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Acoustic Emission Events Interpreted in Terms of Source Directivity

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During inversion for the source mechanisms of laboratory acoustic emission events, relatively high misfit values (expressed as the RMS of the inverted equations) were observed. Our experiment was performed on Westerly Granite. A processed set of data consisting of 2405 acoustic emission events was used and a semi-homogeneous velocity model was considered. A correction for sensor radiation patterns and individual sensor constants was taken into account, and an acausal attenuation model was assumed. Source mechanisms were inverted for the moment tensor. The application of a more sophisticated medium model improved inversion quality only for some events. Introducing the source directivity, a standard approach for earthquakes with magnitudes larger than approximately 4, increased the number of successfully inverted events. Directivity was introduced using a Haskell source model; optionally unilateral and bilateral versions of the source were considered. Lower values of RMS for the Haskell source model were considered to justify the directivity approach. This formalism enables us to select the preferable fault from the two nodal planes within the fault plane solution. The rupture directions were observed to tend to the dip direction of the preferred fault. They were found to be preferably subparallel to slip directions for acoustic emissions with a significant DC component. The source time functions retrieved from the seismograms are in agreement with finite source theory.

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Fig. 1

source time function (STF) shape variation with dependence of angle θ between the rupture direction and the direction of the observation (right). A synthetic example of modification of the STF; STF amplitude decreases with increasing  angle θ. As the STF area is invariant, the STF duration changes in the opposite sense. These modifications are consequently transformed into amplitudes of signals radiated from the source. The figure is inspired by Fig. 4.3–4 of Stein and Wysession (2003)

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source with an opposite rupture direction, and the red line indicates a bilateral source—a summation of two opposite unilateral sources. All of the values are arbitrary normalized for an angle θ = 90°, i.e. for a direction perpendicular to the rupture direction

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  1. If detailed finite circular source modeling is required, the stopping-phase theory could be applied. According to this theory (Bernard and Madariaga 1984), the finite circular seismic source effectively only generates signals from the beginning rupture point at the source centrum and from two rupture stopping points located at the source edge. Signals radiated from other source points (Boatwright 1980) are effectively mutually destroyed by interference. Additionally, these two stopping phases are in a mutual convolutional relationship. This feature can be used for interpretation in seismograms. This theory was used in works considered (Bernard and Madariaga 1984; Imanishi et al. 2004; Imanishi and Takeo 1998, 2002; Kolář 2015; Kolář and Růžek 2015).

  2. \(RMS=\sqrt{\frac{1}{n}{\sum }_{i=1}^{n}{\left({x}_{i}^{modeled}- {x}_{i}^{observed}\right)}^{2}}.\)

  3. Minimum source distance and maximum incidence angle criteria do not apply for signal onset and all stations or readings, respectively. Those signals can be used, for example, for event location.

  4. BIC differences greater than 2 are recommended in the quoted BIC reference as a minimum level for a positive statistical significance of the performed test.


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This study was partially supported by the Czech Science Foundation, research Grants 16-03950S and 18-08826S; by the Czech Academy of Sciences, project RVO 67985831; by the Institute of Geophysics; and by the Institute of Geology (authors’ home institutions). Loading experiments were performed in the Laboratory of Mechanical Properties of Rocks of the Institute of Geology of the Czech Academy of Science, located in Prague, Czech Republic. We are thankful to G. Kwiatek for providing comments that strengthened the presentation of results.

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Correspondence to Petr Kolář.

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Kolář, P., Petružálek, M., Lokajíček, T. et al. Acoustic Emission Events Interpreted in Terms of Source Directivity. Pure Appl. Geophys. 177, 4271–4288 (2020).

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