Geophys. J. Int. (2001) 146, 795–800

Isotropic and anisotropic Pn velocity inversion of regional earthquake traveltimes underneath Germany L.-P. Song,1 M. Koch,1 K. Koch2 and J. Schlittenhardt2 1

Department of Geohydraulics and Engineering Hydrology, University of Kassel, Kurt-Wolters-Strasse 3, D-34109 Kassel, Germany. E-mail: [email protected] 2 Bundesanstalt fu¨r Geowissenschaften und Rohstoffe, Stilleweg 2, D-30655 Hannover, Germany

Accepted 2001 April 23. Received 2001 April 18; in original form 2000 December 6

SUMMARY This paper investigates the uppermost mantle velocity beneath Germany using regional earthquake traveltime data. 2149 Pn traveltimes corresponding to 220 events recorded at 70 stations covering the region of 47uN–52uN latitude, 5uE–15uE longitude result in a satisfactory ray-path distribution. Three methods with increasing degree of complexity are used to analyse the Pn traveltime data: a straight-line fit; the classical time-term method; and a modified time-term method including azimuthal anisotropy. First, from the straight-line fit to the data set, an average Pn velocity of 7.98 km sx1 is inferred. Second, the classical time-term method yields a mean uppermost mantle velocity of 7.99 km sx1. The most important feature in this analysis is the azimuth-dependent pattern of the residuals, indicating some evidence of velocity anisotropy in the upper mantle. The time-term method achieves about 55 per cent variance reduction relative to the straight-line fit. Third, two modified ‘anisotropic’ time-term methods provide an average Pn velocity of 8.09 km sx1, with a further data variance reduction of 64 and 20 per cent relative to the straight-line fit and the classical time-term method, respectively. The estimated anisotropy level is about 3.5–4 per cent, with maximum and minimum velocities of 8.24–8.27 km sx1 and 7.95 km sx1. Our estimated maximum velocity direction of yN25uE coincides with those of previous anisotropic studies on the uppermost mantle in this region based on seismic refraction data. The results from the present study thus support the idea that Pn-wave anisotropy is a large-scale lithospheric feature over much of central Europe. Key words: earthquake traveltimes, Germany, seismic velocity anisotropy, upper mantle.

1

INTRODUCTION

Since the 1960s, Pn-wave traveltimes have been used extensively to investigate both isotropic and anisotropic uppermost mantle velocity structures beneath the oceans and continents on local and regional scales (e.g. Hess 1964; Backus 1965; Morris et al. 1969; Raitt et al. 1969; Hearn 1984, 1996). Similar studies of Pn velocity within Germany have been conducted, although they mostly focused on the area of southern or only southwestern Germany (e.g. Bamford 1973, 1977; Fuchs 1977, 1983; Koch 1993; Enderle et al. 1996). With these refraction-based investigations, the existence of Pn velocity anisotropy in the uppermost mantle underneath Germany has been widely recognized. Vinnik et al. (1994) inferred the mantle anisotropy from observations of shear wave splitting in SKS data recorded at German Regional Seismic Network (GRSN) stations. More recently, Schlittenhardt (1999), using traveltimes of regional earthquakes recorded by the GRSN, showed qualitatively that an azimuthal dependence of Pn residuals for a homogeneous isotropic Pn # 2001

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velocity could be an indication of uppermost mantle anisotropy in the region. Using regional earthquake traveltime data from several thousand events, from which a subset was studied by Schlittenhardt (1999), we intend to map quantitatively the uppermost mantle velocity underneath Germany, including possible anisotropic effects. This, in conjunction with a full 3-D simultaneous tomographic inversion, presently being carried out by the authors, should provide new valuable information for an enhanced understanding of regional tectonics and the geodynamic evolution of the area. As a first step of this ongoing work, we describe here the data, the analysis methods used— namely modifications of the time-term analysis method—and some preliminary results. 2

DATA

The data used in this study were compiled at BGR (Hannover) (Henger & Leydecker 1975 and later). BGR gathers local and

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Figure 1. Earthquakes and explosions (circles) recorded between 1975 and 1999 at seismic stations (triangles) in Germany and neighbouring countries.

regional earthquake arrival-time data from seismic stations in Germany and neighbouring countries, and issues a yearly bulletin of the seismic activity in this region. Depending on the geographical location and magnitude of a seismic event, the hypocentre is either determined by local seismic station networks (university institutes or state geological surveys) or re-determined independently by BGR. In this study we use arrival times from earthquakes and explosions for the years 1975–1999. The majority of the events are located between 47uN and 52uN, and 5uE and 15uE (Fig. 1). Although nearly 7000 seismic events were reported during this time, based on the selection criteria relevant for the objectives of this study, namely the analysis of Pn traveltimes, only a few hundred suitable events could be used. First of all, it was necessary to identify the Pn phases correctly from the arrival-time data set for each event. It turned out that many phases that were originally labelled Pn proved not to be so. This can be seen from Fig. 2, which shows the originally labelled Pn traveltimes versus epicentral distances, using a reduction velocity of 8.0 km sx1. Only those phases scattered around the horizontal branch beyond an epicentral distance of about 150 km can really be considered Pn. Numerous mislabelled Pn phases are observed along the sloping line, corresponding to a reduced traveltime with a velocity of 6.0 km sx1, typical for the crustal Pg. Based on this plot, we first determine a major branch of Pn phases with epicentral distances above 150 km and reduced traveltimes between 4 and 8 s. We expect that the Pn in this branch is almost always the first arrival, and minimize the possible misidentification between Pn and Pg. Although this misidentification certainly will have an effect on the hypocentral

Figure 2. The original Pn data and the selected branch of Pn arrivals within the superimposed window.

location of the event, its effect in the present study on Pn data will be negligible. Errors introduced by the mislocation of events are partially absorbed into the event time terms defined in the time-term analysis presented in the next section. Second, we restrict our Pn data set by the constraints that each event (with a Pn phase) should be recorded by at least five stations, and each station should have at least five Pn recordings. These constraints were introduced in order to allow application of the time-term method. Using these constraints, we obtained a total of 2149 Pn traveltimes from the raw 9031 Pn data shown #

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Pn velocity inversion underneath Germany in Fig. 2, corresponding to 220 events recorded at 70 stations. The selected Pn traveltimes are implicitly corrected for station elevations during the following analysis. The average event depth of these 220 events, which is also absorbed in the following analysis, is 6.8 km with a maximum depth of 23 km. These events and the recording stations distributed within the 47uNx52uN latitude, 5uEx15uE longitude geographical rectangle are shown in Fig. 3 with the corresponding Pn ray-path coverage. Except for the northeastern portion of the outlined area, the ray-path coverage (i.e. both ray-path density and azimuths) is satisfactory. Relative to previous Pn studies within Germany (e.g. Bamford 1977; Enderle et al. 1996), our data set not only contains over two times to almost four times the number of ray paths used before, but also covers a larger area.

3

METHODS OF ANALYSIS

Three approaches with increasing complexity will be used in this research to analyse the Pn traveltime data: (a) a linear regression model for an isotropic Pn velocity; (b) the time-term method to incorporate event and station corrections; and (c) a modified time-term method to include azimuthal anisotropy in two different representations. (a) The linear regression model, the basic technique in refraction seismology to determine layer velocities and depths (Telford et al. 1976), uses the well-known formula tij ¼ t0 þ Dij S ,

(1)

where tij is the total traveltime between event i and station j, Dij is the epicentral distance, S=1/Vref is the refractor slowness,

and t0 is the intercept time, which are used to determine layer velocities and depths. (b) The classical time-term method (Scheidegger & Willmore 1957) was introduced in refraction seismology a few decades ago and allows for a most effective analysis of regional refraction traveltimes. The basic equation of the time-term method expresses the total traveltime tij between event i and station j as tij ¼ ai þ bj þ Dij S ,

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(2)

where now ai is the ith event delay term and bj is the jth station delay term, replacing the intercept term in the linear regression model and accounting for the times spent in the crust by the refracted wave. The delay terms characterize the combined effect of the crustal thickness, crustal velocity and the subsurface velocity beneath each station. Compared with small slowness perturbations, crustal thickness changes have the largest effect on the delay terms (e.g. Bamford 1973; Hearn & Ni 1994). For regional earthquake traveltimes, the event delays cannot be used for a reasonable structural interpretation since they also contain the unknown errors in source depth and origin time, both of which, contrary to studies using refraction seismic data, are not precisely known. Station static delays may be interpreted in terms of Moho topography and near-station velocity structure, but may also reflect systematic errors probably induced by clock errors, timing errors or phase identification errors. In summary, in the present study the delay time terms serve to isolate the unknown errors efficiently from the Pn traveltimes (c) The modified time-term method, an extension of the classical time-term method above that allows for the determination of both uppermost mantle isotropic and anisotropic lateral velocity variation is obtained by modifying eq. (2) as

Figure 3. Ray-path distribution for the 2149 Pn arrivals. Circles represent the events. Triangles represent the stations. #

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follows (e.g. Morris et al. 1969; Raitt et al. 1969). Introducing a small perturbation DS to the constant refractor slowness S0 and expanding eq. (2) around S0 to first order, one obtains tij ¼ ai þ bj þ Dij S0 þ ðDij ÿ Fi ÿ Fj Þ*S ,

(3)

where Fi and Fj are the horizontal offsets at the source and receiver sides, respectively, and are calculated easily from wellknown refraction seismic formulae. In the use of eq. (3) it is usually assumed that Fi=Fj=const. for all i and j in order to simplify the problem (Raitt et al. 1969; Bamford 1973, 1977). Depending on its explicit form, the slowness perturbation DS may be treated either as an uppermost isotropic mantle velocity variation, or as an anisotropy term, or as a combination of both in the 2-D horizontal sub-Moho plane. For weakly orthorhombic media, an approximate formula in a symmetry plane (Song & Every 2000) describes the directional variation of the Pn group slowness S as S2 ¼ A þ B cos 2ðr ÿ hÞ þ C cos 4ðr ÿ hÞ ¼ A þ D cos 2r þ E sin 2r þ F cos 4r þ G sin 4r ,

(4)

where the coefficients A to C are combinations of the longitudinal elastic constants, w is the azimuthal angle of the ray direction, h is the direction of the maximum velocity axis, D=B cos 2h, E=B sin 2h, F=C cos 4h, and G=C sin 4h. Eq. (4) expresses the squared group slowness S2 as two parts: one isotropic part A=S02, and one anisotropic part consisting of the last four terms, which may be treated as a perturbation of the isotropic part. This formula works well even for a fairly large degree of anisotropy (Song & Every 2000). When the cos 4(w xh) term is neglected, eq. (4) reduces to an elliptical equation for the group velocity (e.g. Michelena et al. 1993; Song et al. 1998) assuming a low degree of anisotropy. In this case, one may also use a phase velocity formula to approximate the group velocity (Backus 1965). In the current study, we limit ourselves to the analysis of the 1-D gross Pn wave anisotropy across the upper mantle underneath Germany (Fig. 3). With this, the slowness perturbation in eq. (3) can be explicitly written as *S ¼

1 ðD cos 2r þ E sin 2r þ F cos 4r þ G sin 4rÞ , 2S0

(5)

and, depending on whether the 4w terms are included or not, two variants of the modified time-term method are obtained. All three methods of analysis presented above are cast in the form of a general linear least-squares problem with increasing complexity and number m of unknowns. Whereas for the regression method two unknowns are determined, for the classical time-term method mtot=mevent+mstat+1y300,

i.e. event static delays ai, station static delays bj, and refractor velocity, and for the anisotropic time-term method mtot=mevent+ mstat+5y300, with extra anisotropic perturbation parameters D to G. For the solution of the three least-squares problems a standard singular value analysis is used (Lawson & Hanson 1974; Koch 1992).

4

RESULTS AND CONCLUSIONS

Table 1 summarizes the relevant results obtained from the application of the three analysis methods to the data set selected in Section 2. The straight-line fit to the data set produces an average Pn velocity of 7.98 km sx1 and an intercept time of 5.85 s. From these two values and a mean crustal velocity of 5.9 km sx1, the mean crustal thickness is calculated as 25.62 km, which is smaller than that (30 km) of the BGR model (Schlittenhardt 1999). This is clearly indicative of some systematic bias in the traveltime data that cannot be explained by a simple horizontal 2-layer crustal model without accounting for event and station corrections. In fact, the variance of this fit listed in Table 1 is 1.12 s2, which is much higher than the observational arrival-time variance, indicating that much of the residual data variance is not yet explained by the model. Fig. 4(a) shows the traveltime residuals after the straight-line fit as a function of distance. The residuals fluctuate around t2 s and have no significant correlation with distance. This illustrates that a possible mantle velocity gradient may be weak enough to be negligible and the finer structure of the uppermost mantle may not be resolved by our data, in the way as found, for example, by Hirn et al. (1973). The data from our Pn branch may contain both the classical Pn head wave and, for epicentral distances larger than 300 km, the diving mantle P wave, the latter most likely, since we use foci at depth in contrast to seismic refraction studies. As such, we are mapping the upper mantle velocity in a 20–30 km thick layer beneath the Moho. Using the classical time-term method (called time-term 1 in Table 1) a mean uppermost mantle velocity of 7.99 km sx1 and a total mean crustal delay of 5.94 s are obtained. The corresponding data fit variance is 0.5 s2, which represents a variance reduction of about 55 per cent compared with the straight-line fit, but it is still larger than the observed data variance. Fig. 4(b) displays the time residuals versus azimuth. The most important feature is the azimuth-dependent pattern of the residuals in this scatter plot, indicating some evidence of anisotropy in the upper mantle that has not yet been accounted for. Employing now the modified ‘anisotropic’ time-term method, with only the 2w terms included (called time-term 2a in Table 1), a further reduction of the fitted data variance to 0.40 s2 is

Table 1. Results from the straight-line fit, time-term method 1, and time-term methods 2a and 2b. In columns 2 to 4 and column 6 the standard deviations for the estimated velocity values and the direction of Vmax, respectively, are given. Method

Straight line fit Time-term 1 Time-term 2a Time-term 2b

Vref (km sx1)

Vmax (km sx1)

Vmin (km sx1)

Level of Anisotropy (%)

Direction of Vmax (E of North)

Mean Delay (s)

Var (s2)

7.98t1.38r10x2 7.99t1.26r10x2 8.09t1.28r10x2 8.09t1.26r10x2

– – 8.24t1.55r10x2 8.27t1.64r10x2

– – 7.95t1.39r10x2 7.95t1.47r10x2

– – 3.5 3.9

– – 25ut3u 23ut4u

5.85 5.94 6.27 6.26

1.12 0.50 0.40 0.39

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Pn velocity inversion underneath Germany

Figure 4. Traveltime residuals as (a) a function of epicentral distance for the straight-line fit, and (b, c) as a function of azimuth after application of the time-term methods. (b) residuals for time-term method 1 (classical method); (c) residuals for time-term method 2a (only 2w terms included). Note the systematic decrease of the data variance with increasing complexity of the analysis method.

achieved. This amounts to a variance reduction of 64 and 20 per cent, relative to the straight-line fit and to time-term method 1, respectively, indicating that some amount of upper mantle Pn anisotropy is able to explain the observed traveltime residuals better. In fact, the scatter plot of the residuals versus azimuth in Fig. 4(c) illustrates a clear reduction of the undulating correlation of the residuals with azimuth as observed in the middle panel. On the other hand, one notes that an inclusion of the 4w terms (called time-term 2b in Table 1) does not improve the data fit significantly, since the data fit variance is 0.39 s2 only. This may #

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be due to the fact that the anisotropy contained in our data set is not very strong, so there is no need to include 4w terms. In other words, an elliptical approximation seems good enough for fitting the data. Note that Figs 4(b) and (c) show an azimuthal gap of data between 205u and 270u, mainly due to the lack of paths resulting from the Pn selection criteria and the scarcity of seismic events in the northeast. Table 1 shows an average Pn velocity of 8.09 km sx1 and a total mean crustal delay of 6.27 s for time-term method 2a. The estimated maximum and minimum velocities are 8.24 and 7.95 km sx1 and the overall velocity variation is 0.29 km sx1, which corresponds to an anisotropy level of y3.5 per cent. The estimated direction of the angle h of the maximum velocity in eq. (4) is N25uE. For time-term method 2b the estimated maximum and minimum velocities are 8.27 and 7.95 km sx1, i.e. an overall velocity variation of 0.32 km sx1 which corresponds to an anisotropy level of y3.9 per cent. The direction of the maximum velocity is N23uE, which is practically identical to the value above. In fact, the velocity surface in the vicinity of the apex of the ellipsoid is so flat that in a range of t4u around the given maximum direction the velocity changes by less than 0.01 km sx1, a value that corresponds to the estimated standard deviation for Vmax in Table 1. Using refraction data for the estimation of uppermost mantle Pn anisotropy in southern Germany, Bamford (1977) gave an overall velocity variation of about 0.5–0.6 km sx1 (6–7 per cent anisotropy) with a maximum velocity direction of N20uE. Enderle et al. (1996) obtained an overall velocity variation of 0.26 km sx1 (about 3–4 per cent anisotropy), with the fast direction N31uE. Our estimated level of Pn anisotropy is close to that of Enderle et al. (1996) but smaller than that of Bamford (1977). In particular, with regard to the direction of maximum velocity, which is generally considered important for the interpretation of anisotropy in terms of tectonic processes (Hess 1964; Fuchs 1977), the present results are consistent with those of previous Pn studies (Bamford 1973, 1977; Enderle et al. 1996). The mantle anisotropy beneath Germany has also been studied by the analysis of teleseismic shear wave (SKS) splitting (Vinnik et al. 1994; Brechner et al. 1998; Plenefisch et al. 2001). Of course, vertically propagating SKS waves reflect the integrated effects of anisotropy within a larger depth range of the upper mantle, while horizontally propagating Pn waves characterize the uppermost mantle anisotropy. However, the fast propagation direction for Pn waves should coincide with the fast polarization direction for split shear waves in a homogeneous hexagonal medium, which is usually assumed in the study of upper mantle anisotropy. Therefore, some useful constraints on the upper mantle anisotropy could be obtained by comparing Pn anisotropy to SKS splitting results (Silver & Chan 1991; Hearn 1996). In particular for the fast anisotropy directions within this region, the earlier result of SKS study with one anisotropic layer in southern Germany (Vinnik et al. 1989, 1992) differs from the previous Pn anisotropy directions (Bamford 1977; Fuchs 1983) and also from ours. In later SKS studies, the variation of the SKS splitting parameters with respect to the azimuth of the incoming waves demanded the use of a two-layer anisotropic model (Vinnik et al. 1994; Brechner et al. 1998; Plenefisch et al. 2001). The upper layer anisotropy at the GRF array and GRSN stations in these studies has the fast direction close to the Pn anisotropy. The existing discrepancy in some areas of

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Germany may simply be explained by the fact that the Pn waves are sampling a shallow anisotropic structure, while the SKS results are more sensitive to a lithospheric or upper mantle wide structure resulting from different episodes of deformation. However, the measurements of the SKS waves at many stations can offer good lateral resolution for the upper mantle anisotropy (Brechner et al. 1998; Plenefisch et al. 2001). Hence, it is possibly useful to compare these SKS results with a 2-D anisotropic Pn inversion, a work which is in progress (Koch et al. 2001). In brief, there is some additional convincing evidence from the present study that Pn-wave anisotropy might be a largescale lithospheric feature over much of central Europe, since the anisotropic parameters, derived from different data and methods, for the uppermost mantle in this region vary overall in an acceptable range and are compatible with each other. Although Enderle et al. (1996) and Vinnik et al. (1994) prefer to interpret their Pn or SKS data by the introduction of a velocity– depth gradient or a layered anisotropic model, our data set may not require such an approach. In addition to a further geodynamic and tectonic interpretation of the results obtained so far, efforts are presently underway to image lateral variations of Pn velocity and anisotropy with this regional data set. To remove bias effects due to mislocations of events and lateral velocity variations of the crust, a full 3-D simultaneous tomographic inversion of the crust and mantle and source locations (SSH) will be attempted for the study region (over most of Germany), including possible anisotropic upper mantle effects. Eventually this might allow us to answer the classical question of whether traveltime residuals could be explained by lateral velocity variations and/or by anisotropy. ACKNOWLEDGMENTS This study is being supported by the DFG through a grant to the authors. We are grateful for Prof. K. Fuchs’ encouraging and very constructive comments and for one anonymous reviewer’s useful comments. We also thank Dr G. Bock for his helpful suggestions. REFERENCES Backus, G.E., 1965. Possible forms of seismic anisotropy of the uppermost mantle under oceans, J. geophys. Res., 70, 3429–3439. Bamford, D., 1973. Refraction data in western Germany—A time term interpretation, Z. Geophys., 39, 907–927. Bamford, D., 1977. Pn velocity anisotropy in a continental upper mantle, Geophys. J. R. astr. Soc., 49, 29–48. Brechner, S., Klinge, K., Kru¨ger, F. & Plenefisch, T., 1998. Backazimuthal variations of splitting parameters of teleseismic SKS phases observed at the broadband stations in Germany, Pure appl. Geophys., 151, 305–331. Enderle, U., Mechie, J. & Sobolev, Fuchs, S., K., 1996. Seismic anisotropy within the uppermost mantle of southern Germany, Geophys. J. Int., 125, 747–767. Fuchs, K., 1977. Seismic anisotropy of the subcrustal lithosphere as evidence for dynamical processes in the upper mantle, Geophys. J. R. astr. Soc., 49, 167–179. Fuchs, K., 1983. Recently formed elastic anisotropy and petrological models for the continental subcrustal lithosphere in southern Germany, Phys. Earth planet. Inter., 31, 93–118.

Hearn, T., 1984. Pn travel times in southern California, J. geophys. Res., 89, 1843–1855. Hearn, T., 1996. Anisotropic Pn tomography in the western United States, J. geophys. Res., 101, 8403–8414. Hearn, T. & Ni, J., 1994. Pn velocities beneath continental collision zones: The Turkish-Iranian Plateau, Geophys. J. Int., 117, 273–283. Henger, M. & Leydecker, G., eds, 1975. (and following years), Data Catalogue of Earthquakes in the Federal Republic of Germany and Adjacent Areas, Bundesanstalt fu¨r Geowissenschaften und Rohstoffe, Hannover, Hannover. Hess, H.H., 1964. Seismic anisotropy of the uppermost mantle under oceans, Nature, 203, 629–631. Hirn, A., Steinmetz, L., Kind, R. & Fuchs, K., 1973. Long range profiles in western Europe: II. Fine structure of the lower lithosphere in France (southern Bretagne), Z. Geophys., 39, 363–384. Koch, M., 1992. The optimal regularization of the linear seismic inverse problem, in Geophysical Inversion, pp. 170–234, eds Bednar, J.B., Lines, L., Stolt, R.H., Weglein, A.B., SIAM, Philadelphia. Koch, M., 1993. Simultaneous inversion for 3-D crustal structure and hypocentres including direct, refracted and reflected phases—III. Application to the southern Rhine Graben, Germany, Geophys. J. Int., 112, 429–447. Koch, M., Song, L.-P., Koch, K. & Schlittenhardt, J., 2001. 1D and 2D isotropic and anisotropic Pn velocity inversion of regional earthquake traveltimes underneath Germany, in EGS XXVI General Assembly, Nice, France, ed. Ritchie, A.K., Geophys. Res. Abs., 3. Lawson, C.L. & Hanson, R.J., 1974. Solving Least Squares Problems, Prentice Hall, Englewood Cliffs. Michelena, R.J., Muir, F. & Harris, J.M., 1993. Anisotropic travel time tomography, Geophys. Prospect., 41, 381–412. Morris, G.B., Raitt, R.W. & Shor, G.G., 1969. Velocity anisotropy and delay-time maps of the mantle near Hawaii, J. geophys. Res., 74, 3095–3109. Plenefisch, T., Klinge, K. & Kind, R., 2001. Upper mantle anisotropy at the transition zone of the Saxothuringicum and Moldanubicum in southeast Germany revealed by shear wave splitting, Geophys. J. Int., 144, 309–319. Raitt, R.W., Shor, G.G., Francis, T.J.G. & Morris, G.B., 1969. Anisotropy of the Pacific upper mantle, J. geophys., Res., 74, 3095–3109. Scheidegger, A.E. & Willmore, P.L., 1957. The use of a least squares method for the interpretation of data from seismic surveys, Geophysics, 22, 9–22. Schlittenhardt, J., 1999. Regional velocity models for Germany: a contribution to the systematic travel-time calibration of the international monitoring system, in 21st Seismic Res. Symp, pp. 263–273, Dept. of Defense/Dept of Energy, Las Vegas. Silver, P.G. & Chan, W.W., 1991. Shear wave splitting and subcontinental mantle deformation, J. geophys. Res., 96, 16 429–16 454. Song, L.-P. & Every, A.G., 2000. Approximate formulae for acoustic wave group slownesses in weakly orthorhombic media, J. Phys. D: Appl. Phys., 33, L81–L85, 2519, Erratum page. Song, L.-P., Liu, H., Sha, C., Song, Z. & Zhang, S.-Y., 1998. Mapping an underground rock mass by anisotropic acoustical transmission tomography, Ultrasonics, 36, 1009–1012. Telford, W.M., Geldart, L.P., Sheriff, R.E. & Keys, A.D., 1976. Applied Geophysics, Cambridge University Press, New York. Vinnik, L.P., Kind, R., Kosarev, G.L. & Makeyeva, L.I., 1989. Azimuthal anisotropy in the lithosphere from observations of longperiod S-waves, Geophys. J. Int., 99, 549–559. Vinnik, L.P., Makeyeva, L.I., Milev, A. & Usenko, A.Yu., 1992. Global patterns of azimuthal anisotropy and defromations in the continental mantle, Geophys. J. Int., 111, 433–447. Vinnik, L.P., Krishna, V.G., Kind, R., Bormann, P. & Stammler, K., 1994. Shear wave splitting in the records of the German Regional Seismic Network, Geophys. Res. Lett., 21, 457–460.

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