Retrieval of the Seismic Moment Tensor from Joint Measurements of Translational and Rotational Ground Motions: Sparse Networks and Single Stations

Abstract

Seismic moment tensors help us to increase our understanding about e.g. earthquake processes, tectonics, Earth or planetary structure. Based on ground motion measurements of seismic networks their determination is in general standard for all distance ranges, provided the velocity model of the target region is known well enough. For sparse networks in inaccessible terrain and planetary seismology, the waveform inversion for the moment tensor often fails. Rotational ground motions are on the verge of becoming routinely observable with the potential of providing additional constraints for seismic inverse problems. In this study, we test their benefit for the waveform inversion for seismic moment tensors under the condition of sparse networks. We compare the results of (1) inverting only traditional translational data with (2) inverting translational plus rotational data for the cases of only one, two, and three stations. Even for the single station case the inversion results can be improved when including rotational ground motions. However, from data of a single station only, the probability of determining the correct full seismic moment tensor is still low. When using data of two or three stations, the information gain due to rotational ground motions almost doubles. The probability of deriving the correct full moment tensor here is very high.

Appendix: Analytical Solution for the Radiation Pattern of a Double-Couple Source

To further analyse the event station geometry with respect to the radiation pattern of a seismic source, in the following, we show the analytical solution of a double-couple source, i.e. the tectonic part of the seismic moment tensor.

The far-field displacement $u(\vec {x},t)$ due to a double-couple point-source in an infinite, homogeneous, isotropic medium in a spherical coordinate system is (Aki and Richards 2002):

$$\begin{aligned} {\mathbf {u}}({\mathbf {x}},t) = \frac{1}{4\pi \rho \alpha ^3}{\mathbf {A}}^{FP}\frac{1}{r}\dot{M}_0\left( t - \frac{r}{\alpha } \right) + \frac{1}{4\pi \rho \beta ^3}{\mathbf {A}}^{FS}\frac{1}{r}\dot{M}_0\left( t - \frac{r}{\beta } \right) \end{aligned}$$

(6)

where $\alpha$, $\beta$, and $\rho$ are the velocities for P- and S-wave and the density of the medium, respectively. r is the source-receiver distance and $M_0(t)$ the seismic moment function. $A^{FP}$ and $A^{FS}$ are the far-field radiation patters for the P- and S-wave in spherical coordinates:

$$\begin{aligned} \varvec{A}^{FP}&= \sin {2\theta }\cos {\phi }{\varvec{\hat{r}}} \nonumber \\ \varvec{A}^{FS}&= \cos {2\theta }\cos {\phi }\varvec{\hat{\theta }} - \cos {\theta }\sin {\phi }\varvec{\hat{\phi }} \end{aligned}$$

(7)

Similarly, the far-field rotation can be described as

$$\begin{aligned} \varvec{\omega }({\mathbf {x,}}t) = \frac{-1}{8\pi \rho \beta ^4} \varvec{A}^{R}\frac{1}{r} \ddot{M_0}\left( t - \frac{r}{\beta } \right) \end{aligned}$$

(8)

with

$$\begin{aligned} \varvec{A}^R = \cos {\theta }\sin {\phi }\varvec{\hat{\theta }} + \cos {2\theta }\cos {\phi }\varvec{\hat{\phi }} \end{aligned}$$

(9)

as the rotational radiation pattern (Cochard et al. 2006). Figure 9 shows the (a) translational and (b) rotational radiation patterns for a source in the XY-plane. Blue, red, and pink are the radiation patterns for the P-wave, S-wave, and rotation, respectively. The black, red, and blue circles mark positions of equally spaced stations in the XY-, XZ-, and YZ-plane around the source, respectively. It is easy to see that the biggest amplitude variations with azimuth are to be expect for stations along the black ring in the XY-plane, i.e. in plane with the source.

Fig. 9

Radiation patterns for slip in the XY-plane with slip in X direction in a homogeneous half-space. a, b are the radiation patterns for translational and rotational ground motions, respectively. In a, the blue and red radiation patterns are for P- and S-waves, respectively. Black, red, and blue circles mark positions of equally spaced stations in the XY-, XZ-, and YZ-plane, respectively. The colours and numbers correspond to the waveform colours and trace numbers in Figs. 10 and 11

The Figs. 10 and 11 show the theoretical waveforms at the stations around the source shown in Fig. 9 in cartesian (ZXY) and spherical (r $\theta$ $\phi$) coordinate systems, respectively. The colour-coding and numbering of the waveforms correspond to the station rings. As expected, the black waveforms for stations in-plane with the source show a clear signal on the majority of the traces compared to the waveforms of the other ring stations. The stations 17, 21, 25, and 29 are located on the nodal planes of the mechanism. In Fig. 11 there is only energy on their $\theta$ component, corresponding to the SH-wave energy of the source. The stations 19, 23, 27, and 31, rotated by 45° to the nodal planes, show the opposite behaviour with energy on the radial (P-wave energy) and the $\phi$ component (SV-wave energy) but no energy on the $\theta$ component.

Fig. 10

Waveforms for the double-couple source of Fig. 9 in the ZXY coordinate system. Vertical numbers and waveform colours correspond to the three station rings around the source in Fig. 9. First three columns show translational displacement, while last three columns show rotational ground motions. Black dotted lines mark the time of the P-and S-wave arrivals. Waveforms are normalized preserving the ratios between the components. Rotational waveforms are additionally amplified by a factor $1.5\times 10^4$

Similar patterns can be found for the rotational waveforms. In the spherical coordinate system, the stations 21 and 29 have no rotation energy at all on all three components (nodal planes along rupture direction), while they show a maximum of rotation energy for the $\phi$ component at the stations 17 and 25 (nodal planes perpendicular to rupture direction). Slightly different, but still azimuth dependant energy patters are visible for the other two station rings.

Fig. 11

Same as Fig. 10, the waveforms are now shown in the spherical coordinate system (r, $\theta$, and $\phi$) as used by e.g. Aki and Richards (2002)

These amplitude patterns in the form of ratios between the different components are the information needed to resolve the mechanism of an earthquake during inversion. When including rotational ground motion data to the inversion, 15 instead of only 3 amplitude ratios can be determined. Therefore, the position of the station within the radiation pattern can be much better constrained. As a consequence, the mechanism can be determined much more reliably, at least in cases of tectonic events where the double-couple part of the moment tensor should dominate.