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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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⁎ 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
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Phase velocity
Group velocity
dC/dVs
dU/dVs
0
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10
10
20
20
30
30
40
40
0
0
10
10
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20
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30
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Period: 15s
60
0
Depth, km
Depth, km
Period: 6s
Period: 15s 60
0
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Depth, km
Depth, km
Period: 6s
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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).
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inversion strategies, Tectonophysics (2013), http://dx.doi.org/10.1016/j.tecto.2012.12.030