Point‐Source Decay of Ground‐Motion Amplitudes at <10 km

We use a unique database of recorded ground motions from 13 small events (moment magnitude, M < 3) that occurred around a hard‐rock underground mine in Australia to examine the decay of ground motion amplitudes for hard‐rock sites (shear‐wave velocity $VS>3000m/s$⁠) at short hypocentral distances (0.15–30 km). The decay rate of ground‐motion amplitudes is an essential and uncertain element in ground‐motion models (GMMs) used in seismic hazard analyses (e.g., Atkinson and Boore, 2006; Yenier and Atkinson, 2015a,b) and is critical for the determination of source parameters (Boore et al., 2010). The near‐distance attenuation is particularly consequential for the evaluation of ground motions associated with induced seismicity, because shallow events may be experienced on the surface at close distances (Atkinson, 2015; Atkinson et al., 2016). Amplitude decay rates are relatively well determined empirically for regional distances (>10 km) but have large uncertainties at close distances due to the paucity of observations.

The amplitude decay rate at distances near the rupture is largely controlled by the interaction between apparent geometric spreading and near‐distance saturation effects (nonlinearity is also a contributing factor for soil sites but is not a significant factor for rock). Geometric spreading for a point source in the Fourier domain is independent of magnitude. The theoretical decay rate is 1/R (in which R is distance) for body wave spreading from a point source in a whole space. However, the rate is steeper for real earth models due to the effects of velocity layering, and may become complex at distances >∼50 km due to multiple reflections and refractions (e.g., Ojo and Mereu 1986; Burger et al., 1987; Chapman, 2012). Observations in eastern and western North America support a geometric spreading rate in the Fourier domain (at <∼50 km) of $R−1.3$ (Atkinson, 2004; Atkinson and Boore, 2006; Atkinson and Morrison, 2009). In the response spectra domain, in which the ground‐motion parameter is the maximum response of a damped single‐degree‐of‐freedom oscillator, the decay rate due to geometric spreading effects depends on magnitude (steeper at smaller magnitudes) and frequency (steeper at higher frequencies); this dependency has been modeled within an equivalent point‐source stochastic modeling framework by Yenier and Atkinson (2015a,b) and Hassani and Atkinson (2018). For small‐to‐moderate events, the apparent geometric spreading rate in the response spectral domain is quite steep. For example, the observed rate of decay of 1–2 Hz response spectral amplitudes in California (Next Generation Attenuation‐West2 [NGA‐W2] database) for events of M∼3–5 at <30 km is $∼R−1.5$ (Atkinson, 2015).

The point‐source geometric spreading rate at near‐source distances is difficult to determine observationally. For M < 5 events the closest distance to the rupture for observations on the surface is typically >5 km, except for rare shallow events. For larger earthquakes, for which ruptures may reach the surface, saturation effects due to fault finiteness obscure the underlying geometric spreading; these effects can be clearly seen in empirical GMMs such as the NGA west GMMs (Abrahamson et al., 2014; Boore et al., 2014; Campbell and Bozorgnia, 2014; Chiou and Youngs, 2014; e.g., Fig. 1).

This study presents new empirical evidence that supports the assumption of $R−1.3$ geometric spreading in the Fourier domain, mapping into steep response spectral decay rates (Hassani and Atkinson, 2018), with no apparent near‐distance saturation for events of M < 3 to distances as close as 0.1 km. Because the application of the results is geared to GMM development and hazard assessment, we focus the data analysis on response spectra observations.

The database comprises recordings of 13 events having rupture scales given by log potency (P) $>0.5m3$ (in which potency is the seismic moment divided by shear modulus; see Ben‐Zion, 2001). The events were recorded on nine instruments located at the surface and/or cemented into shallow boreholes (10–15 m deep) and 68 underground instruments within hard rock, in an area ∼2.7 km × 2.7 km at depths of 500–1500 m. Shear‐wave velocity of underground sites ranges from 3100 to 3300 m/s based on measurements within the mine rock. Three of the surface sites are cemented into rock, with the remaining six being on soil. Soil velocities in an area of the Tailings reservoir vary from ∼360 to 500 m/s, according to a 3D model developed from passive tomography, but velocity profiles specific to the surficial recording sites are not available. Table 1 lists event attributes and the hypocentral distance range of observations; events are recorded at remarkably short hypocentral distances due to their shallow focus and the underground instrumentation.

This study uses 5% damped pseudoacceleration response spectra (PSA) for the geometric mean horizontal component. The PSA database, along with event and station parameters, were provided by Institute of Mine Seismology (IMS, personal comm., 2021). The near‐surface instruments are a mix of 4.5 Hz geophones (from IMS), broadband seismometers (Nanometrics Trillium Compact), and accelerometers (Nanometrics Titan). All instruments record at 1000 or more samples/s and can recover frequencies up to 200 Hz or greater. To avoid limitations on the accuracy of ground‐motion amplitudes associated with the short‐period geophones at low frequency, the study is focused on PSA from 2 to 100 Hz, for which preliminary examinations showed that ground motions from all instrument types are mutually consistent. The data were examined with regards to their noise characteristics using information provided by IMS. The minimum event size threshold for the study was chosen to ensure adequate signal‐to‐noise ratio (>∼2) over all frequencies.

The moment magnitudes for the study events were estimated using two alternative formulae: (1) an estimate from IMS given by M = 0.667 logP + 0.95 (assuming a shear modulus of 33 GPa at the mine site); and (2) an estimate based on an empirical study by Ben‐Zion and Zhu (2002). For each event, the average value of M is adopted as representative. The estimated values of M are not critical for the purposes of this study, which focuses on amplitude decay rates.

The hypocentral distance for local mining events is accurate to within a few meters, due to the density of the network at the underground mine. By contrast, location uncertainties for events >10 km away (events 1–3) may be hundreds of meters, and their depths are more uncertain. This study is focused on the hypocentral distance range from 0.1 to 8 km, which comprises the accurately located shallow local mining events (events 4–13).

The mining events include a mixture of crush‐type and slip‐type events (Ryder, 1988). Crush‐type events result from the dynamic stress fracturing of a rock mass in areas of the mine with low confinement (next to voids) and under high compressive stresses (Malovichko, 2020), usually pillars or tunnel walls and faces; these events are generally of limited magnitude. Slip‐type events are associated with shear failure on geological structures that are unclamped or loaded by stress field redistribution imposed by the excavation, and are analogous to natural earthquakes; these events tend to be less frequent and may have larger magnitudes (Gibowicz and Kijko, 1994). The two types of events are typically distinguished by their seismic moment tensor solution and stress drop; slip‐type events have dominant double‐couple components and generally higher stress drops, whereas crush‐type events have dominant negative isotropic and negative calibrated linear vector dipole components (Malovichko, 2020) with generally lower stress drops.

In Figure 1, the predicted GMM from the Hassani and Atkinson (2018; HA18) model is plotted, assuming that $b1=−1.3$ with no near‐distance saturation; the amplitude level of the HA18 line is prescribed by (⁠$FE+Fκ0+C$⁠) of equation (1), using the applicable values for this event as discussed later (stress parameter 1 MPa; kappa = 0.006 s; C = 0). We note that $b1=−1.3$ is the geometric spreading rate determined by Hassani and Atkinson (2018) and Yenier and Atkinson (2015a) for California and by Yenier and Atkinson (2015b) for eastern North America—in both the cases for events of M ≥ 3 at depths >∼5 km. We conducted a check on a subset of events and confirmed that the inferred spreading rate of $b1=−1.3$ is consistent with the decay of Fourier amplitudes for these small shallow events. At this stage, no adjustments of the observed amplitudes to account for site terms (⁠$FS$⁠) have been made.

Figure 1 also compares the near‐distance attenuation rate of ground‐motion amplitudes with that predicted by typical GMMs used in engineering applications. The GMMs for comparison are the NGA‐W2 GMMs and the GMM developed by Atkinson (2015) for small‐to‐moderate events at close distances. Their evaluation at M 2.8 constitutes a slight extrapolation, as they were developed using data of M ≥ 3. All GMMs (except the HA18 GMM) are plotted for National Earthquake Hazards Reduction Program (NEHRP) B/C boundary (average shear‐wave velocity of 760 m/s in the top 30 m), because they are not well constrained for hard rock. No attempt at conversion for site conditions was made. As shown by Boore (2015), the conversion of PSA GMMs from 3000 to 760 m/s depends on the assumed velocity profile and the kappa values for both the B/C profile and the rock site and on magnitude and distance. For the conditions of this study, the conversion ratio would be approximately constant and could be about a factor of two.

There are some relevant initial conclusions to be drawn from inspection of Figure 1, which have broader implications for both hazard assessment and finite‐fault modeling. Overall, the steep slope of the observations is consistent with the HA18 model and with the Atkinson (2015; A15) GMM, but not with the NGA‐W2 GMMs. This is because the NGA‐W2 GMMs were developed primarily for larger magnitude events of most engineering concern and did not adequately model the short‐distance scaling of small events. Beyond 5 km, the NGA‐W2 and A15 GMMs are generally comparable in their amplitude predictions. We conclude that the observed PSA decay rates are consistent with those predicted by the HA18 model for a geometric spreading of $R−1.3$ in the Fourier domain, even though the HA18 model was derived from deeper events. A similar conclusion was reached by Novakovic et al. (2018) for induced events in Oklahoma. There is no evidence of near‐distance saturation within 10 km for the small events of our dataset. An inference is that observed saturation within 2 km for small events on softer sites in other studies (e.g., Douglas et al., 2013; Atkinson, 2015; Atkinson et al., 2016) may be driven chiefly by the effects of soil nonlinearity. Figure 1 was constructed for PSA at high frequencies, but similar conclusions regarding the shape of the amplitude decay are reached from examination across the entire frequency range of this study (2–100 Hz).

The agreement of the observed attenuation trend in Figure 1 with the HA18 model supports its use as a tool to study the source, attenuation, and site components of these events in more detail. Over the magnitude range of this study, the amplitude decay slope, $FZ$⁠, depends only weakly on magnitude, so that the frequency‐dependent value of $FZ$ as evaluated for $M≈3$ is a reasonable approximation of the expected amplitude decay for all study events. Specifically, the error that may result from this assumption is <0.1 ln units (∼10%) over the magnitude–distance range of this study at all frequencies. This suggests a very simple analysis strategy to determine the source and site effects as parameterized by equation (1). The key is determination of the site terms (⁠$FS$⁠) so that we can remove their effects from the observed amplitudes, allowing us to utilize the surface, borehole, and underground data together to characterize the observed ground motions. We define the reference condition for the analysis as hard rock on (or near) the surface. From preliminary inspection of the data, we postulate that this condition is reasonably represented by the three rock sites on the surface or in shallow boreholes. We, therefore, constrain the analysis such that the average of the site terms over these three stations must equal 0 ln units at all frequencies.

The analysis proceeds as follows: (1) $FZ$ is evaluated for each record (equation 2) assuming M 3 and $b1=−1.3$ (no saturation). (2) Assuming anelastic attenuation is negligible, $Fγ≈0$⁠, an initial estimate of the apparent source spectrum at reference distance R = 1 km ((⁠$FE+Fκ0+C$⁠) of equation 1) is obtained for each event as the average of (⁠$lnPSA−FZ$⁠) taken over all stations. This initial estimate of the source terms assumes that the average effect of site terms is negligible (e.g., in equation 1, $FS=0$⁠). (3) An estimate of $FS$ at each station is then obtained by averaging residuals by station (e.g., difference in source term for an observation relative to the event average, in ln units). (4) We adjust the input values of PSA for each observation for the corresponding estimate of $FS$ and repeat the analysis, iterating until subsequent iterations do not provide any further significant changes in either the source or site terms. The process converges rapidly, resulting in stable estimates of individual source (⁠$FE+Fκ0+C$⁠) and site terms (⁠$FS$⁠) in two to three iterations.

Plots of final residuals allow the inspection of deviations from the assumed $R−1.3$ geometric spreading, with the perturbations revealing any finer structure not captured by the assumed linear decay trend. Figure 2 shows the residuals for ln PSA at 5 Hz. Trends are very similar at all frequencies, with the average residual being strongly positive (∼1 ln units) at a hypocentral distance of 0.2 km, weakly negative (∼−0.4 ln units) at 1 km, and zero for distances of 2–30 km. It should be noted that the zero residuals for the observations at 20–30 km are not diagnostic, because these residuals come from the deeper regional events (events 1–3) that were observed only at regional distances, and, thus, the residual trends are only meaningful for distances <10 km. These trends suggest that source terms (defined at a reference distance of 1 km) will tend to be slightly overestimated, whereas motions at very near distances (<1 km) will be underestimated. The model is unbiased for hypocentral distances from ∼2 to 10 km, which would be the near‐distance observational distance range for events having a focal depth ≥2 km.

As shown in Figure 3, this product is remarkably consistent with the average differences between surficial and underground hard‐rock sites.

The apparent source spectra (at R = 1 km), after removal of the effects of station terms (⁠$FS$⁠) and geometric spreading (⁠$FZ$⁠), but including the effects of kappa for the reference condition, (⁠$FE+Fκ0+C$⁠), are shown in Figure 4 for the largest four events. The absolute amplitude level of the two midcrustal regional events are more uncertain, because they were recorded only at distances >10 km, and are thus based on assuming the distance correction to source. The spectral shape for the two deeper regional events differs from that for the two shallow mining‐related events, suggesting higher stress parameter for the deeper events (i.e., greater high‐frequency content relative to low‐frequency level). For comparison with typical earthquake spectral shapes, the predicted apparent source spectra of Hassani and Atkinson (2018) are plotted for M 2 for a stress parameter of 1 MPa and for M 3 for a stress parameter of 6 MPa, in both the cases assuming $κ0=0.006s$ and C = 0 (the application of the model to M < 3 is an extrapolation). The values for stress parameters were selected for this visual comparison to show the range of shapes and apparent corner frequencies (∼10–20 Hz). The effects of kappa at high frequencies can be seen in the gradual decay of high‐frequency amplitudes above 20 Hz, with the value of $κ0=0.006s$ for hard‐rock sites on the surface being consistent with the observed high‐frequency decay trend. Taken together with the observations from Figure 3, we infer that the average kappa for the intact rock underground is zero, with $κ0=0.006s$ for hard‐rock sites on the surface likely being attributable to near‐surface weathering of the rock.

The predicted apparent source spectra in Figure 4 are less than the largest observed motions for the mine events, because the source terms are evaluated for the reference distance of R = 1 km (by definition), whereas observations are at distances as close as 0.15 km. As seen in Figure 1, the amplitude decay from 0.1 to 1 km is significant. We can infer from Figures 1 and 3 that median PSA values on the surface (on rock) could be $∼1000cm/s2$ at very close distances, for events of M∼3 and thus for some observations would be even larger due to random variability.

The hazard implications of such motions for critical infrastructures at surface are unclear, given their very high frequencies and short duration. Furthermore, these motions decay rapidly with distance, and consequently the hazard from such events is limited in its reach. Moreover, the hazard on the surface may also be limited by the type of event, and/or by the relationship between stress and focal depth. For example, the M 2.8 event is a crush‐type event, which typically has a low stress drop. Slip‐type events, by contrast, are analogous to natural earthquakes, which have been observed to have stress parameter that increases with depth (e.g., Yenier and Atkinson, 2015a,b). The events which have higher implied stress (Fig. 4) are at depths >5 km. Thus, the high‐frequency hazard that results from high‐stress events is offset by the greater distance of the events from the surface. For soil sites, motions may be significantly altered by site‐response effects that amplify motions at lower frequencies but damp high‐frequency motions.

The decay of ground‐motion amplitudes from small events (M 2–3) on hard‐rock sites is consistent with a geometric spreading rate of $R−1.3$ for hypocentral distances from 1 to 10 km, with no apparent saturation effects. At hypocentral distances <0.3 km, median PSA values may reach $∼1000cm/s2$ at high frequencies (>10 Hz) for events of M < 3. These results are useful as point‐source input components to finite‐fault modeling of larger events and for the assessment of induced‐seismicity hazards.

The ground‐motion data used in this study may be available upon request from Newcrest Mining (contact Gisela Viegas).

The authors acknowledge that there are no conflicts of interest recorded.

Newcrest Mining provided the data and support for this study. The authors are grateful to Denver Birch of Institute of Mine Seismology (IMS) for his support in providing the data and for helpful discussions. The authors thank the Editor and two reviewers, Norm Abrahamson and Grace Parker, for their constructive suggestions.