*2.1. Comprehensive Consideration of the Scale*

An interesting analogy can be made to AE events by comparing it to earthquakes, which can be regarded as largest natural occurring emission sources. The principal mechanism does not depend on the scale: The rapid release of elastic energy by processes of crack growth or deformation within the rock generates transient pulses of elastic wave energy as AE events. In an even larger scale, the same is true for earthquakes. Similar to AE sources, seismic waves propagate through the earth, and are detected with a global network of seismometers located around the world. Therefore, most of the theory of earthquakes can be transferred to AE sources [11,12]. Figure 1 shows the dependences between measured corner frequency *fC*, the moment magnitude *MW*, the seismic moment *M*0, and the source radius *r*<sup>0</sup> for studies in earthquake seismology, microseismicity, in situ AE monitoring, and AE in laboratory (see more details in Box 1) [13]. In addition, in situ AE monitoring is reported from rock face monitoring e.g., on construction sites [14,15].

The largest and thereby the longest events, namely earthquakes, are found at the lowest end of the frequency scale (Figure 1). The focal length and displacement of an earthquake can amount to more than several hundred kilometers and up to several meters, respectively. Whereas global seismology exploits frequencies less than 1 Hz, local seismology focuses on the analysis of frequencies of hundreds of Hertz or even 1000 Hz. Recordings of microseismicity in dense local (e.g., borehole) networks or underground are limited to frequencies up to a few kilohertz. In these cases pendulum-based geophones or seismometers are utilized. The corner frequency in the field of microseismic measurements ranges between 5 Hz and some hundred Hertz with magnitudes from approximately 4.0 down to −2.0. Above these frequencies, the range of in situ AE monitoring in rock begins with applications in mines. In the frequency rang of about 1 kHz up to 200 kHz accelerometers or special piezoelectric AE sensors were developed for in situ application in mines. Source-receiver distances realized depend strongly on rock type, but can reach up to 150 m in rocks with very low wave attenuation. The covered areas may have linear dimensions of up to 200 m. The expected magnitudes are in the range between −2.0 and −5.0.

**Figure 1.** Relations between measured corner frequency *fC*, moment magnitude *MW*, seismic moment *M*0, and the source radius *r*<sup>0</sup> for studies in earthquake seismology, microseismicity, In situ AE monitoring and AE in laboratory (see more details in Box 1) (modified from [13]).

On the other hand, the highest frequencies up to several MHz are utilized in laboratory studies. These frequencies are generated by AE events in the microscopic region, for instance by microcracking of rock or dislocation movement. Only sensitive piezoelectric sensors for laboratory applications are able to measure such high frequencies. In the laboratory, the source radius may extend to some micrometers only and the displacement (Burgers vector) is to be measured in nanometers. The expected magnitude of such small AE events is below −6.0 [11]. A detailed study on the theoretical limits on detection and analysis of small earthquakes in dependency on the sensitivity and frequency coverage of the monitoring network was published by Kwiatek and Ben Zion [16].

Figure 2 taken from Bohnhoff et al. [17] shows the co-seismic stress drop plotted with key earthquake source parameters over the entire bandwidth of observed rupture processes, extending from large natural earthquakes to AEs in the laboratory. The source parameters were calculated from individual data sets of natural earthquakes, induced seismicity in mines and reservoirs, and volcano seismicity [18–22]. AE data are unpublished results from the rock-deformation laboratory in the German research Centre for Geosciences, Potsdam, (GFZ)-Section 4.2. The Madariaga [23] circular source model is assumed to calculate source radii and the lines of constant static stress drop from 0.01 MPa up to 100 MPa and *vs* = 3500 m/s (see Box 1).

**Figure 2.** Dependence between seismic moment, moment magnitude, source radius, average fault slip, and corner frequency for natural earthquakes (gray rectangles) in comparison with data from other studies in the low-magnitude range. The Madariaga [18] circular source model is assumed to calculate source radii and the lines of constant static stress drop from 0.01 MPa up to 100 MPa and *vs* = 3500 m/s (see Box 1). This figure is an updated figure from Kwiatek et al. [16] and was first published in [17]. Reproduced with permission.

**Box 1.** Estimation of source radius, static stress drop, and moment magnitude.

Seismic moments *M*<sup>0</sup> and corner frequencies *fc* in Figure 2 were estimated from the spectral level of ground velocity or displacement spectra corrected for instrument response and wave propagation effects, as described in detail in the references. It follows that the dependency in Figure 2 between seismic moment and corner frequency are based on data-driven observables. The seismic moment is a measure of how much "work" an earthquake does in sliding when rock slips off other rock. It is necessary to take into account that the physical unit of *<sup>M</sup>*<sup>0</sup> is given in N·m (corresponds to 107 dyne·cm).

The estimation of source radii, the lines of constant static stress drop, and the average fault slip are based on the model of Madariaga [23] and Brune [24]. Both modeled a circular fault and use the corner frequency *fc* to calculate the source radius r0 and the static stress drop Δ*σ*. The stress drop and source radius is defined as <sup>Δ</sup>*<sup>σ</sup>* <sup>=</sup> 7/16·*M*0/*r*<sup>3</sup> <sup>0</sup> and *r*<sup>0</sup> = *c*·*vs*/2*π fc*, respectively (*c* = 2.34 for Brune's model, *c* = 1.32 for Madariaga's modell). The moment magnitude *MW* is calculated using the standard relation for tectonic events developed by Hanks et al. [25] *MW* = (*log*10*M*<sup>0</sup> − 9.1)/1.5. The formula is empirically established for tectonic earthquakes only. Extrapolation over so many orders of magnitude seems questionable, but is useful for visualization.

The question to which extent AE events represent double-couple shear events or events with a dominant isotropic component remains open. Owing to the AE sensor calibration problem (see Box 2), few studies exist that investigate reliably the source mode of AE events e.g., calculating the full moment tensor. Results from hydraulic fracturing studies suggest that AE events may represent sources with a major double-couple component [26,27], whereas studies related to stress-induced AE events find indications for isotropic components or mixed mode events [28–31].
