submitted to Geophys. J. Int.
Databases of surface wave dispersion
Simona Carannante (1,2), Lapo Boschi (3)
(1) Dipartimento di Scienze Fisiche, Università Federico II, Naples, Italy
(2) Istituto Nazionale di Geofisica e Vulcanologia, Bologna, Italy
(3) E.T.H. Zürich, 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 frequency, or, in other words, individual harmonic components of the surface wave seismogram
(individual modes) propagate over the globe at different speeds. For any given earthquakeseismometer 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., Ekström 2000). Over the last decade, two large intermediate-period (35-150 s) dispersion databases have been assembled by Trampert & Woodhouse (1995; 2001) and Ekström 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 & Ekström 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 Ekström 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 Ekström 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.
Ekström et al. (1997) and Trampert & Woodhouse (1995; 2001) find synthetic seismograms in slightly different, but equivalent ways; both groups write the corrections to
amplitude and phase, with respect to PREM (Dziewonski & Anderson 1981), as linear combinations of a set of cubic splines fi (ω), with coefficients bi and di , respectively,
δA(ω) =
N
X
bi fi (ω),
(3)
di fi (ω),
(4)
i=1
δc(ω) =
N
X
i=1
and the dispersion measurement is reduced to the determination of bi , di (i = 1, ..., N).
The outlined inverse problem is non-linear, and Ekström et al. (1997) accordingly choose
to solve it through the non-linear SIMPLEX method (e.g., Press et al. 1994). They parameterize 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 Ekström et al. (1997) and Trampert & Woodhouse (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 higherfrequency 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 Ekström et al. (1997) and Trampert & Woodhouse (1995), and
nicely illustrated by figure 3 of Ekström et al. (1997).
We show in figures 1 and 2 histograms of Trampert & Woodhouse’s (2001) and Ekström
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 Ekström 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; Ekström et al. 1997; Boschi & Ekström 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 Ekström 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 Ekström 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 Ekström 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; figure 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 Ekström 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 datafit (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 Ekström et al.’s (1997)
data is systematically better; this might mean that Ekström 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 Ekström 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. Ekström 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 nonlinear 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 Ekström 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 Ekström 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 Ekström et al.’s (1997) database; the problem, however,
deserves further analysis, possibly from authors with deeper insight in the measuring techniques, rather than tomographic applications.
ACKNOWLEDGMENTS
We are particularly grateful to prof. Paolo Gasparini for his help and encouragement. Thanks
also to Göran Ekström, Domenico Giardini, Jeannot Trampert, Aldo Zollo. During his stay
at Università di Napoli Lapo Boschi was funded by Ministero dell’Istruzione, Università e
Ricerca. All figures were done with GMT (Wessel and Smith 2001).
REFERENCES
Antolik, M., Gu, Y. J., Ekström, G. & Dziewonski, A. M., 2003. J362D28: a new joint model of
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–
25,594.
Boschi, L. & Ekström, 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.
Boschi, L., Ekström, G. & Kustowski, K., 2004. Multiple resolution surface wave tomography: the
Mediterranean basin, Geophys. J Int., 157, 293–304 doi:10.1111/j.1365-246X.2004.02194.x.
Carannante, S., 2004. Velocità di fase delle onde sismiche di superficie: immagini tomografiche
globali a risoluzione variabile, Tesi di laurea, Università di Napoli Federico II, Naples, Italy.
Ekström, G., 2000. Mapping the lithosphere and asthenosphere with surface waves: Lateral structure and anisotropy, in The History and Dynamics of Global Plate Motions, edited by M.
Richards et al., AGU monograph.
Databases of surface wave dispersion
9
Ekström, G., Tromp, J. & Larson, E. W. F., 1997. Measurements and global models of surface
wave propagation, J. geophys. Res., 102, 8137–8157.
Jordan, T. H., 1978. A procedure for estimating lateral variations from low-frequency eigenspectra
data, Geophys. J. R. astr. Soc., , 52, 441–455.
Mooney, W. D., Laske, G., and Masters, G., 1998. Crust 5.1: a global crustal model at 5x5 degrees,
J. geophys. Res., 103, 727–747.
Spetzler, J., Trampert, J. & Snieder, R., 2002. The effect of scattering in surface wave tomography,
Geophys. J Int., 149, 755–767.
Tarantola, A. & Valette, B., 1982. Generalized non-linear inverse problems solved using the leastsquares criterion, Rev. Geophys. Space Phys., 20, 219–232.
Trampert, J. & Woodhouse, J. H., 1995. Global phase velocity maps of Love and Rayleigh waves
between 40 and 150 seconds, Geophys. J Int., 122, 675-690.
Trampert, J. & Woodhouse, J. H., 2001. Assessment of global phase velocity models, Geophys. J
Int., 144, 165–174.
Tromp, J. & Dahlen, F. A., 1992. Variational principles for surface wave propagation on a laterally
heterogeneous Earth, II, Frequency-domain JWKB theory, Geophys. J Int., 109 599–619.
Tromp, J. & Dahlen, F. A., 1993. Maslov theory for surface wave propagation on a laterally
heterogeneous Earth, Geophys. J Int., 115 512–528.
Wessel, P., and W. H. F. Smith, Free software helps map and display data, 1991. Eos Trans. AGU,
72, 445–446.
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) Ekström 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) Ekström 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)
Ekström et 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 (Ekström 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 regularization scheme were applied throughout. Results of inversions not discussed above are also
included.
Figures
20
36
18
16
14
12
10
8
6
24
20
16
12
4
8
2
4
0
-200
-100
0
100
data(s)
0
-120 -80
200
48
-40
0
40
data(s)
80
120
80
120
48
(c)
40
32
24
16
32
24
16
8
8
0
-120 -80
0
-120 -80
-40
0
40
data(s)
(d)
40
frequency(%)
frequency(%)
(b)
28
frequency(%)
frequency(%)
32
(a)
80
120
-40
0
40
data(s)
Figure 1. Histograms of Love wave phase delays (seconds) observed by (blue) Ekström et al.
(1997) and (green) Trampert & Woodhouse (2001), at (a) 35 s, (b) 60 s, (c) 100 s, and (d) 150
s periods.
24
36
16
12
8
24
20
16
12
8
4
4
0
0
-200
-100
0
100
data(s)
200
-120 -80 -40
48
0 40
data(s)
80 120
60
(c)
40
(d)
50
32
frequency(%)
frequency(%)
(b)
28
frequency(%)
frequency(%)
32
(a)
20
24
16
8
40
30
20
10
0
0
-120 -80 -40
0 40
data(s)
80 120
Figure 2. Same as figure 1, Rayleigh wave phase delays.
-120 -80 -40
0 40
data(s)
80 120
LOVE WAVES
TRAMPERT
EKSTROM
(35s)
(35s)
(40s)
(40s)
(51s)
(50s)
(62s)
(60s)
-4.5
0.0
4.5
Figure 3. Tomographic maps of Love wave phase velocity (percent perturbations to PREM)
from the databases of (left) Trampert & Woodhouse (2001) and (right) Ekström et al. (1997),
at short to intermediate periods (given in brackets at each panel).
RAYLEIGH WAVES
TRAMPERT
EKSTROM
(35s)
(35s)
(40s)
(40s)
(51s)
(50s)
(62s)
(60s)
-4.5
0.0
4.5
Figure 4. Same as figure 3, Rayleigh wave phase velocity maps.
LOVE WAVES (35s)
TRAMPERT
EKSTROM
10
TRAMPERT
EKSTROM
9
Amplitude(%)
8
7
6
5
4
3
2
1
0
0
2
4
6
8 10 12 14 16
Spherical Harmonic degree
18
20
22
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)
Ekström et al. (1997). It is at this Love wave frequency that images resulting from the two data
sets are most different.
1.4
variance reduction (%)
LOVE WAVES
1.2
Ekstrom
Trampert
1.0
0.8
0.6
0.4
0.2
0
80
160
period (s)
240
320
240
320
1.4
variance reduction (%)
RAYLEIGH WAVES
1.2
Ekstrom
Trampert
1.0
0.8
0.6
0.4
0.2
0
80
160
period (s)
Figure 6. Fit of the two databases (Ekström 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 regularization scheme were applied throughout. Results of inversions not discussed above are also
included.