Measuring the Polarization

of Long-Period Surface Waves

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).

Why measure surface wave arrival angles?

When seismic surface waves travel through heterogeneous structure, the wave packets get refracted laterally, away from the source-receiver great circle. The angle between the arriving packet and the receiver-source azimuth is the arrival angle. For the surface wave phase, Fermat's principle applies and there exists a linear relationship between the phase perturbation and the structure along the source-receiver great circle.
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.

How can we see that surface waves get refracted?

On the spherically symmetric earth, Love and Rayleigh waves are de-coupled and appear on separate components in a properly rotated 3-component seismogram (Z,N,E are rotated into Z,R,T). Love waves are then only visible on the T (transverse) component, while Rayleigh waves appear on the Z and R (radial) components.
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 (--> ***).

How do we measure surface wave arrival angles?

The easiest way to measure arrival angles is to look at particle motion plots in the horizontal plane (hodograms). In isotropic media with no dipping interfaces, Love waves are linearly polarized.The particle motion of Rayleigh waves is elliptical in the Z-R plane but is also linear in the R-T plane. The arrival angle can then be measured quite easily. In reality, particle motion in a 3-component seismogram is more complex (partly due to wave propagation in complex structure but also due to the presence of noise -- e.g. from overtones -- in the seismogram).

The Multitaper Technique

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.

What does a typical dataset look like?

For quality control purposes it is helpful to plot the measurements for each station prior to an inversion for Earth structure. We plot the data in rose diagrams. For a typical station and a typical dataset (i.e. reasonably good data coverage), the arrival angles cluster around the great circle direction (i.e. around zero in the rose diagrams). Only in special cases does the clustering occur around non-zero values (e.g. all the ray paths pass a certain region or originate in the same source region). An example shows Love wave measurements at six GSN stations.

Some datasets are quite difficult to interpret. An example is the dataset for station ELK (Elko, Nevada) of the U.S. National Seismic Network (USNSN). The data for this station appear to cluster around at least two different means. It turns out that the clustering is time-dependent (i.e. data cluster around different means at different times). For example, data prior to Sep 11, 1997 clustered around a large positive angle (about 30 degrees) while data after Oct 27, 1997 have been clustering around a small negative angle. There is a gap in the data between these two dates. It is therefore likely that a network operator visited this station, (repared) and reoriented to seismic sensor. Since there is no information available from the network operators, this is just an assumption, but a quite reasonable one.

What do we do with these data?

Ultimately, we want to use these data to constrain the small-scale structure in global phase velocity maps. Before we can do this, we need to correct our data for misalignments of the seismic sensor (because the signal from Earth structure is typically only a few degrees). The contribution of the misalignment to our data is non-linear but can easily be linearized and the inversion converges quickly (typically after 2-3 iterations).

To determine the misalignment of the sensors accurately, we typically perform independent inversion for 3 periods (6, 8 and 10 mHz), for both Rayleigh and Love waves. The average values for each station, over all inversions, is then the final misorientation value. Negative values (of apparent North) imply a clockwise rotation of the sensor package. We give a complete list of stations analyzed so far. Note that we determined these values before network operator began to report station misalignments (i.e. other workers include this information) so our values are "true apparent North".

NB: A value of "Apparent North" could include systematic errors from sources other than a simple rotation of the whole sensor package. These include calibration errors (e.g. when one component has a different gain factor). Our technique also presupposes that the horizontal components are orthogonal. This is not always the case. For example, the North component at station AAK was rotated clockwise by 6 degrees while the East component was correct. It is case, our value of -6.8deg is probably coincidence.

Some of the value we published were later confirmed by network operators. For example, in the case of PPT, the -5deg misalignment was confirmed. For TERRAscope station ISA, a value of 20deg is now reported (so our 16deg seems to be an underestimate).

Download Section

Click here to get a README and download the files with Fortran 77 computer code. We provide the essential subroutines and computer code to work with data using the gas file format, as well as our in-house screen I/O leolib codes. Please check the README file for instructions.
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 (


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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