SHORT CUT: Seismic station misalignment table

This web site summarizes our motivation and technique to measure surface wave arrival angles. A more detailed description can be found in Laske et al. (1994) and Laske (1994).

A similar relationship (the linear path integral approximation of Woodhouse and Wong, 1986) can be obtained for the tangent of the arrival angle and the lateral gradient of structure along the source-receiver great circle. Assumptions are that structure is smoothly varying along the true ray path which is assumed to be weakly perturbed from the great circle. The sensitivity to the lateral gradient rather than structure itself implies enhanced sensitivity of such data to shorter wavelength structure.

In a heterogeneous Earth, this is no longer true and Love wave signals can occur on the R component (or even on the Z component, e.g. in the presences of dipping interfaces) and Rayleigh waves on the T component (--> ***).

In order to measure arrival angles more precisely and to obtain error bars on the measurements, we apply the multitaper spectral analysis of Park et al. (1987). We seek the principle complex polarization vector of our 3-component time series. To achieve this, we taper the 3-component time series with a set of K tapers, calculate the Fourier Transforms and compose a 3 x K matrix. The SVD (singular value decomposition) of this matrix yields the principle polarization vectors. We get 3 complex orthogonal eigenvectors each of which represents motion of a signal in a plane in 3-D space. The most general signal that can be described by these three vectors is ellipsoidal motion. In the ideal case of only one well polarized signal (that occurs in a plane), only one singular value is non-zero. For our analysis, we use the prolate spheroidal wave function eigentapers (Slepian, 1978) to extract our wave packets from the seismograms. These tapers are the solution of an eigenvalue problem that concentrates the maximum amount of energy within a chosen frequency band. The tapers are orthogonal so spectral estimates obtained with these tapers are independent. Using these estimates we then obtain statistical error bars for our measurements. There is a trade-off between resistance of these tapers to bias caused by the presence of incoherent noise and spectral leakage for the chosen frequency band. The number of tapers and the retention band width P of the P-pi tapers have to be chosen accordingly (-->**). An example shows the three singular values, ellipticity and the arrival angles for a typical Rayleigh and Love wave measurement.

The codes work on a Sun workstation and little changes should be necessary to make them work on other platforms.

Sorry, we do not provide codes for other data formats, and it is up to the user to make the necessary changes.

Please contact us if you have questions (glaske@ucsd.edu).

Laske, G., 1995. Global observation of off-great circle propagtaion of long-period surface waves. Geophys. J. Int., 123, 245-259.

Laske, G., Masters, G. and Zürn, W., 1994. Frequency-dependent polarization measurements of long-period surface waves and their implications for global phase velocity maps. Phys. Earth Planet. Int., 84, 111-137.

Laske, G. and Masters, G., 1996. Constraints on global phase velocity maps from long-period polarization data. J. Geophys. Res., 101, 16,059-16,075.

Park, J., Vernon III, F. and Lindberg, C.R., 1987. Frequency dependent polarization analysis of high-frequency seismograms. J. Geophys. Res., 92, 12,664-12,674.

Park, J., Lindberg, C.V. and Vernon III, F., 1987. Multitaper spectral analysis of high-frequency seismograms. J. Geophys. Res., 92, 12,675-12,684.

Slepian, D., 1978. Prolate spheroidal wave functions, Fourier analysis, and uncertainty. V: The discrete case. Bell Systems Tech., J., 57, 1371-1430.

Woodhouse, J.H. and Wong, Y.K., 1989. Amplitude, phase and path anomalies of mantle waves. Geophys. J. R. astr. Soc., 87, 753-773.

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Gabi Laske ( glaske@ucsd.edu)

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