Earthquake rupture tracking with six 1 degree-of-freedom ground motion observations : a 2 synthetic proof of concept 3

With the availability of new instrumentation for more complete ground motion measurements, as rotation or strain measurements using optical technology, novel application opportunities in seismology arise. Back azimuth information can be determined from combined measurements of rotations and translations at a single site. Such six degree-of-freedom (6-DoF) measurements are reasonably stable in delivering similar information compared to a small-scale array of three-component seismometers. Here we investigate whether a 6-DoF approach is applicable for imaging earthquake rupture propagation. While common approaches determining the timing and location of energy sources generating seismic waves rely on the information of P-waves, here we take S-waves into account. We analyze 2-D and 3-D synthetic cases of unilateral but complex rupture propagation. The back azimuths of directly arriving SH-waves in the 2-D case, and P-converted SV-waves and SH-waves in the 3-D case are tracked. For data analysis in terms of wave polarity we compare a cross-correlation approach using a grid-search optimization algorithm with a polarization analysis method using point measurements. We successfully recover rupture path and rupture velocity with only one station, under the assumption of an approximately known fault location. Using more than one station, rupture imaging in space and time is possible without a priori assumptions. We demonstrate robustness of the approach in resolving relatively small variations of rupture velocity, and rupture jumping across off set fault segments. We discuss the effects of rupture directivity, supershear rupture velocity, source-receiver geometry as well as potential and challenges for the method.


Introduction
The difficulty of retrieving BAz (source directivity) from 3-D observations with solely 150 translational motions is due to two challenges: on vertical components. We can therefore take advantage of the fact that two horizontal 164 rotational components contain exclusively SV-or Rayleigh-waves. 165 Without the interference of other types of waves, the ratio between the two horizontal 166 rotational components is directly related to the BAz according to: 168 where ω n and ω e denote the north-south and east-west components of rotation (or rota-169 tional rate in this study). This simple relationship is specially useful for estimating source 170 directivity and it is independent of any possible radiation pattern that the source might 171 have (Langston & Liang, 2008).  Each station records the horizontal accelerations a x , a y and the vertical rotation rate 215ω z of the directly incoming P-and S-waves (middle panels of Fig. 3 by moving a sliding window of 1.5 s length through the SH-wave signal of the seismograms.

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In each window the BAz is estimated by the polarization analysis and the CC method   can propagate at sub-Rayleigh or at intersonic speeds (e.g., Archuleta, 1984;Gabriel et al., 250 2012) and a speed of 150% v s means that the rupture is propagating faster than the radiated 251 SH-waves. This effect is referred to as super-shear rupture speeds.

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In Fig. 4 we estimate the BAz changes for each case of velocity variation at station A in 253 the same way as in Fig. 3. Each column represents the result for a specific velocity jump. The

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BAz results are divided into three subwindows, in which the rupture velocity is calculated, We expect that 6-DoF rupture tracking is more difficult in heterogeneous materials, 265 since reflected and scattered energy will contaminate the directly arriving SH-waves. We

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Previous studies neglect such directivity effects expecting only small deviations.

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First, we verify that a 6-DoF method provides accurate results for simple and more 303 complex fault geometry then we discuss the importance of directivity effects. We apply a 304 time correction to the BAz estimates by assuming that the start and the end of the directly 305 arriving SH-waves is visible in the signal. This is done in the rotational component of ground 306 motions due to its high sensitivity to shear motions.

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In the following, we track a simple unilateral rupture at five stations. The stations are 308 placed in an asymmetrical pattern around the rupture with different distances to the source.

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The stations are situated in such a way that the resolution is about the same for both spatial  Fig. 6c. Animation S1 visualises the continuous 315 rupture imaging (Movie S1: ms01.mov).

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In the first subplot of Fig. 6 We repeat the same experiment as presented in Fig. 7 for a more complex rupture ge-  The following description is a purely geometrical concept to demonstrate the expected 381 scaling of opening angles. If we define the rupture path as a straight line on a sphere, we 382 can describe the geometry between the receiver and rupture by a large triangle. In Fig. 9 383 we illustrate the opening angle α for fault lengths between 100 and 1000 km (blue lines) at   Middle: recorded seismograms of horizontal accelerations ax , ay and vertical rotation rateωz.
Bottom: the BAz is estimated by two different methods in the direct SH-arrivals. The estimations are divided into three sub-windows in which the rupture speed is determined. The true rupture velocity is 80% vs.
-24-  Figure 4. Tracking variations in rupture velocity at station A. The first half of the rupture breaks with 80% of the shear-wave velocity vs, the second half breaks with 40%, 60% or 150% vs. The BAz variation is estimated in the SH-arrivals at station A (see Fig. 3