unfortunately, there is usually more noise in Love-wave measurements. For the northern Kaapvaal Craton, errors in the Love-wave phase-velocity curve of Adam and Lebedev (2012) appear to be up to 0.2–0.3% at 5–50 s and up 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) did not provide robust solutions for the Moho depth. We also attempted joint Love and Rayleigh inversions, allowing for radial anisotropy, but Love-wave data did not contribute usefully to constraining the Moho depth, due to the higher levels of noise in them. 6. Noise in the data: how much is too much for the Moho to 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 inversion of the data. This will occur regardless of what inversion approach is used. The trade-offs between the Moho depth and crustal and mantle seismic velocities make the signal of the Moho depth in the data very subtle: ~ 0.1–0.2% of the phase velocity values. If the inversion of the data accounts for the trade-offs correctly and if no a priori information on seismic velocities is available, then an amount of noise that is similar to or higher in amplitude than the signal of the Moho may bias the results of the inversion. The resolvability of the depth of a continental Moho with surfacewave, phase-velocity data alone (and with no a priori information) is not warranted if the noise level exceeds ~ 0.2%. With stronger noise, the Moho may or may not be resolved correctly, depending on the character of the noise. We illustrate the effects of different noise patterns in Fig. 14. (For completeness, misfits as a function of periods are presented in Supplementary Fig. 1). Five different synthetic phase-velocity curves were inverted in different model-space-map inversion tests. One of the five curves was computed for a cratonic seismic-velocity profile (Fig. 14A, dashed line), and the other four were obtained by an addition of different patterns of noise to this curve. Please cite this article as: Lebedev, S., et al., Mapping the Moho with seismic surface waves: A review, resolution analysis, and recommended inversion strategies, Tectonophysics (2013), http://dx.doi.org/10.1016/j.tecto.2012.12.030 S. Lebedev et al. / Tectonophysics xxx (2013) xxx–xxx S wave velocity, km/s 4.0 4.5 5.0 A 50 5 10 B Period, s 20 50 "High-frequency" noise, <0.2-0.4% 0.0 100 Depth, km 150 C Complete estimated noise, <0.3-0.6% D "Ramp" noise, -0.3% E "Ramp" noise, -0.5% -0.8 0.8 0.0 200 250 0.8 -0.8 0.8 Noise, % 3.5 0 13 0.0 300 350 Global reference model True reference model 400 -0.8 0.8 0.0 -0.8 1.2 32 36 Moho depth, km 40 44 48 52 32 36 Moho depth, km 40 44 48 52 1.4 1.0 1.3 1.2 0.6 Misfit x104 Misfit x104 0.8 1.1 0.4 0.2 1.0 G F 0.0 No noise "Ramp" noise, -0.3% "Ramp" noise, -0.5% True reference model Global reference model "High-frequency" noise, <0.3% Complete estimated noise, 0.3-0.6% True Global reference reference model model 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 reference model” (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 before their 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 reference 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. Complete presentation 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 the model-space-map inversion (this effect is due to the damping applied in the gradient searches), each of the five dispersion curves was inverted twice, first with the correct, “true reference model”, and then with a substantially different, global reference model (Fig. 14A, solid line). These tests confirmed that the influence of the reference model is limited, much smaller than that of the noise in each case (Fig. 14F, G). The first two noise patterns were estimated from real data, measured across the Limpopo Belt (Adam and Lebedev, 2012). The “complete estimated noise” (Fig. 14C) is the difference between the measured and synthetic phase velocities, the latter computed for a “preferred,” smooth 1-D profile obtained in a damped, gradientsearch inversion of the data. The “high-frequency noise” is estimated in the same way but with the gradient-search inversion underdamped, and the 1-D profile showing some unrealistic roughness, most likely introduced by the inversion so as to fit the smoother components of the noise. Regardless of how accurately these noise patterns represent the actual errors in the Limpopo measurements, they are reasonable estimates and are well suited for the purposes of our tests as examples of noise distribution with period. The “high-frequency noise” pattern is characterised by random errors oscillating and changing sign every few seconds along the period axis. This noise pattern increases the misfits but has little effect on the Moho-depth values yielded by the inversions (misfit-curve minima in Fig. 14G). The influence of such random-noise patterns was investigated previously by Bartzsch et al. (2011) in their surface-wave inversions for the depth of the lithosphere–asthenosphere boundary. If the errors were random (did not persist over broad period ranges), then their effect on the best-fitting parameter values was small, even if the noise amplitude was as high as 1.0%. Here, we use noise estimates not from a random-number generator, as Bartzsch et al. (2011), but from real data. We arrive, however, to the same conclusion: random, “high-frequency” noise will be unlikely to bias the results of the inversion, in our case for the Moho depth. In contrast, the complete estimated noise (Fig. 14C) contains errors that persist over relatively broad period ranges. This noise pattern Please cite this article as: Lebedev, S., et al., Mapping the Moho with seismic surface waves: A review, resolution analysis, and recommended inversion strategies, Tectonophysics (2013), http://dx.doi.org/10.1016/j.tecto.2012.12.030 14 S. Lebedev et al. / Tectonophysics xxx (2013) xxx–xxx causes large errors in the Moho depth values. Although the noise is only up to 0.3% in the 11–80 s period range and up to 0.6% in the 5–11 s period range, it causes a 10-km error in the Moho depth (Fig. 14G). One component of this noise pattern is a sharp phase-velocity increase at 10–11 s. Approximating this component using “ramp” functions with 0.3% and 0.5% amplitudes (Fig. 14D, E), we find that it alone gives rise to Moho-depth errors of a few kilometres (Fig. 14F). It is clear from these tests that errors in phase-velocity data of only 0.3–0.5% can prevent the determination of the Moho depth with useful accuracy, particularly if the errors vary slowly along the period axis. 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? Unfortunately, the most damaging errors are the ones that vary smoothly with period, and such errors can map, to a large extent, into artificial structure in seismic-velocity models. If period ranges with relatively large suspected errors can be identified – for example, by an observed increase in data-synthetic, dispersion-curve misfit, by an increase of standard deviations or standard errors, or by an increase in frequency-dependent waveform misfit – excluding such period ranges from the inversion would be most effective (Fig. 13). Another way to improve the resolvability of the Moho is to include a priori information in the inversions, as is done often. The errors tested in this section are relatively small. Most surface-wave studies of the Moho to date have used data with errors larger than these. Does this mean that the results of all these studies should be in doubt? Not if they incorporated accurate additional, a priori information, explicitly or implicitly. For example, in our inversions shown in Fig. 13 we included no a priori information and allowed the lower-crustal and uppermost-mantle sheer speeds to vary in very broad ranges of 3.5–4.2 km/s and 4.2–4.6 km/s, respectively. Often, geological or geophysical constraints will be available to make these ranges much narrower. This will steepen the misfit valleys and allow the Moho to be resolved even with noise higher than a few tenths of a percent. The Moho can thus be resolved even with relatively noisy data if accurate a priori information on the crustal and mantle seismic velocities (or on the difference between the two) is available. Such constraints have been used extensively in surface-wave crustal studies, either explicitly, with a clear analysis of the ranges of values consistent with existing data, or implicitly, through the choice of a reference or starting model. It is important to keep in mind that the accuracy of the Moho depth yielded by such a constrained inversion will depend directly on the accuracy of the a priori constraints. 7. Recommended inversion strategies Basic strategies for the surface-wave inversion for the Moho depth are the same for the different types of surface-wave observables (phase velocities, group velocities or waveforms) and are dictated by the sensitivity of surface waves and the trade-offs between the discontinuity depth and seismic-velocity structure. Resolving the Moho depth with surface waves alone is possible but is guaranteed to work only if the errors of the measurements are very small (e.g., up to ~0.2% of phase velocities for mapping the Moho beneath continents). Data with larger errors can still be used effectively if accurate a priori information is available that can reduce the possible ranges of seismic velocities in the crust and uppermost mantle or of the velocity contrast across the Moho. (More information translates into better resolution in the presence of the same errors.) Similarly, relatively noisy surface-wave data can be successfully inverted for the Moho jointly with data of other types that provide 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-wave measurements in a period range as broad as possible directly for the Moho depth is not optimal. Because the Moho depth is constrained by very subtle signal in surface-wave data, even small errors at short periods (sensitive primarily to the upper crust) or longer periods (sensitive primarily to the deep mantle lithosphere) can bias the inversion results, because of the trade-offs between seismic velocities at different depths. In order to determine the Moho depth using surface-wave data only, with no a priori information on crustal or mantle structure, the most effective inversion strategy is to first constrain the crustal and mantle structure by an inversion of the data in a broad period range and then, as a second step, to find best-fitting Moho depths in an inversion in a narrow period range with the most sensitivity to the Moho, using the results of the first step as a reference model (see Section 5.3 for details). 7.2. A priori information: include whenever available! A successful inversion of surface waves alone for the Moho depth requires high accuracy of the measurements, because the strong sensitivity of surface waves to the Moho is reduced, in inversions, by severe trade-offs of the depth of the Moho and the shear-speed structure above and below it. Accurate a priori constraints on crustal and mantle structure will reduce the trade-offs and should be sought when possible. Such constraints come, for example, from controlled-source crustal imaging in the region, from receiver-function studies, or from crustal xenoliths (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 can reduce model uncertainties, making Moho mapping possible even when the accuracy of surface-wave measurements alone is insufficient for the purpose. Surface-wave and receiver-function data complement each other especially well: both types of data are yielded by passive imaging methods and can be obtained from a small array of broadband stations situated virtually anywhere in the world. And, whereas surface waves with their broad depth sensitivity kernels (Fig. 1) provide strong constraints on shear speeds within depth ranges (i.e., on smooth variations of shear speed with depth), receiver functions 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 well established, recent developments in computational petrological and geophysical modelling now also make increasingly feasible joint inversions of surface-wave and other geophysical and geological data. Fullea et al. (2012) developed a joint inversion of surface-wave dispersion curves and topography and applied it to data from central Mongolia, also incorporating constraints from surface heat flow measurements, controlled-source seismic data, and crustal xenolith analysis. Designed for determination of the thermal structure of the lithosphere with a high vertical resolution, such petro-physical inversion also constrains the depth of the Moho, using the complementary sensitivities of the surface-wave and other data. 8. Discussion In our investigation of the resolution of surface-wave inversions for the Moho depth, we considered only one type of surface-wave observables, their phase velocities. Our results, however, characterise general data-model relationships and apply, with obvious adjustments, to inversions of other surface-wave observables, including group velocities and waveforms. Group velocities have a higher sensitivity to the Moho depth, compared to phase velocities, in the absence of errors in the measurements or with the same error levels (Figs. 6–11). This advantage of group velocities is offset, normally, by the larger actual errors in Please cite this article as: Lebedev, S., et al., Mapping the Moho with seismic surface waves: A review, resolution analysis, and recommended inversion strategies, Tectonophysics (2013), http://dx.doi.org/10.1016/j.tecto.2012.12.030 S. Lebedev et al. / Tectonophysics xxx (2013) xxx–xxx their measurements. In group-velocity tomography, further errors may come from the group delays' off-ray sensitivity to local phasevelocity perturbations and their dispersion (Eq. (2)), in addition to the expected sensitivity to local group velocity perturbations (Dahlen and Zhou, 2006). In spite of this, the high sensitivity of group velocities to the Moho depth has been exploited successfully in Moho mapping, including in joint inversions of group and phase velocities (e.g., Shapiro and Ritzwoller, 2002). Although we have focussed so far on the fundamental-mode surface waves only, higher surface-wave modes also sample the crust and 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 and Woodhouse, 1999; Visser et al., 2007) will offer new types of constraints on the global crustal structure. At a regional scale, the utility of higher modes has already been clearly established: Levshin et al. (2005) measured group velocities of Rayleigh- and Love-wave first crustal overtones in the 7 – 18 s period range and showed that the joint inversion of the fundamental and higher mode measurements increased the resolution of crustal structure. Lateral variations in the Moho depth (Moho topography) can be resolved using tomography based on surface-wave ray theory if they are smooth enough so that the validity of surface-wave ray theory is warranted (Dahlen and Tromp, 1998) or, at least, so that sufficient amount of surface-wave signals in the time-frequency planes can be selected empirically, such that they can be modelled using ray theory (Das and Nolet, 1995; Lebedev et al., 2005; Schaeffer and Lebedev, submitted for publication). Strong lateral changes in the Moho depth result in strong lateral seismic-velocity changes and give rise to scattering and multipathing (e.g., Levshin et al., 1992; Meier and Malischewsky, 2000; Meier et al., 1997). The Moho topography itself, if not accounted for accurately, can also cause biases in the interpretation of surface-wave measurements that average over paths or areas (Levshin and Ratnikova, 1984). In order to resolve small-scale Moho topography these effects have to be taken into account in the inversion for local phase or shear velocities (Wielandt, 1993). The strong trade-offs between the depth of the Moho and the shear speeds just above and just below it – discussed throughout this paper – also have implications for large-scale mantle tomography that uses intermediate- and long-period surface waves. Here, the trade-offs play a positive role, reducing the effect of uncertainties in the Moho depth on mantle models. If the inversion set-up includes a few parameters for wavespeeds within the crust and a reasonably dense parameterisation in the lithospheric mantle, and if the reference model for the tomography is 3-D and includes a realistic crust, such as CRUST2.0 (Bassin et al., 2000) or a more detailed regional model (e.g., Grad et al., 2009; Molinari and Morelli, 2011; Tesauro et al., 2008), then the Moho depth may not need to be perturbed in the inversion at all (e.g., Lebedev and van der Hilst, 2008; Legendre et al., 2012; Schaeffer and Lebedev, submitted for publication). The effects of differences between the true and 3-D-reference Moho depths will 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 in source parameters. 9. Conclusions Surface waves have been used to study the outer layers of the Earth since the early days of modern seismology. Their sensitivity to the crustal thickness (the depth to the Moho) has been established and applied in structural studies already in the first half of the 20th century. The beginning of the 21st century has seen the crustal surface-wave seismology re-energised by the emergence of new techniques for broad-band surface-wave measurements, using the 15 increasingly abundant data from arrays of seismic stations. Thanks to the progress in the precision and bandwidth of the measurements, surface-wave imaging of the crust and the Moho is now reaching a new level of accuracy. Both Rayleigh and Love waves have strong sensitivity to the Moho depth. Tests with synthetic data show that if seismic wavespeeds within the crust and upper mantle can be fixed (assumed to be known), then Moho-depth variations of a few kilometres produce large (>1%) perturbations in phase velocities, varying gradually with period. In inversions of surface-wave data with no a priori information, the Moho depth shows strong trade-offs with shear-wave speeds in the lower crust and uppermost mantle. Adjustments in the crustal and mantle structure can compensate for as much as 90% of the Moho-depth signal in surface-wave data. In the inversion, changes of a few kilometres in the depth of a continental Moho result in oscillatory phase-velocity perturbations that reach a maximum on the order of 0.1% only. For the inversion to be guaranteed to resolve the Moho depth with useful precision, very high accuracy of the measurements is thus required, with errors up to 0.1–0.2% at most. Errors that persist over broad period ranges are particularly harmful; random errors that vary rapidly with period have a smaller effect. An effective strategy for the inversion of surface-wave data alone for the Moho depth, with no a priori information, is to first constrain the crustal and upper-mantle structure by an inversion in a broad period range, and then, as a second step, to find best-fitting Moho depths in an inversion of the data in a narrow period range that has the most sensitivity to the Moho, with the results of the first step used as a reference model. A priori constraints on crustal and mantle structure (from controlledsource imaging, receiver-functions, xenoliths or regional geology) will reduce the trade-offs. Joint analysis or inversion of surface-wave and other data (receiver functions, topography, gravity) can reduce model uncertainties and make Moho mapping possible even when the accuracy of surface-wave measurements alone would be insufficient. Alone or as a part of multi-disciplinary datasets, surface-wave data offer unique sensitivity to the crustal and upper-mantle structure and are becoming increasingly 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 and H. Thybo for their work on this volume. Insightful comments and suggestions by Anatoli Levshin, an anonymous reviewer and one of the editors have helped us to improve the clarity of the paper. Most figures were created with the Generic Mapping Tools (Wessel and Smith, 1998). This work was funded by the Dublin Institute for Advanced Studies and Science Foundation Ireland (grants 08/RFP/ GEO1704 and 09/RFP/GEO2550). References Adam, J.M.C., Lebedev, S., 2012. Azimuthal anisotropy beneath southern Africa, from very-broadband, surface-wave dispersion measurements. Geophysical Journal International 191, 155–174. Agius, M.R., Lebedev, S., 2010. Shear-velocity profiles across the Tibetan Plateau from broadband, surface-wave, phase-velocity measurements. In: Leech, Mary L., Klemperer, Simon L., Mooney, Walter D. (Eds.), Proceedings of the 25th Himalaya–Karakoram–Tibet Workshop: U.S. Geological Survey Open-File Report 2010-1099. Aki, K., Richards, P.G., 1980. Quantitative Seismology, Theory and Methods, Volume 1. W. H. Freeman and Company, San Francisco. Alterman, Z., Jarosch, H., Pekeris, C.L., 1961. Propagation of Rayleigh waves in the Earth. Geophysical Journal of the Royal Astronomical Society 4, 219–241. Angenheister, G., 1906. Bestimmung der Fortpflanzungsgeschwindigkeit und Absorption von Erdbebenwellen, die durch den Gegenpunkt des Herdes gegangen sind. Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-physikalische Klasse. Please cite this article as: Lebedev, S., et al., Mapping the Moho with seismic surface waves: A review, resolution analysis, and recommended inversion strategies, Tectonophysics (2013), http://dx.doi.org/10.1016/j.tecto.2012.12.030 16 S. Lebedev et al. / Tectonophysics xxx (2013) xxx–xxx Angenheister, G., 1921. Beobachtungen an pazifischen Beben – Ein Beitrag zum Studium der obersten Erdkruste. Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-physikalische Klasse. Bartzsch, S., Lebedev, S., Meier, T., 2011. Resolving the lithosphere–asthenosphere boundary with seismic Rayleigh waves. Geophysical Journal International 186, 1152–1164. Bassin, C., Laske, G., Masters, G., 2000. The current limits of resolution for surface wave tomography in North America. EOS, Fall Meeting Supplement F897. Bensen, G.D., Ritzwoller, M.H., Barmin, M.P., Levshin, A.L., Lin, F., Moschetti, M.P., Shapiro, N.M., Yang, Y., 2007. Processing seismic ambient noise data to obtain reliable broad-band surface wave dispersion measurements. Geophysical Journal International 169, 1239–1260. Boschi, L., Ekstrom, G., 2002. New images of the Earth's upper mantle from measurements of surface wave phase velocity anomalies. Journal of Geophysical Research 107 (B4), 2059. http://dx.doi.org/10.1029/2000JB000059. Bostock, M.G., Hyndman, R.D., Rondenay, S., Peacock, S.M., 2002. An inverted continental Moho and serpentinization of the forearc mantle. Nature 417, 536–538. Bozdag, E., Trampert, J., 2008. On crustal corrections in surface wave tomography. Geophysical Journal International 172, 1066–1082. Brilliant, R.M., Ewing, M., 1954. Dispersion of Rayleigh waves across the U.S. Bulletin of the Seismological Society of America 44, 149–158. Brune, J., Dorman, J., 1963. Seismic waves and Earth structure in the Canadian Shield. Bulletin of the Seismological Society of America 53, 167–210. Brune, J.N., Nafe, J.E., Oliver, J.E., 1960. A simplified method for the analysis and synthesis of dispersed wave trains. Journal of Geophysical Research 65, 287–304. Bullen, K.E., 1939. On Rayleigh waves across the Pacific Ocean. Monthly Notices, Royal Astronomical Society, Geophysical Supplement 4, 579–582. Carder, D.S., 1934. Seismic surface waves and the crustal structure of the Pacific region. Bulletin of the Seismological Society of America 24, 231–301. Christensen, N.I., Mooney, W.D., 1995. Seismic velocity structure and composition of the continental crust: a global view. Journal of Geophysical Research 100, 9761–9788. Dahlen, F.A., Tromp, J., 1998. Theoretical Global Seismology. Princeton University Press, Princeton, NJ. Dahlen, F.A., Zhou, Y., 2006. Surface-wave group-delay and attenuation kernels. Geophysical Journal International 165, 545–554. Das, T., Nolet, G., 1995. Crustal thickness estimation using high frequency Rayleigh waves. Geophysical Research Letters 22 (5), 539–542. Das, T., Nolet, G., 1998. Crustal thickness map of the western United States by partitioned waveform inversion. Journal of Geophysical Research 103 (B12), 30,021–30,038. DeLisle, J.F., 1941. On dispersion of Rayleigh waves from the North Pacific earthquake of November 10, 1938. Bulletin of the Seismological Society of America 31, 303–307. Deschamps, F., Lebedev, S., Meier, T., Trampert, J., 2008a. Azimuthal anisotropy of Rayleigh-wave phase velocities in the east-central United States. Geophysical Journal International 173, 827–843. Deschamps, F., Lebedev, S., Meier, T., Trampert, J., 2008b. Stratified seismic anisotropy reveals past and present deformation beneath the East-central United States. Earth and Planetary Science Letters 274, 489–498. Dorman, J., Ewing, M., 1962. Numerical inversion of seismic surface wave dispersion data and crust-mantle structure in the New York-Pennsylvania Area. Journal of Geophysical Research 67, 5227–5241. Dziewonski, A.M., 1970. On regional differences in dispersion of mantle Rayleigh waves. Geophysical Journal of the Royal Astronomical Society 22, 289–325. Endrun, B., Meier, T., Bischoff, M., Harjes, H.P., 2004. Lithospheric structure in the area of Crete constrained by receiver functions and dispersion analysis of Rayleigh phase velocities. Geophysical Journal International 158, 592–608. Endrun, B., Meier, T., Lebedev, S., Bohnhoff, M., Stavrakakis, G., Harjes, H.-P., 2008. S velocity structure and radial anisotropy in the Aegean region from surface wave dispersion. Geophysical Journal International 174, 593–616. Endrun, B., Lebedev, S., Meier, T., Tirel, C., Friederich, W., 2011. Complex layered deformation within the Aegean crust and mantle revealed by seismic anisotropy. Nature Geoscience 4, 203–207. Erduran, M., Endrun, B., Meier, T., 2008. Continental vs. oceanic lithosphere beneath the eastern Mediterranean Sea — implications from Rayleigh wave dispersion measurements. Tectonophysics 457, 42–52. Evernden, J.F., 1954. Love-wave dispersion and the structure of the Pacific basin. Bulletin of the Seismological Society of America 44, 1–5. Ewing, M., Press, F., 1950. Crustal structure and surface-wave dispersion. Bulletin of the Seismological Society of America 40, 271–280. Ewing, M., Press, F., 1952. Crustal structure and surface-wave dispersion, Part II: Solomon Islands earthquake of July 29, 1950. Bulletin of the Seismological Society of America 42, 315–325. Ewing, M., Press, F., 1956. Rayleigh wave dispersion in the period range 10 to 500 seconds. Transactions of the American Geophysical Union 37, 213–215. Ferreira, A.M.G., Woodhouse, J.H., Visser, K., Trampert, J., 2010. On the robustness of global radially anisotropic surface wave tomography. Journal of Geophysical Research 115, B04313. http://dx.doi.org/10.1029/2009JB006716. Fullea, J., Lebedev, S., Agius, M.R., Jones, A.G., Afonso, J.C., 2012. Lithospheric structure in the Baikal–central Mongolia region from integrated geophysical-petrological inversion of surface-wave data and topographic elevation. Geochemistry, Geophysics, Geosystems 13, Q0AK09. http://dx.doi.org/10.1029/2012GC004138. Golitsyn, B.B., 1960. Selected works. Press of the Academy of Sciences of the USSR, Moscow: Seismology, 2. Grad, M., Tiira, T., ESC Working Group, 2009. The Moho depth map of the European Plate. Geophysical Journal International 176, 279–292. Gutenberg, B., 1924. Dispersion und Extinktion von seismischen Oberflächenwellen und der Aufbau der obersten Erdschichten. Physikalische Zeitschrift XXV, 377–381. Harmon, N., Henstock, T., Tilmann, F., Rietbrock, A., Barton, P., 2012. Shear velocity structure across the Sumatran forearc-arc. Geophysical Journal International 189, 1306–1314. Jeffreys, H., 1935. The surface waves of earthquakes. Monthly Notices, Royal Astronomical Society, Geophysical Supplement 3, 253–261. Julià, J., Ammon, C.J., Herrmann, R.B., Correig, A.M., 2000. Joint inversion of receiver function and surface wave dispersion observations. Geophysical Journal International 143, 99–112. Kennett, B.L.N., 1983. Seismic Wave Propagation in Stratified Media. Cambridge University Press, Cambridge. Kennett, B.L.N., 2001. The Seismic Wavefield I. — Introduction and Theoretical Development. Cambridge University Press, Cambridge. Kgaswane, E.M., Nyblade, A.A., Julià, J., Dirks, P.H.G.M., Durrheim, R.J., Pasyanos, M.E., 2009. Shear wave velocity structure of the lower crust in Southern Africa: evidence for compositional heterogeneity within Archaean and Proterozoic terrains. Journal of Geophysical Research 114, B12304. http://dx.doi.org/10.1029/2008JB006217. Kind, R., Yuan, X., Saul, J., Nelson, D., Sobolev, S.V., Mechie, J., Zhao, W., Kosarev, G., Ni, J., Achauer, U., Jiang, M., 2002. Seismic images of crust and upper mantle beneath Tibet: evidence for Eurasian plate subduction. Science 298, 1219–1221. Kissling, E., 1993. Deep structure of the Alps — what do we really know? Physics of the Earth and Planetary Interiors 79, 87–112. Knopoff, L., 1972. Observation and inversion of surface-wave dispersion. Tectonophysics 13, 497–519. Knopoff, L., Berry, J., Schwab, F.A., 1967. Tripartite phase velocity observation in laterally heterogeneous regions. Journal of Geophysical Research 72, 2595–2601. Koulakov, I., Sobolev, S.V., 2006. Moho depth and three-dimensional P and S structure of the crust and uppermost mantle in the Eastern Mediterranean and Middle East derived from tomographic inversion of local ISC data. Geophysical Journal International 164, 218–235. Kovach, R.L., 1978. Seismic Surface waves and crustal and upper mantle structure. Reviews of Geophysics and Space Physics 16, 1–13. Kustowski, B., Dziewonski, A.M., Ekstrom, G., 2007. Nonlinear crustal corrections for normal-mode seismograms. Bulletin of the Seismological Society of America 97, 1756–1762. Landisman, M., Dziewonski, A., Sato, Y., 1969. Recent improvements in the analysis of surface wave observations. Geophysical Journal of the Royal Astronomical Society 17, 369–403. Lebedev, S., Nolet, G., 2003. Upper mantle beneath Southeast Asia from S velocity tomography. Journal of Geophysical Research 108. http://dx.doi.org/10.1029/2000JB000073. Lebedev, S., van der Hilst, R.D., 2008. Global upper-mantle tomography with the automated multimode inversion of surface and S-wave forms. Geophysical Journal International 173, 505–518. Lebedev, S., Nolet, G., van der Hilst, R.D., 1997. The upper mantle beneath the Philippine Sea region from waveform inversions. Geophysical Research Letters 25, 1851–1854. Lebedev, S., Chevrot, S., van der Hilst, R.D., 2002a. Seismic evidence for olivine phase changes at the 410- and 660-kilometer discontinuities. Science 296, 1300–1302. Lebedev, S., Chevrot, S., van der Hilst, R.D., 2002b. The 660-km discontinuity within the subducting NW-Pacific lithospheric slab. Earth and Planetary Science Letters 205, 25–35. Lebedev, S., Nolet, G., Meier, T., van der Hilst, R.D., 2005. Automated multimode inversion of surface and S waveforms. Geophysical Journal International 162, 951–964. Lebedev, S., Meier, T., van der Hilst, R.D., 2006. Asthenospheric flow and origin of volcanism in the Baikal Rift area. Earth and Planetary Science Letters 249, 415–424. Lebedev, S., Boonen, J., Trampert, J., 2009. Seismic structure of Precambrian lithosphere: new constraints from broad-band surface-wave dispersion. Lithos 109, 96–111. Legendre, C., Meier, T., Lebedev, S., Friederich, W., Viereck-Goette, L., 2012. A shearwave velocity model of the European upper mantle from automated inversion of seismic shear and surface waveforms. Geophysical Journal International 191, 282–304. Lekic, V., Panning, M., Romanowicz, B., 2010. A simple method for improving crustal corrections in waveform tomography. Geophysical Journal International 182, 265–278. Levshin, A., Ratnikova, L., 1984. Apparent anisotropy in inhomogeneous media. Geophysical Journal of the Royal Astronomical Society 74, 65–69. Levshin, A.L., Yanovskaya, T.B., Lander, A.V., Bukchin, B.G., Barmin, M.P., Ratnikova, L.I., Its, E.N., 1989. In: Keilis-Borok, V.I. (Ed.), Seismic Surface Waves in a Laterally Inhomogeneous Earth. Kluwer, Norwell, Mass. Levshin, A., Ratnikova, L., Berger, J., 1992. Peculiarities of surface-wave propagation across central Eurasia. Bulletin of the Seismological Society of America 82, 2464–2493. Levshin, A.L., Ritzwoller, M.H., Shapiro, N.M., 2005. The use of crustal higher modes to constrain crustal structure across Central Asia. Geophysical Journal International 160, 961–972. Li, X., Romanowicz, B., 1996. Global mantle shear velocity model developed using nonlinear asymptotic coupling theory. Journal of Geophysical Research 101, 22245–22272. Lin, F.C., Ritzwoller, M.H., Yang, Y., Moschetti, M.P., Fouch, M.J., 2011. Complex and variable crustal and uppermost mantle seismic anisotropy in the western United States. Nature Geoscience 4, 55–61. Love, A.E.H., 1911. Some Problems of Geodynamics. Cambridge University Press, Cambridge. Macelwane, J.B., 1923. A study of the relation between the periods of elastic waves and the distance traveled by them, based upon the seismographic records of the Please cite this article as: Lebedev, S., et al., Mapping the Moho with seismic surface waves: A review, resolution analysis, and recommended inversion strategies, Tectonophysics (2013), http://dx.doi.org/10.1016/j.tecto.2012.12.030 S. Lebedev et al. / Tectonophysics xxx (2013) xxx–xxx California earthquake, January 31, 1922. Bulletin of the Seismological Society of America 13, 13–69. Marone, F., Romanowicz, B., 2007. Non-linear crustal corrections in high-resolution regional waveform seismic tomography. Geophysical Journal International 170, 460–467. Marone, F., van der Meijde, M., van der Lee, S., Giardini, D., 2003. Joint inversion of local, regional, and teleseismic data for crustal thickness in the Eurasia-Africa plate boundary region. Geophysical Journal International 154, 499–514. McEvilly, T.V., 1964. Central U.S. curst-upper mantle structure from Love and Rayleigh wave phase velocity inversion. Bulletin of the Seismological Society of America 54, 1997–2015. Meier, T., Malischewsky, P.G., 2000. Approximation of surface wave mode conversion at a passive continental margin by a mode-matching technique. Geophysical Journal International 141, 12–24. Meier, T., Lebedev, S., Nolet, G., Dahlen, F.A., 1997. Diffraction tomography using multimode surface waves. Journal of Geophysical Research 102, 8255–8267. Meier, T., Dietrich, K., Stockhert, B., Harjes, H.P., 2004. One-dimensional models of shear wave velocity for the eastern Mediterranean obtained from the inversion of Rayleigh wave phase velocities and tectonic implications. Geophysical Journal International 156, 45–58. Meier, U., Curtis, A., Trampert, J., 2007a. Fully nonlinear inversion of fundamental mode surface waves for a global crustal model. Geophysical Research Letters 34, L16304. http://dx.doi.org/10.1029/2007GL030989. Meier, U., Curtis, A., Trampert, J., 2007b. Global crustal thickness from neural network inversion of surface wave data. Geophysical Journal International 169, 706–722. Meissner, E., 1921. Elastische Oberflächenwellen mit Dispersion in einem inhomogenen Medium. Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich 66, 181–195. Meissner, E., 1926. Elastische Oberflächen-Querwellen. Proceedings of the 2nd International Congress for Applied Mechanics, Zürich, pp. 3–11. Mohorovičić, A., 1910. Potres od 8.X 1909 (Das Beben vom 8.X. 1909). Jahrbuch des meteorologischen Observatoriums in Zagreb (Agram) für das Jahr 1909, pp. 1–56 (English translation: 1992, Earthquake of 8 October 1909, Geofizika, 9, 3–55). Molinari, I., Morelli, A., 2011. EPcrust: a reference crustal model for the European plate. Geophysical Journal International 185, 352–364. Montagner, J.P., Jobert, N., 1988. Vectorial tomography II. Application to the Indian Ocean. Geophysical Journal International 94, 309–344. Mooney, W.D., Laske, G., Masters, T.G., 1998. Crust5.1: a global crustal model at 5 × 5. Journal of Geophysical Research 103, 727–747. Moschetti, M.P., Ritzwoller, M.H., Lin, F.-C., Yang, Y., 2010. Crustal shear wave velocity structure of the western United States inferred from ambient seismic noise and earthquake data. Journal of Geophysical Research 115, B10306. http://dx.doi.org/ 10.1029/2010JB007448. Nabelek, J., Hetenyi, G., Vergne, J., Sapkota, S., Kafle, B., Jiang, M., Su, H.P., Chen, J., Huang, B.S., Team, Hi-CLIMB, 2009. Underplating in the Himalaya–Tibet collision zone revealed by the Hi-CLIMB experiment. Science 325, 1371–1374. Nafe, J.E., Brune, J.N., 1960. Observations of phase velocity for Rayleigh waves in the period range 100 to 400 seconds. Bulletin of the Seismological Society of America 50, 427–439. Nair, S.K., Gao, S.S., Liu, K.H., Silver, P.G., 2006. Southern African crustal evolution and composition: constraints from receiver function studies. Journal of Geophysical Research 111. http://dx.doi.org/10.1029/2005JB003802. Nataf, H.C., Ricard, Y., 1996. 3SMAC: an a priori tomographic model of the upper mantle based on geophysical modelling. Physics of the Earth and Planetary Interiors 95, 101–122. Nataf, H.C., Nakanishi, I., Anderson, D.L., 1986. Measurements of mantle wave velocities and inversion for lateral heterogeneities and anisotropy. Part III: inversion. Journal of Geophysical Research 91, 7261–7307. Neumann, F., 1929. The velocity of seismic surface waves over pacific paths. Bulletin of the Seismological Society of America 19, 63–76. Nolet, G., 1990. Partitioned waveform inversion and two-dimensional structure under the Network of Autonomously Recording Seismographs. Journal of Geophysical Research 95, 8499–8512. Nolet, G., 2008. A Breviary of Seismic Tomography: Imaging the Interior of the Earth and the Sun. Cambridge University Press, Cambridge. Oldham, R.D., 1899. Report on the great earthquake of 12th June 1897. Memoirs of the Geological Survey of India 29, 1–379. Oliver, J.E., Ewing, M., Press, F., 1955. Crustal structure and surface-wave dispersion. Bulletin of the Seismological Society of America 66, 913–946. Panning, M., Lekic, V., Romanowicz, B., 2010. Importance of crustal corrections in the development of a new global model of radial anisotropy. Journal of Geophysical Research 115. http://dx.doi.org/10.1029/2010JB007520. Pasyanos, M.E., Walter, W.R., 2002. Crust and upper-mantle structure of North Africa, Europe and the Middle East from inversion of surface waves. Geophysical Journal International 149, 463–481. Pawlak, A.E., Eaton, D., Darbyshire, F., Lebedev, S., Bastow, I.D., 2012. Crustal anisotropy beneath Hudson Bay from ambient-noise tomography: evidence for post-orogenic lower-crustal flow? Journal of Geophysical Research 117, B08301. http://dx.doi.org/ 10.1029/2011JB009066. Pekeris, C.L., 1948. Theory of propagation of explosive sounds in shallow water. Geological Society of America Memoir 27, 117. Polat, G., Lebedev, S., Readman, P.W., O'Reilly, B.M., Hauser, F., 2012. Anisotropic Rayleigh-wave tomography of Ireland's crust: implications for crustal accretion and evolution within the Caledonian Orogen. Geophysical Research Letters 39, L04302. http://dx.doi.org/10.1029/2012GL051014. Press, F., 1956. Determination of crustal structure from phase velocity of Rayleigh waves, Part I: Southern California. Geological Society of America Bulletin 67, 1647–1658. 17 Press, F., Ewing, M., Oliver, J., 1956. Crustal structure and surface-wave dispersion in Africa. Bulletin of the Seismological Society of America 46, 97–103. Prodehl, C., Mooney, W.D., 2011. Exploring the Earth's crust: history and results of controlled-source seismology. The Geological Society of America, Memoir 208. Reid, H.F., 1910. The California Earthquake of April 18, 1906. Report of the State Earthquake Investigation Commission. Rychert, C.A., Shearer, P.M., 2010. Resolving crustal thickness using SS waveform stacks. Geophysical Journal International 180, 1128–1137. Santo, T., Sato, Y., 1966. World-wide survey of the regional characteristics of group velocity dispersion of Rayleigh waves. Bulletin of the Earthquake Research Institute, University of Tokyo 44, 939–964. Sato, Y., 1955. Analysis of dispersed surface waves by means of Fourier transform: Part 1. Bulletin of the Earthquake Research Institute, University of Tokyo 33, 33–47. Schaeffer, A. J., Lebedev, S., submitted for publication. Global shear speed structure of the upper mantle. Geophysical Journal International. Seidl, D., Müller, S., 1977. Seismische Oberflächenwellen. Zeitschrift für Geophysik 42, 283–328. Shapiro, N.M., Campillo, M., 2004. Emergence of broadband Rayleigh waves from correlations of the ambient seismic noise. Geophysical Research Letters 31, L07614. http://dx.doi.org/10.1029/2004GL019491. Shapiro, N.M., Ritzwoller, M.H., 2002. Monte-Carlo inversion for a global shear-velocity model of the crust and upper mantle. Geophysical Journal International 151, 88–105. Shapiro, N.M., Campillo, M., Stehly, L., Ritzwoller, M.H., 2005. High-resolution surface wave tomography from ambient seismic noise. Science 307, 1615–1618. Shen, W., Ritzwoller, M.H., Schulte-Pelkum, V., Lin, F.-C., 2013. Joint inversion of surface wave dispersion and receiver functions: a Bayesian Monte-Carlo approach. Geophysical Journal International 192, 807–836. Stankiewicz, J., Chevrot, S., van der Hilst, R.D., de Wit, M.J., 2002. Crustal thickness, discontinuity depth, and upper mantle structure beneath southern Africa: constraints form body wave conversions. Physics of the Earth and Planetary Interiors 130, 235–251. Stoneley, R., 1925. Dispersion of seismic waves. Geophysical Journal International 1, 280–282. http://dx.doi.org/10.1111/j.1365-246X.1925.tb05375.x. Stoneley, R., 1926. The effect of the ocean on Rayleigh waves. Monthly Notices, Royal Astronomical Society, Geophysical Supplement 1, 349–356. Tams, E., 1921. Über die Fortpflanzungsgeschwindigkeit der seismischen Oberflächenwellen längs kontinentaler und ozeanischer Wege. Centralblatt der Mineralogie, Geologie und Paläontologie 2-3, 44–52 (75-83). Tesauro, M., Kaban, M.K., Cloetingh, S., 2008. EuCRUST-07: a new reference model for the European crust. Geophysical Research Letters 35, L05313. http://dx.doi.org/ 10.1029/2007GL032244. Thybo, H., 2001. Crustal structure along the EGT profile across the Tornquist Fan interpreted from seismic, gravity and magnetic data. Tectonophysics 334, 155–190. Tkalčić, H., Rawlinson, N., Arroucau, P., Kumar, A., Kennett, B.L.N., 2012. Multistep modelling of receiver-based seismic and ambient noise data from WOMBAT array: crustal structure beneath southeast Australia. Geophysical Journal International 189, 1680–1700. Toksöz, M.N., Ben-Menahem, A., 1963. Velocities of mantle Love and Rayleigh waves over multiple paths. Bulletin of the Seismological Society of America 53, 741–764. Van der Lee, S., Nolet, G., 1997. Upper-mantle S-velocity structure of North America. Journal of Geophysical Research 102, 22,815–22,838. Van Heijst, H.J., Woodhouse, J., 1999. Global high-resolution phase velocity distributions of overtone and fundamental-mode surface waves determined by mode branch stripping. Geophysical Journal International 137, 601–620. Visser, K., Lebedev, S., Trampert, J., Kennett, B.L.N., 2007. Global Love wave overtone measurements. Geophysical Research Letters 34, L03302. http://dx.doi.org/ 10.1029/2006GL028671. Wessel, P., Smith, W.H.F., 1998. New, improved version of generic mapping tools released. Eos Transactions AGU 79 (47), 579. http://dx.doi.org/10.1029/98EO00426. Wiechert, E., 1899. Seismometrische Beobachtungen im Göttinger Geophysikalischen Institut. Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-physikalische Klasse. Wielandt, E., 1993. Propagation and structural interpretation of non-plane waves. Geophysical Journal International 113, 45–53. Wilson, J.T., 1940. The Love waves of the South Atlantic earthquakes of August 28, 1933. Bulletin of the Seismological Society of America 30, 273–301. Wilson, J.T., Baykal, O., 1948. Crustal Structure of the North Atlantic Basin as determined from Rayleigh wave dispersion. Bulletin of the Seismological Society of America 38, 41–53. Woodhouse, J.H., Dziewonski, A.M., 1984. Mapping the upper mantle: threedimensional modeling of earth structure by inversion of seismic waveforms. Journal of Geophysical Research 89, 5953–5986. Yang, Y., Li, A., Ritzwoller, M.H., 2008. Crustal and uppermost mantle structure in southern Africa revealed from ambient noise and teleseismic tomography. Geophysical Journal International 174, 235–248. Yang, Y., Ritzwoller, M.H., Jones, C.H., 2011. Crustal structure determined from ambient noise tomography near the magmatic centers of the Coso region, southeastern California. Geochemistry, Geophysics, Geosystems 12, Q02009. http://dx.doi.org/ 10.1029/2010GC003362. Yang, Y., Ritzwoller, M.H., Zheng, Y., Levshin, A.L., Xie, Z., 2012. A synoptic view of the distribution and connectivity of the mid-crustal low velocity zone beneath Tibet. Journal of Geophysical Research 117, B04303. http://dx.doi.org/10.1029/2011JB008810. Yao, H., Beghein, C., van der Hilst, R.D., 2008. Surface-wave array tomography in SE Tibet from ambient seismic noise and two-station analysis: II — Crustal and upper mantle structure. Geophysical Journal International 173, 205–219. Please cite this article as: Lebedev, S., et al., Mapping the Moho with seismic surface waves: A review, resolution analysis, and recommended inversion strategies, Tectonophysics (2013), http://dx.doi.org/10.1016/j.tecto.2012.12.030 18 S. Lebedev et al. / Tectonophysics xxx (2013) xxx–xxx Yao, H., Gouédard, P., Collins, J.A., McGuire, J.J., van der Hilst, R.D., 2011. Structure of young East Pacific rise lithosphere from ambient noise correlation analysis of fundamental- and higher-mode Scholte–Rayleigh waves. Comptes Rendus Geoscience 343, 571–583. Zhang, X., Paulssen, H., Lebedev, S., Meier, T., 2007. Surface wave tomography of the Gulf of California. Geophysical Research Letters 34. http://dx.doi.org/10.1029/ 2007GL030631. Zhang, X., Paulssen, H., Lebedev, S., Meier, T., 2009. 3D shear velocity structure beneath the Gulf of California from Rayleigh wave dispersion. Earth and Planetary Science Letters 279, 255–262. Zhu, L., Kanamori, H., 2000. Moho depth variation in southern California from teleseismic receiver functions. Journal of Geophysical Research 105, 2969–2980. Please cite this article as: Lebedev, S., et al., Mapping the Moho with seismic surface waves: A review, resolution analysis, and recommended inversion strategies, Tectonophysics (2013), http://dx.doi.org/10.1016/j.tecto.2012.12.030