submitted to Geophys. J. Int.

Databases of surface wave dispersion

Simona Carannante (1,2), Lapo Boschi (3)

(1) Dipartimento di Scienze Fisiche, Universita Federico II, Naples, Italy

(2) Istituto Nazionale di Geofisica e Vulcanologia, Bologna, Italy

(3) E.T.H. Zurich, Switzerland.

Received

SUMMARY

We derive phase velocity maps from two global databases of surface wave dispersion,

and compare them to assess the relative quality of the observations. Although based

upon similar sets of seismic records, the databases show some significant discrepancies,

which we attempt to trace to the different automated measurement techniques employed

by their authors. This exercise is relevant to our current understanding of the Earth’s

upper mantle, whose elastic structure is most importantly constrained by surface wave

observations like the ones in question.

Key words: seismic tomography, upper mantle, lateral resolution

1 INTRODUCTION

Seismic surface waves are dispersive; their speed of propagation is a function of their fre-

quency, or, in other words, individual harmonic components of the surface wave seismogram

(individual modes) propagate over the globe at different speeds. For any given earthquake-

seismometer couple, one can isolate from the seismogram each harmonic component, and

measure its average speed (phase velocity) between source and receiver. We call “dispersion

curve” a plot of phase velocity against frequency. From a large set of measured dispersion

curves, with sources and stations providing a coverage as uniform as possible of the Earth,

2 Carannante and Boschi

local phase velocity heterogeneities can be mapped as a function of longitude and latitude.

At each location, perturbations in phase velocities with respect to a given reference model

are weighted averages of seismic heterogeneities in the underlying mantle (Jordan 1978).

Measurements of surface wave dispersion provide, through the steps outlined above, the

most important global seismological constraint on the nature of the Earth’s upper mantle

(e.g., Ekstrom 2000). Over the last decade, two large intermediate-period (35-150 s) disper-

sion databases have been assembled by Trampert & Woodhouse (1995; 2001) and Ekstrom et

al. (1997) from global networks of broadband seismic stations, and are now (at least to some

extent) available to the public. Both are currently employed in global tomography studies

(e.g., Boschi & Ekstrom 2002; Spetzler et al. 2002; Antolik et al. 2003; Beghein 2003; Boschi

et al. 2004). The two databases are in good, but not complete agreement, and their relative

quality is under debate.

The main difference between the work of Trampert & Woodhouse (1995; 2001) and that

of Ekstrom et al. (1997) rests in the automated algorithms used to determine dispersion

curves from individual seismograms. In both cases, theoretical (“synthetic”) seismograms

are calculated from theoretical dispersion curves, and compared to the measured one; an

inverse problem is then solved to find the theoretical dispersion curve that minimizes the

misfit between measured and synthetic seismograms; such optimal dispersion curve is the

seeked dispersion measurement; as we shall illustrate below, the two groups have tackled

this inverse problem in very different ways.

In the following, we apply the same algorithm to derive phase velocity maps, at a discrete

set of frequencies, from both databases. We identify specific discrepancies, and attempt to

trace them to differences in the measurement procedures.

2 MEASURING SURFACE WAVE DISPERSION

Following Ekstrom et al.’s (1997) notation, we start by writing the surface wave seismogram

u(ω) in the form

u(ω) = A(ω)eiΦ(ω), (1)

Databases of surface wave dispersion 3

where amplitude A and phase Φ are functions of frequency ω. Φ(ω) and the phase velocity

c(ω) are related,

Φ(ω) =ωX

c(ω), (2)

with X denoting propagation path length measured along the great circle. u(ω) can be

calculated, e.g., by surface wave JWKB theory as described by Tromp & Dahlen (1992;

1993).

Given an observed seismogram, surface wave dispersion between the corresponding source

and receiver is measured by finding the curves A(ω) and c(ω) (hence Φ(ω)) that minimize

the misfit between the observed seismogram and u(ω). The best-fitting c(ω) is the seeked

dispersion curve.

Ekstrom et al. (1997) and Trampert & Woodhouse (1995; 2001) find synthetic seis-

mograms in slightly different, but equivalent ways; both groups write the corrections to

amplitude and phase, with respect to PREM (Dziewonski & Anderson 1981), as linear com-

binations of a set of cubic splines fi(ω), with coefficients bi and di, respectively,

δA(ω) =N∑

i=1

bifi(ω), (3)

δc(ω) =N∑

i=1

difi(ω), (4)

and the dispersion measurement is reduced to the determination of bi, di (i = 1, ..., N).

The outlined inverse problem is non-linear, and Ekstrom et al. (1997) accordingly choose

to solve it through the non-linear SIMPLEX method (e.g., Press et al. 1994). They param-

eterize their dispersion curves in terms of only N = 6 splines; this number being relatively

small, curves tend to be smooth, and no further regularization is needed.

Here the approach of Trampert & Woodhouse (1995, 2001) is quite different; first, their

dispersion curves are described by as many as N = 36 splines (Trampert & Woodhouse 2001).

Secondly, they accomodate the non-linearity of the inverse problem in a very different way:

since group velocities “are secondary observables easily calculated from the seismograms”,

they prefer to first measure those, and then derive phase velocity dispersion curves from

4 Carannante and Boschi

them, now exploiting their linear relationship with the least-squares algorithm of Tarantola

& Valette (1982).

To our understanding, the procedures of Ekstrom et al. (1997) and Trampert & Wood-

house (1995, 2001) are, in all other respects, equivalent to each other. It is worth mentioning

one issue that both groups had to consider while developing their algorithms; notice from

equation (1) that the surface wave seismogram is a periodic function of Φ with period 2π;

when dispersion is measured, values of Φ different of exactly 2π will provide an equally good

fit to the data; and perturbations δΦ larger than 2π cannot simply be discarded, because at

intermediate periods (say, < 100s), true heterogeneities in the Earth’s structure can suffice to

cause phase anomalies of more than one cycle. Both groups solve this ambiguity in the same

way: at long periods (> 100s) the exact number of full cycles in phase can be determined

without ambiguity; this part of the seismic spectrum is then inverted first, and the higher-

frequency portion of the dispersion curve is found through subsequent iterations, with the

requirement, based on physical considerations, that the dispersion curve be smooth. This is

discussed in detail both by Ekstrom et al. (1997) and Trampert & Woodhouse (1995), and

nicely illustrated by figure 3 of Ekstrom et al. (1997).

We show in figures 1 and 2 histograms of Trampert & Woodhouse’s (2001) and Ekstrom

et al.’s (1997) phase delay measurements, grouped by surface-wave type (Rayleigh, Love)

and period (data at 35, 60, 100 and 150 s only are shown, to represent the entire databases).

At longer periods, both databases’ standard deviations decrease; at 35 s, where the data are

more sensitive to strong crustal or lithospheric heterogeneities, observed phase delays are,

as expected, stronger; more importantly, figures 1a and 2a (Love and Rayleigh phase delays,

respectively, at 35 s) show the most significant differences between the two databases.

3 PHASE-VELOCITY MAPS

We derive Love and Rayleigh wave phase velocity maps at all periods (35 to 150 s) where

measurements are available; a separate tomographic inversion is performed at each period,

to find an independent phase velocity map; we then compare maps obtained from Trampert

Databases of surface wave dispersion 5

& Woodhouse’s (2001) database with those obtained from Ekstrom et al.’s (1997) one. In

order for this comparison to be meaningful, we employ in both cases the same tomographic

parameterization (approximately equal-area pixels, of size 5◦ × 5◦ degree at the equator),

regularization (a combination of norm and roughness damping), and inversion algorithm

(LSQR) (Boschi & Dziewonski, 1999; Boschi, 2001). The regularization scheme (constant

throughout this work) was chosen after a number of preliminary inversions (Carannante

2004).

Our results (from both databases) are in general agreement with those of other authors

(e.g., Trampert & Woodhouse 1995, 2001; Ekstrom et al. 1997; Boschi & Ekstrom 2002);

shorter period data are predominantly sensitive to crustal structure, and the corresponding

phase velocity maps are dominated by the ocean-continent signature; as period increases,

lithospheric features (upwellings of hot material at mid-oceanic ridges, fast continental roots)

become dominant. Also, as a general rule, at any given period Love wave maps are affected

by structure at shallower depths than Rayleigh wave ones.

Maps derived from the two databases are similar at all periods (Carannante 2004). The

agreement increases with increasing period. Figures 3 and 4 show phase velocity maps, from

both databases, at relatively short periods (35 to 60 s); relatively strong discrepancies are

visible at 35 s and 40 s for Love waves (figure 3), and 35 s only for Rayleigh waves (figure

4). Particularly at those periods, maps resulting from Ekstrom et al.’s (1997) database are

more clearly dominated by the contrast between slow continents and fast oceans; mid-oceanic

ridges, and heterogeneities of various nature within the continents, are only minor, if visible

at all. In the maps that we have obtained at the same short periods from Trampert &

Woodhouse’s (2001) database, instead, the dominant ocean-continent signal is accompanied

by a more significant shorter spatial wavelength component, associated with the mentioned

geophysical features.

In practice, particularly for Love waves, phase velocity heterogeneities imaged from Ek-

strom et al.’s (1997) data tend to depend on period more strongly than those found from

Trampert & Woodhouse’s (2001) data.

6 Carannante and Boschi

The most evident discrepancies are those between the 35 s Love wave phase velocity

maps of figure 3 (enlarged in figure 5), in Tibet and in the Atlantic Ocean. The map derived

from the data of Ekstrom et al. (1997) is dominated by the main features of crustal thickness

(Mooney et al. 1998; Bassin et al. 2000), while the one from Trampert and Woodhouse’s

(2001) database is more correlated to longer period phase velocity heterogeneity (i.e. upper

mantle structure).

To quantify the above observations, we have carried out a spectral analysis of phase

velocity maps, finding their spherical harmonic coefficients through least-squares fits; fig-

ure 5 shows the least well correlated spectra, found (as expected) from the 35 s Love

wave maps. The spectral peaks at harmonic degrees 2 and 5, naturally associated with

oceanic/continental plates, are much stronger according to Ekstrom et al.’s (1997) than

Trampert & Woodhouse’s (2001) database; the opposite is true of higher degree coefficients

(9 through 14 in figure 5).

A last useful information to discriminate between the databases in question is the data-

fit (variance reduction) achieved when inverting the observations; we give it in figure 6,

plotted as a function of period for all the periods considered by Carannante (2004). As

already noticed by the compilers of the databases, variance reduction of both Love and

Rayleigh wave data decreases with increasing period. The fit to Ekstrom et al.’s (1997)

data is systematically better; this might mean that Ekstrom et al. (1997) have selected

the data more conservatively, succeeding to clean them from effects of noise that might

still be present in Trampert & Woodhouse’s (2001) database, or that the latter database

includes information on heterogeneities of shorter spatial frequency, that our 5◦ × 5◦ pixel

parameterization cannot resolve.

4 CONCLUSIONS

The dispersion databases of Trampert & Woodhouse (2001) and Ekstrom et al. (1997)

provide a very important constraint on the shallow seismic structure of the Earth. They lead

to similar, but not entirely consistent images of surface wave phase velocities. Discrepancies

Databases of surface wave dispersion 7

we have found are strong enough to alter our understanding of lithosphere and upper mantle;

they might result from differences in the networks of seismic stations, and time windows

chosen by the authors to collect the original seismic records; more realistically–since the

phase velocity maps shown here are increasingly better correlated at intermediate to long

periods–they could be explained in terms of the different algorithms designed by the authors

to measure dispersion curves from recorded seismograms.

As discussed in section 2, the main difference between the algorithms in question resides

in the parameterization of dispersion curves, and in the inversion methods employed to derive

them from recorded seismograms. Ekstrom et al. (1997) chose a coarser parameterization

of the dispersion curve (6 splines only), but did not require any further smoothing than

that implicit in the parameterization itself, and found dispersion curves through fully non-

linear inversions. At 35 s period, our histogram of their Love wave phase delay observations

(figure 1) is characterized by a larger standard deviation than Trampert & Woodhouse’s

(2001); the latter authors, who derive dispersion curves with a linear, damped least squares

algorithm, might have over-smoothed them; this would explain the systematically lower

variance reduction achieved when inverting their data, and the stronger correlation, in figure

1, between the Love wave phase velocity maps at 35 and 40 s obtained from their data, vs.

those obtained from Ekstrom et al.’s (1997) data. In other words, the short period end of

Trampert & Woodhouse’s (2001) dispersion curves might be too strongly “anchored” to

longer periods.

On the other hand, maps resulting from Trampert & Woodhouse’s (2001) database are

characterized by a stronger power spectrum at shorter spatial wavelengths (e.g., figure 5);

their data might contain meaningful information associated with geophysical features of

small spatial extent, that Ekstrom et al.’s (1997) data cannot resolve. Accordingly, the

variance reduction plots of figure 6 could simply indicate that our intermediate resolution

(5◦ × 5◦ pixels) parameterization cannot adequately describe such narrow heterogeneities.

This interpretation, if correct, would invalidate the previous speculations.

Carannante (2004) has performed an additional set of nonuniform resolution inversions,

8 Carannante and Boschi

whose results appear to favour Ekstrom et al.’s (1997) database; the problem, however,

deserves further analysis, possibly from authors with deeper insight in the measuring tech-

niques, rather than tomographic applications.

ACKNOWLEDGMENTS

We are particularly grateful to prof. Paolo Gasparini for his help and encouragement. Thanks

also to Goran Ekstrom, Domenico Giardini, Jeannot Trampert, Aldo Zollo. During his stay

at Universita di Napoli Lapo Boschi was funded by Ministero dell’Istruzione, Universita e

Ricerca. All figures were done with GMT (Wessel and Smith 2001).

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compressional and shear velocity in the mantle, Geophys. J Int., 153, 443–466.

Bassin, C., Laske, G., and Masters, G., 2000. The current limits of resolution for surface wave

tomography in North America, Eos Trans. AGU, 81, F897.

Beghein, C., 2003. Seismic anisotropy inside the Earth from a model space search approach, PhD

thesis, Utrecht University, Utrecht, Netherlands.

Boschi, L., 2001. Applications of linear inverse theory in modern global seismology, PhD thesis,

Harvard University, Cambridge, Massachusetts.

Boschi, L. & Dziewonski, A., 1999. “High” and “low” resolution images of the Earth’s mantle -

Implications of different approaches to tomographic modeling, J. geophys. Res., 104, 25,567–

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Boschi, L. & Ekstrom, G., 2002. New images of the Earth’s upper mantle from measurements of

surface-wave phase velocity anomalies, J. geophys. Res., 107, doi:10.129/2000JB000059.

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Mediterranean basin, Geophys. J Int., 157, 293–304 doi:10.1111/j.1365-246X.2004.02194.x.

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globali a risoluzione variabile, Tesi di laurea, Universita di Napoli Federico II, Naples, Italy.

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ture and anisotropy, in The History and Dynamics of Global Plate Motions, edited by M.

Richards et al., AGU monograph.

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wave propagation, J. geophys. Res., 102, 8137–8157.

Jordan, T. H., 1978. A procedure for estimating lateral variations from low-frequency eigenspectra

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squares criterion, Rev. Geophys. Space Phys., 20, 219–232.

Trampert, J. & Woodhouse, J. H., 1995. Global phase velocity maps of Love and Rayleigh waves

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Trampert, J. & Woodhouse, J. H., 2001. Assessment of global phase velocity models, Geophys. J

Int., 144, 165–174.

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This paper has been produced using the Blackwell Scientific Publications GJI LaTEX style file.

Figure Captions

Figure 1. Histograms of Love wave phase delays (seconds) observed by (blue) Ekstr¨om et al.

(1997) and (green) Trampert & Woodhouse (2001), at (a) 35 s, (b) 60 s, (c) 100 s, and (d) 150

s periods.

Figure 2. Same as figure 1, Rayleigh wave phase delays.

Figure 3. Tomographic maps of Love wave phase velocity (percent perturbations to PREM)

from the databases of (left) Trampert & Woodhouse (2001) and (right) Ekstr¨om et al. (1997),

at short to intermediate periods (given in brackets at each panel).

Figure 4. Same as figure 3, Rayleigh wave phase velocity maps.

Figure 5. Spectral analysis of 35 s Love wave phase velocity maps derived from the databases

of (top map, green spectrum) Trampert & Woodhouse (2001) and (bottom map, blue spectrum)

Ekstromet al. (1997). It is at this Love wave frequency that images resulting from the two data

sets are most different.

Figure 6. Fit of the two databases (Ekstr¨om et al. (1997 in blue, Trampert and Woodhouse

(2001) in green) achieved in our inversions, as a function of period, for both Love (top panel)

and Rayleigh (bottom panel) wave observations. The same inversion algorithm and regular-

ization scheme were applied throughout. Results of inversions not discussed above are also

included.

Figures

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Figure 1. Histograms of Love wave phase delays (seconds) observed by (blue) Ekstr¨om et al.

(1997) and (green) Trampert & Woodhouse (2001), at (a) 35 s, (b) 60 s, (c) 100 s, and (d) 150

s periods.

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Figure 2. Same as figure 1, Rayleigh wave phase delays.

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Figure 3. Tomographic maps of Love wave phase velocity (percent perturbations to PREM)

from the databases of (left) Trampert & Woodhouse (2001) and (right) Ekstr¨om et al. (1997),

at short to intermediate periods (given in brackets at each panel).

(35s)

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(40s)

(51s)

(62s)

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Figure 4. Same as figure 3, Rayleigh wave phase velocity maps.

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Figure 5. Spectral analysis of 35 s Love wave phase velocity maps derived from the databases

of (top map, green spectrum) Trampert & Woodhouse (2001) and (bottom map, blue spectrum)

Ekstromet al. (1997). It is at this Love wave frequency that images resulting from the two data

sets are most different.

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Figure 6. Fit of the two databases (Ekstr¨om et al. (1997 in blue, Trampert and Woodhouse

(2001) in green) achieved in our inversions, as a function of period, for both Love (top panel)

and Rayleigh (bottom panel) wave observations. The same inversion algorithm and regular-

ization scheme were applied throughout. Results of inversions not discussed above are also

included.