method for the P- and S-wave onset determination (Leonard ... determine source parameters (Andrews, 1986). ... The seismic moment is (Andrews, 1986). 3. M.
Bulletin of the Seismological Society of America, Vol. 95, No. 3, pp. 1011–1026, June 2005, doi: 10.1785/0120040170
Source Parameters and Scaling Relations for Mining-Related Seismicity within the Pyha¨salmi Ore Mine, Finland by Volker Oye, Hilmar Bungum, and Michael Roth
Abstract To investigate source-scaling relations for small earthquakes (Mw ⳮ1.8 to 1.2) we have determined source parameters for numerous events (1500) from the 1400-m-deep Pyha¨salmi ore mine in Finland. In addition to a spectral integration approach, we have fitted Brune, Boatwright, and Haskell spectral-shape models to the observed spectra and investigated attenuation influences. Of three different constant Q models (200, 350, and 800), a Q of 350 in combination with the Brune spectral model satisfied the data best. We have also investigated the frequency dependence of Q using the spectral decay method and found that Q increases with frequency. For selected events from two distinct clusters, we compared source parameters derived from constant Q models with source parameters using the multiple empirical Green’s function (MEGF) approach. By using constant Q models, the apparent stress seems to increase with magnitude, whereas results based on the MEGF approach indicate constant apparent stress with magnitude. In comparison with results from other studies that cover a larger-magnitude scale, we find apparent stresses that are about 1 to 2 orders of magnitude smaller than most of those. A modified M0 fcⳮ(3 Ⳮ e) scaling relation allows for increasing apparent stress with magnitude and can hence combine this study’s results with apparent stresses found for large earthquakes. However, within the limited-magnitude range of our data, apparent stresses seem constant. Introduction The question of whether apparent stress scales or is constant with event size is highly debated and is still essentially unresolved. The static scaling relation M0 fcⳮ3 between the seismic moment M0 and the spectral corner frequency fc or, equivalently, the rupture length, is now commonly accepted (Brune, 1970, 1971; Hanks, 1979; Abercrombie and Leary, 1993). However, this does not necessarily imply that apparent stress is independent of magnitude (Aki, 1987; Bungum and Alsaker, 1991; Kanamori and Rivera, 2004). For small earthquakes, say below Mw 3, estimates for source parameters are difficult to obtain and, hence, scaling relations are less conclusive. Station and path effects (including attenuation) strongly influence the determination of M0, fc, and the radiated seismic energy Erad. Depending on the chosen attenuation model and source spectral shape, large variations in the source parameters may consequently arise (Abercrombie, 1995; Ide and Beroza, 2001; Prejean and Ellsworth, 2001; Ide et al., 2003). One of the classical problems within seismology for a long time has been the formulation of the scaling laws for earthquakes (Aki, 1967; Hanks, 1977; Scholz, 2002). This research was revived with the introduction of the concept of self-organized criticality (e.g., Bak and Tang, 1989),
whereas several studies for small earthquakes claimed a breakdown in self-similarity for earthquakes below about magnitude 3 (e.g., Archuleta et al., 1982). High-frequency observations in deep boreholes have subsequently demonstrated that the scaling invariance may extend to well below Mw 0. (Abercrombie and Leary, 1993; Ide et al., 2003). For more than 30 years, mining-induced seismicity has strongly contributed to the discussion of scaling relations (McGarr and Simpson, 1997). Studies from deep South African gold mines (Spottiswoode and McGarr, 1975; McGarr 1994; Richardson and Jordan, 2002) and from the Strathcona mine in Canada (Urbancic and Young, 1993) range from about Mw ⳮ2 to Mw 3, whereas events from the URL (Underground Research Laboratory, Canada) have an upper limit of about Mw ⳮ2 (Gibowicz et al., 1991). Mininginduced seismicity occurs at scales between those in the laboratory and those observed regionally from tectonic faults and can help to unite their scaling relations (McGarr and Fletcher, 2003). Because of the new generation of in-mine geophones and their installation in the bedrock close to the events, high-frequency seismic data can be provided (Mendecki, 1997). Under strongly heterogeneous conditions such as in a
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mine, with tunnels, stopes and different rock types, the coupling of the stations varies and seismic waves are scattered, resulting in focusing and defocusing effects, depending on which path they travel. It is important to compensate for such station and path effects because those effects could otherwise strongly affect (bias) the inferred source parameters. Constant Q models are often used to quantify the effect of intrinsic attenuation (Abercrombie, 1995; Roth and Bungum, 2003). However, a frequency-dependent Q is more likely, even though the determination of Q(f) is often ambiguous (Hough et al., 1988; Kvamme et al., 1995; Brockman and Bollinger, 1992). The empirical Green’s function method (Berckhemer, 1962; Mueller, 1985; Hough, 1997) and its extension to the multiple empirical Green’s function method (MEGF) (Hough, 1997) is a promising tool to compensate for path and site effects. Ide et al. (2003) showed, by using an MEGF algorithm, that a constant Q assumption can cause an apparent break in earthquake scaling because of path and site effects, even when based on high-quality recordings in deep boreholes. In the first part of this study, we introduce the methods applied to determine source parameters for data from a mine in Finland. We then discuss different ways to determine and compensate for attenuation, and finally we present our results on how the determined source parameters scale.
stoping, achieving an annual production of about 1.3 Mt of ore. Before the start of the mining below 1000 m, stresses were measured at a depth of 1225 m. The main stress is horizontal compression at N310E, with vertical stress rV ⳱ 33 MPa, maximum horizontal stress rH ⳱ 65 MPa, and minimum horizontal stress rh ⳱ 41 MPa (Ledger, 1999). This is largely in agreement with tectonic stress directions from the world stress map (Reinecker et al., 2004), even though secondary effects of mining might have disturbed the stress field as compared with intact rock conditions.
The Pyha¨salmi Data The monitoring network within the Pyha¨salmi mine consists of 16 sensors as shown in Figure 1, which also displays the ore body and the tunnel systems. The geophones are cemented in 10.5-m-long boreholes (76 mm in diameter) that were drilled vertically into the roof of the tunnels. The geophone’s natural frequency is at 4.5 Hz, and the instrument response is flat for frequencies greater than 10 Hz. In general, sampling rates are at 3000 Hz, in few cases, however, down-sampling to 1000 Hz or 500 Hz was carried out to save storage space. The network has been operational in a continuous mode since January 2003 and, until March 2004, about 18,000 events were detected and recorded.
The Pyha¨salmi Ore Mine The Pyha¨salmi ore mine in central Finland extends to a depth of 1.4 km and is the deepest of its kind in western Europe. Massive sulfide ores form a potato-shaped body, which seems to be located in the hinge of a large synform fold (Puustja¨rvi, 1999). The ore body is medium to coarse grained and consists of sphalerite-pyrite with some zinc (2%–10%) in the outer rim and in an inner rim of chalcopyrite-pyrite with a copper content of 1%–6%, whereas the innermost part of the ore body is uneconomical pyrite. The ore body is surrounded mainly by quartz-rich felsic volcanics (80%–90%) with amphibolite bands that vary in width from millimeters to meters. The whole fold hinge is defined by distinct schistosity. Cracks are most prominent in the schistosity/banding direction. There are numerous pegmatite veins, varying in size and orientation. The veins are common near the ore contacts. There are no distinct faults in the active mining area between 1100 and 1450 m depth that could have been activated by mining. However, pegmatites might play a role for the stability of the mining area, because pegmatite veins have been identified as fault planes. Furthermore, severe shearing has occurred in the geological past in Pyha¨salmi, and active faults exist outside the current mining area and near the surface. The complete mining infrastructure with its tunnels, stopes, and passes has been surveyed by laser scanning, and a model has been built using a mine design software. This software allows to update the model in response to successive mining. The mine is quarried by sublevel and bench
Figure 1. Model of the Pyha¨salmi ore body, the surrounding infrastructure, and the seismic stations. We observe most seismic activity close to KN1, which is a pass for quarried ore (Fig. 3). Geophones are from Integrated Seismic System International Ltd, South Africa, type ISS SM6-B geophones (4 threecomponent [3C] and 12 vertical-component [1C] geophones) and the monitoring is conducted by INMET Mining.
Source Parameters and Scaling Relations for Mining-Related Seismicity within the Pyha¨salmi Ore Mine, Finland
About two-thirds of these events were identified as blasts related to the mining operations whereas the remaining 6000 were seismic events. Further selection of the seismic events based on signal-to-noise ratio (3), number of traces that report a detection (12), location error (10 m), rms time residual (5 msec), and visual control of onset pick quality lead to a smaller data set of about 1500 seismic events. Because intervals between events can be very short, we reject the events that occur closer than 200 msec after each other to determine proper source parameters for single events. The cumulative and the discrete number of earthquakes during the 15 months give an overview of the detected seismic events within the monitoring network (Fig. 2). Events range in moment magnitude from ⳮ1.8 to 1.2, whereas most events have Mw from ⳮ1.5 to ⳮ0.5. The distribution of seismic-event magnitudes indicates a self-similar behavior (b-value ⳱ 1), at least in the range from Mw ⳮ1 to 0. Hence, the ratio between the observed distribution and the b-value unity provides a rough estimate for the detectability, indicating a 90% detectability at Mw ⳮ1.1 and a 50% detectability at Mw ⳮ1.6. Microseismic events occur throughout the tunnel system and in stopes after excavation. The events located with the highest accuracies are of course those in the center of the network. Here we find two clusters (C1 and C2 in Fig. 3) of higher seismic activity than other regions in the mining infrastructure. Events in C1 are associated with the location of an ore pass called KN1 (Fig. 1), where the ore is dropped in from each production level (at up to 1150 m depth) and falls down to the crusher level at 1400 m depth. KN1 has expanded over the past 2 years from its original 3-m diameter to almost 30 m in a northeast–southwest direction, which is similar to the minimum horizontal stress direction. Although about half of the C1 events occurred during daytime (working hours), the automatic and visual selection criteria ensured that the events are truly seismic and did not originate from falling rocks, even though they might be induced by such activity. In addition to the high accuracy of the events determined along the pass KN1, Figure 3 also shows the spatiotemporal extension of the pass, because early events only occurred in the southern center of the original pass and then extended further north. The second cluster C2 corresponds to the most active mining area. Two stopes at 1220 and 1225 m depth are completed; most of the microseismicity occurs here. Recently, two new adjacent stopes were opened in the depth levels of 1250 and 1325 m. Between the old and the new stopes is a narrow pillar and its surroundings have also been seismically active. Source-to-receiver distances in the Pyha¨salmi mine, in general, range from about 60 to 400 m. An example of an Mw ⳮ0.2 event, located within C1, is plotted as a seismic section in Figure 4. The P- and S-wave signals are compact and impulsive at short distances, whereas already at distances of 200 m large parts of the energy are transferred into
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Figure 2. Discrete and cumulative number of visually checked seismic events from January 2003 to March 2004. b-values are close to one for moment magnitudes Mw ⳮ1 to 0.
P- and S-wave coda. This is probably related to a high level of scattering in the heterogeneous mining area, which is characterized by strong-velocity contrasts between the rock types and the extensive tunnel systems (Fig. 1). At larger distances the S-wave onsets, in particular, become unclear or diffuse, creating problems for onset picking at 1C geophones, because a rotation into the ray coordinate system and hence an improved signal-to-noise ratio is only possible for 3C geophones. To compare locations with catalog events determined from the mining company, we use the same constant velocity values for P and S waves (␣ ⳱ 5500 m/sec and b ⳱ 3500 m/sec, respectively). The homogeneous velocity model explains the observed phase arrival times fairly well with residuals of a few time samples. Given the above-mentioned heterogeneity within the Pyha¨salmi mine, however, a 3D velocity model should be expected to improve location accuracies. Table 1 shows a compilation of the most abundant rock types within the active mining area together with their laboratory-determined physical properties (Puustja¨rvi, 1999). A complete 3D model with 2-m grid distance is currently in development and will eventually replace the homogeneous model (Bungum et al., 2004).
Determination of Source Parameters We processed all the Pyha¨salmi events automatically with a modified version of a microseismic-monitoring software package developed at NORSAR (Oye and Roth, 2003). The processing steps include a detection algorithm with event identification, and association of the traces to the event, an autoregressive AIC (Akaike Information Criteria) method for the P- and S-wave onset determination (Leonard and Kennett, 1999), P-wave polarization analysis, and a lin-
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Figure 3.
Map view and depth section of about 1500 seismic events within the Pyha¨salmi ore mine, Finland. Four three-component (red triangles) and 12 onecomponent (vertical) geophones (red diamonds) are installed within the mine infrastructure. The color scale in the main pictures represents the location error (blue, 5 m to red, 30 m). Most events occur in cluster C1, which is related to the location of the ore pass KN1, indicated as a black line in the insets (K. Sahala, INMET Mining, personal comm., 2004). The color scale in the insets represents origin time (blue to red), indicating the extension of the pass to the north; all events in the insets have location errors smaller than 10 m.
earized inversion solving for location and origin time with P- and S-wave arrival times. The general form of the source-displacement spectra used in this analysis is given as follows: X( f ) ⳱
X0 f 1Ⳮ fc
冤
cn 1/c
冢冣冥
(1)
where X0 is the low-frequency spectral level, f is the frequency, fc is the corner frequency, n is the high-frequency level fall-off rate, and c is a constant. For n ⳱ 2 and c ⳱ 1, equation (1) is the classical Brune spectrum (Brune, 1970, 1971). Under the assumption that our data are well described by a Brune spectral shape, we use integrals of spectra to determine source parameters (Andrews, 1986). We determine the source parameters from the S-wave signals of the event, because it is the most energetic part of the seismogram. To calculate the displacement-amplitude spectrum, we Fourier transform the displacement seismograms for a discrete time window. The windows for all events and components are always 200 msec long; they start at the S-wave onset time and are extended by 20 msec cosine tapered time windows at both ends (Fig. 5). To compensate the spectra for geometrical spreading and intrinsic attenuation, we multiply the spectra with each geophone’s source distance R and the attenuation factor exp(pfR/Qb), where b is the shearwave velocity and Q is assumed to be frequency indepen-
dent. The signal is usually above the noise level from 15 to 20 Hz up to the Nyquist frequency (Fig. 6) and, because the instrument response is flat for frequencies greater than 10 Hz, there is no need to correct for the instrument response. We attempt to compensate for radiation-pattern effects by stacking all corrected spectra. We now square and integrate the corrected and stacked amplitude displacement (D), velocity (V), and acceleration (A) spectra from 20 Hz (f1) up to 90% of the Nyquist frequency (f2): f2
SX2 ⳱ 2
冮X
2
( f ) df
for X ⳱ (D, V, A) .
(2)
f1
The radiated seismic energy Erad is determined from the integrated velocity spectrum by Erad ⳱ 4pqbSV2 .
(3)
The average density used in this study is q ⳱ 4000 kg/ m3 (Table 1). Theoretically, the integration limits in equation (2) are f1 ⳱ 0 and f2 ⳱ , whereas in practice we are limited to the frequency bandwidth at which the signal is clearly above the noise, as well as to the instrument’s spectral bandwidth. Ide and Beroza (2001) showed that, when assuming a Brune’s spectral shape, only about 20% of the radiated seismic energy is carried by frequencies smaller than the
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Source Parameters and Scaling Relations for Mining-Related Seismicity within the Pyha¨salmi Ore Mine, Finland
Figure 4. Seismogram section of an event that occurred in cluster C1 at 8342 m E, 2364 m N and 1327 m depth, showing z components of all stations. P- and S-wave onsets are fitted fairly well by the homogeneous velocity model used. The maximum trace amplitudes are given for each trace in units of millimeters per second, whereas in the plot the trace amplitudes are multiplied by epicentral distance.
Table 1
The Brune stress drop is (Eshelby, 1957; Brune, 1970)
Typical Rock Types within the Pyha¨salmi Ore Mine and Its Properties Rock Type
Volcanite, felsic Volcanite, mafic Pegmatite Ore, zinc Ore, copper Pyrite
E (GPa)
n
std
m
n
std
q
␣
b
68 76 63 98 139 120
33 18 4 18 22 12
24 21 13 37 33 21
0,24 0,26 0,23 0,32 0,30 0,34
33 18 4 18 25 12
33 27 25 38 31 45
2500 2900 2700 4200 4600 5000
5662 5663 5201 5778 6378 6078
3312 3225 3080 2973 3409 2993
E is Young’s modulus, n is the number of samples taken, std is one standard deviation, m is Poisson’s Ratio, q is density, ␣ is P-wave velocity ⳱ sqrt(E/q * (1 ⳮ m)/((1 Ⳮ m)(1 ⳮ 2 m))) and b is S-wave velocity ⳱ ␣ * sqrt((0.5 ⳮ m)/(1 ⳮ m)).
corner frequency, and that by integrating up to 10 * fc one can expect to enclose about 90% of the theoretical energy release. Therefore, we corrected the radiated seismic energy with respect to the estimated corner frequency and the upper integration limit. The Brune spectrum in equation (1) is defined by two parameters: the corner frequency and the low-frequency spectral level (Andrews, 1986) fc ⳱
1 2p
X02 ⳱ 4
冪S
SV2
(4)
D2
冪
3 SD2 SV2
(5)
The seismic moment is (Andrews, 1986) M0 ⳱ 4pqb3 X0 /C , with C ⳱ 冪2/5 the rms radiation pattern for S waves.
(6)
DrB ⳱
冢
7 M0 7 2pfc M0 3 ⳱ 16 r 16 2.34 b
3
冣
(7)
where r is the radius of the circular fault plane. We insert equation (6) in (7) and express fc and X0 with equations (4) and (5), respectively, and thus obtain the stress drop in terms of integrals over velocity and displacement spectra (Andrews, 1986; Snoke 1987):
DrB ⳱
q(2pfc)3 X0 2q ⳱ 2.34 C 2.34 C
冪
4
5 SV2 3 . SD2
(8)
Alternatively to the integration method we determine the best-fitting source parameters fc and X0 for the general formulation of the displacement spectrum in equation (1) using constant Q values of 200, 350, and 800 (see the next section) (Abercrombie, 1995; Prejean and Ellsworth, 2001). Again we stack all distance- and Q-corrected displacement spectra of a seismic event recorded at the individual stations i. This time we fit three different displacement spectral models to the stack, for frequencies from 20 Hz to 90% of the Nyquist frequency: (1) the classical x2 model (n ⳱ 2, c ⳱ 1) (Brune, 1970); (2) the modified x2 model (n ⳱ 2, c ⳱ 2) after Boatwright (1980) that is characterized through a sharper corner; and (3) the x3 model (n ⳱ 3, c ⳱ 2/3) after Haskell (1964) with a stronger high-frequency fall-off rate. We solved the nonlinear inversion problem (equation 9) with the Matlab routine “lsqnonlin” that is based on the Levenberg–Marquart algorithm.
冢 冫冤1 Ⳮ 冢 ff 冣 冥 冣 cn 1/c
log DS( f ) ⳱ log X0
c
(9)
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Figure 5. Three-component seismogram and window length applied for spectral analysis with extended windows that are cosine tapered (dashed in gray). The maximum amplitudes are given for each trace.
Figure 6.
Displacement amplitude spectrum calculated for a trace without Q correction (black dashed) and with constant Q of 350 (gray dot-dashed). The Q effect is small and only seen at high frequencies. The corresponding noise spectra computed for a time window before the signal onset is about two decades lower (gray thin line). The stack of all components of the event (black thick line) and the fit for an x2 model (gray thick line) are also indicated.
The start value for fc is half the Nyquist frequency and the start value for X0 is the mean value in the low-frequency part of spectrum D. Independent of all selected Q values, the lowest misfits were achieved with the x2 model except for few data (less than 5%) that seem to match better with the modified x2 model. The strong high-frequency fall-off rate of the x3 model is not observed in the data, resulting in highmisfit values compared with the other two models (Fig. 7).
(Roth and Bungum, 2003). All data further discussed from the two clusters have a sampling rate of 3000 Hz. As earlier, we compensate all displacement spectra Di(f) observed at stations i for geometrical spreading by a multiplication with the hypocentral distance Ri. Station and source-radiation effects are assumed to be frequency independent and are corrected by a constant factor Fi , which we estimate from the low-frequency part of each spectrum, that is more or less unaffected by attenuation:
Estimation of Q To get an estimate for the attenuation within the mining area, we fit spectra for events from the two clusters C1 and C2 to an x2 model as given in equation (1), while minimizing the residual by varying frequency-independent Q values
misfit ⳱
兺i 冢log冤Di ( f ) • Ri • exp(pRi f/bQ) • Fi冥
冤 冫
ⳮ log X0 1 Ⳮ
冢 ff 冣 冥冣 . 2
c
(10)
Source Parameters and Scaling Relations for Mining-Related Seismicity within the Pyha¨salmi Ore Mine, Finland
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Figure 7. Stacked displacement spectra for three events, corrected for constant Q (200, gray; 350, light gray; 800, dark gray). Black lines show the fit for Q ⳱ 350 for the x2 (solid), the modified x2 (dashed), and the x3 models (dash-dotted). A Q value of 350 in combination with the x2 model results in the best fit for the spectra.
We solve the nonlinear system of equation (10) for 20 Q values ranging in logarithmic scale from 75 to 1500. The derived Q values for events from clusters C1 and C2 vary from about 200 to 500, whereas some events seem to be unaffected by attenuation, resulting in Q values larger than 1500 (Fig. 8). There is no apparent connection between the location of events and Q values. The presence of a strong variation in Q values is possibly caused by the complexity of the mine structure (voids and tunnels). We conducted all processing with three different constant Q values, 200, 350, and 800, even though the resulting differences in source parameters turned out to be minor (Table 2). Constant Q models are often used because they provide a first estimate for the general attenuation of the investigated area. However, several studies have shown that a frequencydependent Q might explain the data better, especially at frequencies greater than 100 Hz, where little is known about attenuation (Hough et al., 1988; Kvamme et al., 1995; Brockman and Bollinger, 1992). To investigate the dependence of Q on frequency we apply the spectral decay method (Campillo et al., 1985; Sato and Fehler, 1998). We select station pairs in a way that rays from an event cluster have
similar travel paths to the two stations. Hereby we compensate for source radiation pattern, source spectral shape and path effects, but station effects remain, which we assume to be frequency independent. Then, the spectral amplitude decreases on the travel path between the two stations only because of geometrical spreading and attenuation: R1 D1 ( f ) ⳱ D0( f ) exp[ⳮfR1 /Q( f )b] and R2D2( f ) ⳱ D0( f ) exp[ⳮfR2 /Q( f )b] .
(11)
Taking the logarithm of the ratio from equations (11), we solve for Q(f) Q( f ) ⳱
pf • b
R2 ⳮ R1 Ⳮ K. R D (f) ln 1 Ⳮ ln 1 R2 D2 ( f )
(12)
where station effects and differences of the assumed geometrical spreading of 1/R from the actual geometrical spreading are gathered in the constant K. For statistical reasons we need to repeat the spectral
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Figure 8. Estimated constant Q values for events of cluster C1 (solid) and cluster C2 (dashed). For most events we find a minimum misfit for Q values between 200 and 500, whereas some events seem to have higher Q values than 1500, independent of the event’s location.
decay method for many events within the clusters. The geophone configuration, however, limits the amount of suitable station pairs. We can therefore use only two pairs of geophones (geophone 3 to 8 and geophone 2 to 5) that record events from cluster C1 (Fig. 9). Because distances between the selected station pairs are only about 100 m, the ratios between spectra are close to 1, and we apply smoothing (D log f ⳱ 0.3) before we divide the spectra. The short distance between the two stations also limits the applicability of the method to frequencies greater than about 100 Hz, because this is the equivalent to about three wavelengths. Despite the high scatter in the results in Figure 9, the results indicate a clearly increasing Q value with increasing frequency for both station pairs. The conclusion we draw from the spectral decay method is that the data deviate from a constant Q model such that Q is increasing with frequency.
Spectral Ratios For colocated events one can apply the empirical Green’s function method (Berckhemer, 1962; Mueller, 1985; Hough, 1997) to determine source parameters without assuming path and site effects. We apply the MEGF method (Hough, 1997; Ide et al., 2003) to a set of 19 earthquakes within C1 and to another set of 18 earthquakes within C2. We then determine corner frequencies and relative seismic moments between the events within each cluster. As in the spectral decay method, we also apply smoothing (D log f ⳱ 0.3) before we compute the spectral ratios. Taking the ratio between spectra of two events at a single station eliminates path and station effects. We can express the ratio as:
log
冢
冢 冣冣
D1 ( f ) M01 f ⳱ log Ⳮ log 1 Ⳮ fc2 D2 ( f ) M02
冢
2
ⳮ log 1 Ⳮ
(13)
冢f 冣 冣. f
2
c1
In addition to the approach taken by Hough (1997) and Ide et al. (2003), who only used single-station data, we stack the determined ratios for all traces and receive one stacked ratio for each event of the cluster. For a cluster of N events sampled at K frequencies, we have to solve K • N(N ⳮ 1)/2 equations to determine N corner frequencies and N seismic moments. Because the MEGF approach can only provide relative seismic moments, we add one more equation to fix the determined seismic moments to the logarithmic average of the seismic moments that were determined by the constant Q analysis. Figure 10 shows a comparison between observed and calculated spectral ratios for events in C1 and C2. In the constant Q analysis we calculate the radiated energy from the stacked distance- and Q-corrected velocitydisplacement spectra (equation 3). Using the MEGF method, we need to determine a frequency-dependent attenuation curve to correct the velocity spectra before calculating the radiated energy. The attenuation curves are different for each receiver and determined by the ratio between the observed and the calculated spectra for each trace. Because we assume similar source mechanisms for all events in the clusters, we can average the attenuation curves for all events in each cluster (Fig. 11) and then recalculate the radiated energy for the case of MEGF by using equation (3).
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Source Parameters and Scaling Relations for Mining-Related Seismicity within the Pyha¨salmi Ore Mine, Finland
Table 2 Source Parameters for Events from Clusters C1 and C2 Shown in Figure 3, with Locations Based on the Same Coordinate System as in the Figure M0 (N m * 106)
Location (m) Event
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
fc (Hz)
Erad /M0 (* 10ⳮ6)
Erad (J)
x
y
z
A
B
C
D
E
F
A
B
C
D
E
F
8347 8336 8348 8341 8345 8341 8335 8340 8344 8339 8345 8339 8349 8339 8348 8341 8342 8339 8341 8401 8393 8394 8405 8407 8393 8409 8392 8396 8398 8420 8393 8396 8392 8399 8396 8410 8390
2365 2342 2367 2357 2369 2363 2347 2363 2355 2364 2363 2361 2350 2367 2357 2369 2364 2363 2368 2291 2286 2283 2282 2291 2293 2289 2281 2283 2276 2292 2286 2294 2299 2288 2284 2293 2293
1325 1306 1322 1323 1326 1332 1380 1325 1332 1326 1338 1317 1321 1313 1350 1337 1327 1333 1340 1196 1215 1204 1176 1201 1227 1207 1216 1199 1200 1219 1195 1227 1228 1183 1187 1191 1187
17 18 39 72 79 82 83 101 107 179 221 234 289 401 507 612 641 639 841 29 29 36 49 55 60 60 63 65 81 86 130 144 164 192 293 399 684
17 18 39 75 84 86 86 106 113 185 232 243 299 419 529 645 663 670 895 29 30 37 51 59 62 61 64 67 83 89 138 150 172 205 311 428 739
17 18 39 74 82 84 85 104 111 182 228 239 295 412 520 632 654 658 873 29 30 37 50 57 61 61 64 66 82 88 134 148 168 199 303 415 714
18 19 42 75 88 86 87 105 116 186 225 246 310 418 535 629 662 661 865 31 35 41 53 62 63 66 73 71 90 90 152 154 184 214 320 430 746
15 17 36 64 71 74 74 91 97 161 195 210 262 357 449 537 571 577 745 26 26 33 45 48 55 55 55 59 73 78 118 132 146 165 254 339 578
14 16 35 69 67 74 65 92 97 183 198 238 322 385 515 614 656 651 876 28 29 30 50 53 51 56 63 63 78 72 155 145 162 220 316 439 765
304 610 425 167 162 217 239 151 176 242 145 262 225 180 168 155 176 176 136 362 261 292 348 148 347 286 239 247 279 225 202 291 161 132 140 127 108
237 429 328 139 129 173 187 124 140 195 119 199 184 146 137 127 147 143 111 263 199 228 251 116 256 219 186 190 212 176 153 216 128 105 111 101 87
261 492 363 149 141 189 207 134 153 212 128 223 200 159 149 137 158 155 120 299 222 253 287 128 290 244 207 212 237 194 171 244 140 116 122 111 95
228 416 306 143 128 176 192 126 139 193 124 205 178 148 138 132 147 144 115 258 185 217 250 116 259 214 181 190 207 180 147 217 128 109 116 110 93
252 448 338 150 140 186 204 133 151 207 130 219 194 158 149 139 156 153 122 286 219 240 274 131 277 232 205 205 227 190 169 235 141 120 125 116 101
269 449 346 153 166 192 201 146 167 209 138 248 178 175 151 142 162 158 126 286 228 273 278 137 275 350 215 237 262 226 173 280 159 114 125 111 98
A
B
C
D
6 3 4 4 64 22 33 33 94 43 58 57 20 13 15 16 21 12 15 15 56 31 39 39 78 43 54 54 27 17 20 21 49 27 34 35 357 199 250 249 117 75 89 90 791 417 531 529 756 440 543 539 744 450 547 547 963 589 712 717 1117 707 842 852 1773 1098 1322 1349 1717 1010 1236 1259 1407 896 1057 1070 31 13 18 18 12 6 8 8 27 14 18 18 84 33 48 47 8 5 6 6 125 53 74 74 68 33 44 43 46 25 32 31 53 27 35 35 117 55 75 73 68 37 47 47 109 54 71 71 413 183 252 250 92 55 67 67 71 46 54 55 199 123 148 149 293 193 227 230 510 358 411 416
F
A
B
C
F
11 73 128 26 22 84 77 33 45 305 123 1157 661 837 971 1337 1656 1602 1421 24 15 32 44 8 39 72 32 35 73 44 67 289 83 53 149 221 396
0.38 3.52 2.40 0.28 0.26 0.68 0.94 0.27 0.46 1.99 0.53 3.39 2.61 1.85 1.90 1.82 2.77 2.69 1.67 1.10 0.42 0.75 1.71 0.14 2.08 1.14 0.73 0.82 1.45 0.79 0.84 2.86 0.56 0.37 0.68 0.73 0.75
0.18 1.19 1.10 0.17 0.15 0.36 0.50 0.16 0.24 1.08 0.32 1.71 1.47 1.08 1.11 1.10 1.66 1.51 1.00 0.43 0.21 0.38 0.65 0.08 0.86 0.53 0.39 0.40 0.67 0.41 0.39 1.22 0.32 0.22 0.40 0.45 0.48
0.24 1.82 1.49 0.21 0.18 0.46 0.64 0.20 0.31 1.37 0.39 2.22 1.84 1.33 1.37 1.33 2.02 1.88 1.21 0.63 0.27 0.50 0.95 0.10 1.22 0.72 0.50 0.53 0.91 0.53 0.53 1.71 0.40 0.27 0.49 0.55 0.58
0.76 4.47 3.65 0.37 0.33 1.14 1.19 0.36 0.47 1.66 0.62 4.86 2.05 2.18 1.89 2.18 2.53 2.46 1.62 0.84 0.52 1.06 0.88 0.16 0.77 1.29 0.51 0.56 0.93 0.61 0.43 2.00 0.51 0.24 0.47 0.50 0.52
Seismic moment M0, corner frequency fc, radiated energy Erad, and the energy-to-moment ratio Erad /M0 are given for all or some of the following models: A, Q ⳱ 200; B, Q ⳱ 800; C, Q ⳱ 350; D, Q ⳱ 350 and spectral integration method; E, Q ⳱ 350 and Boatwright model; F, Q(f) from MEGF.
Resulting Scaling Relations The static scaling relation between the seismic moment and the corner frequency is well established and commonly accepted (Aki, 1967). The results of about 1500 analyzed events using Q ⳱ 350 and fitting an x2 model (Fig. 12), in general, agree with this scaling relation M0 f c3 constant (Brune, 1970, 1971; Hanks and Wyss, 1972). Even so, we do observe differences in Brune stress drop from about 0.002 MPa up to 30 MPa, indicating that there are strong factors which contribute to significant deviations from constant stress drop and the M0 f c3 relation (Fig. 12a and b). On the other hand, the bulk of events have a stress drop from 0.01
to 1 MPa, independent of M0 and are fitted quite well with a constant stress drop. The radiated energy is related to the velocity spectrum, thereby reflecting the dynamics of the source process and, in general, the ratio Erad /M0 is used as a dynamic scaling parameter. Alternatively, the dynamic scaling relation is often expressed by the apparent stress rapp, which is defined as lErad /M0, where the rigidity is l ⳱ qb2. We compute l as 4.9 • 104 MPa from the average q and b values (Table 1). The ratio Erad /M0 is plotted against Mw (Fig. 12c) and varies from about 0.05 • 10ⳮ6 to 5 • 10ⳮ6, slightly increasing with Mw. Equivalently, rapp varies from about 0.0025 to 0.25 MPa.
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Figure 9.
Spectral decay method to determine Q(f) for two geophone pairs (geophones 3 and 8, black circles, 100 m distance; geophones 2 and 5, gray pluses, 200 m distance). An increase of the Q value is observed with increasing frequency for both geophone pairs. The Q values for frequencies less than 100 Hz are less reliable, because for these frequencies less than three to six wavelengths fit between the stations where attenuation is estimated. The relatively low Q values may be caused by station effects (equation 12). The red line indicates an estimate of Q(f) that is later used as a general example for attenuation correction.
Using the spectral integration method (Andrews, 1986; Snoke, 1987) we obtain similar values for M0 and Erad as compared with the results from the x2 fit, using the same Q ⳱ 350. The obtained values for fc are slightly lower using the spectral integration method. Because the differences are small, we only show a selection of the data in Table 2, even though all associated spectra were fitted with different spectral models (x2, modified x2, and x3) considering different Q values (Q ⳱ 200, 350, and 800). When fitting the x2 model for the three different Q values, M0 remains almost unchanged, whereas estimates for fc are slightly higher using Q ⳱ 200 and slightly lower using Q ⳱ 800 (Fig. 13; Table 2). The estimates for Erad differ by about a factor of 2, comparing results for Q ⳱ 200 and Q ⳱ 800, and this difference is rather constant throughout the data. Fitting the Haskell x3 source model results in significantly higher fc and lower M0 values, whereas the estimation of Erad is independent of the assumed spectral shape (equation 3). Because the data are not fitted well by the Haskell model (Fig. 7), we disregard the resulting source parameters. Using the Boatwright modified x2 model, we get significantly lower M0 values, whereas the estimated fc values are similar to the estimates from Brune’s x2 model (Table 2). Because the bulk of our data are fitted best with Brune’s x2 model, we do not discuss further the results achieved from Boatwright’s modified x2 model. The results for the M0 –fc relation derived from the MEGF method differ systematically from the results using
Figure 10. Observed (gray) and calculated (black) spectral ratios for events in C1 (A) and C2 (B) by using the MEGF method. Ratios are taken relative to the event with the highest corner frequency of each cluster.
Source Parameters and Scaling Relations for Mining-Related Seismicity within the Pyha¨salmi Ore Mine, Finland
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Figure 11. Attenuation curves at selected receivers derived from MEGF, from events in clusters C1 (black) and C2 (gray dashed). These curves are used to correct the velocity spectra at each receiver and determine the radiated seismic energy. Epicentral distance is marked at the right end of each curve in meters. Receivers at short distances seem to have unusual shapes in attenuation curves compared with constant Q-attenuation curves.
different constant Q values (Fig. 13a). The MEGF changes the slope of the M0–fc relation, whereas the results from constant Q do not. The best agreement between the results from different constant Q values and results from the MEGF method are achieved with Q ⳱ 350. For small events (M0 108 N m), the MEGF method gives lower moments, whereas for large events (M0 108 N m) the MEGF yields similar or higher moments. The corner frequencies become slightly larger for high M0 events, whereas they show no distinct trend for low M0 events (Fig. 13). In comparison with Q ⳱ 350, the MEGF method changes the slope of the M0 f c3 relation toward M0 f c3Ⳮe, where e is a constant between 0 and 1 (Kanamori and Rivera, 2003). However, the scatter in the data is very large, and more reliable data are needed to clarify this question. The scaling relations discussed so far are independent of the location, that is, whether they occurred in C1 or C2. Regarding the radiated energy derived from MEGF, events from C2 are only slightly changed by the attenuation correction compared with the Q ⳱ 350 model, whereas the radiated energies determined for C1 events are much higher; for small events the energies approximately double (Table 2). In Figure 13b, the ratio Erad /M0 is more or less constant at about 10ⳮ6 for all data. However, in comparison with the results from the MEGF approach, the results determined by constant Q models seem to increase slightly with M0.
Figure 12. Data points shown are determined by using constant Q ⳱ 350 and fitting an x2 spectral model. (A) Data fit the general M0 fc3 scaling relation, where lines of constant M0 fc3 indicate constant stress drops. (B) Brune stress drop varies from about 0.01 to 1 MPa, slightly increasing with M0. (C) Energy-tomoment ratio seems to increase with Mw.
Discussion and Conclusions The determination of source parameters is always affected by ambiguities and trade-offs, in particular, with respect to the connection between fc and Q. The determined constant Q values between events within one cluster vary
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Figure 13. (A) M0–fc relation for events in Table 2, calculated by different methods: Ⳮ, Q ⳱ 200 (x2 fit); open triangle, Q ⳱ 350 (x2 fit); ⳯, Q ⳱ 800 (x2 fit); filled triangle, Q(f) based on MEGF. The results from the MEGF method indicate a slightly different slope for the M0 fcⳮ(3Ⳮe) relation. Solid lines are e ⳱ 0; dashed lines stand for e ⳱ 1. (B) Energyto-moment ratio based on MEGF is almost constant, whereas estimates from methods with constant Q may indicate an increasing ratio with M0.
from 200 to more than 1500 (Fig. 8), and the spectral decay method indicates an increasing Q with frequency (Fig. 9). Therefore, we applied the MEGF method to compensate for attenuation. The choice of the appropriate attenuation or Q model is even more crucial for the estimation of Erad than for fc (Fig. 13b). A compilation of amplitude velocity spectra for different attenuation corrections reveals the different nature between constant Q and frequency-dependent Q(f) (Fig. 14). The velocity spectra are recorded at station 3, where we also estimated Q(f) with the “two-station” spectral decay method. From Figure 9 we assume a simple Q(f), which is increasing linearly from 10 at 10 Hz to 500 at 1000 Hz, in a log–log scale. The velocity spectrum corrected with the frequency-dependent Q is close to the spectrum corrected via the MEGF method. The velocity spectra that are corrected for constant Q cannot achieve a spectral shape similar to the
velocity spectrum that is derived from MEGF (Fig. 14). Since we assume that station and path effects are compensated by MEGF, the estimates for source parameters derived from MEGF can probably treat attenuation best. Ide and Beroza (2001) showed that the increase of apparent stress with Mw could be caused by the underestimation of the radiated energy because of limitations of the frequency bandwidth. Ide et al. (2003) re-evaluated data using the MEGF method, which previously were interpreted by Prejean and Ellsworth (2001) using constant Q models. The re-evaluation concludes with constant apparent stress, whereas the previous work stated an increasing apparent stress with increasing seismic moment. Also our results for a constant Q ⳱ 350 show slightly increasing Erad /M0 with increasing seismic moment (Fig. 12c), and constant Erad /M0 applying the MEGF method (Fig. 13b). However, the max-
Source Parameters and Scaling Relations for Mining-Related Seismicity within the Pyha¨salmi Ore Mine, Finland
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Figure 14. Corrections for velocity amplitude spectra that are used to calculate the radiated seismic energy. The example is from an event within cluster C1 (C1 1) as recorded at station 3. Thin solid line, original velocity spectrum; thin dashed lines, corrected for Q ⳱ 800, 350, and 200; thick dash-dotted line, corrected with Q(f) from spectral decay method; thick solid line, corrected for Q(f) from MEGF method. The slopes of the MEGF and the spectral decay methods are similar at high frequencies.
imum difference of Erad /M0 that we observe for a constant Q, and the frequency-dependent Q(f) is only about a factor of 3. The data processed with the MEGF method cover only a relatively small-magnitude range (ⳮ1.2 Mw 0), but they supplement earlier studies that focused mainly on larger events. Figure 15 shows our results together with data obtained by Richardson and Jordan (2002), Ide et al. (2003, 2004), Gibowicz et al. (1991) and with results determined and compiled by Kanamori and Rivera (2004). The upper bound for Erad /M0 is relatively well defined at 10ⳮ4 for all data sets, whereas the lower bound decreases with decreasing Mw for many data sets. Because we have a greater confidence in the Erad /M0 values that are determined by the MEGF method, we only include these in the composite diagram in Figure 15. Since our results are about 1 to 2 orders of magnitude smaller than the Erad /M0 ratios for large earthquakes, an increase of Erad /M0 with Mw could link our results to the ones of large earthquakes. Such an increase in Erad / M0 can be achieved by a change in the scaling relation M0 f c3 to M0 f c3Ⳮe as mentioned above in association with Figure 13a (Kanamori and Rivera, 2004).
On the other hand, not all data in Figure 15 follow this modified scaling relation. Aftershocks of the Izutani and Kanamori (2001) data, analyzed by Ide et al. (2004), indicate that the trend of decreasing Erad /M0 with decreasing Mw may not continue for Mw 3. Moreover, the results of Abercrombie (1995) and Richardson and Jordan (2002) are based on constant Q models, so that a direct comparison with our results needs to be done with care. Our study reveals reliable Erad /M0 values of about 5 • 10ⳮ7 at Mw ⳮ1, which is about 1 to 2 orders of magnitude smaller than results from Ide et al. (2003, 2004) at Mw 1 to 3. One possible explanation for this difference could be the mechanism of the earthquakes analyzed in this study. Since the events in the cluster C1 occurred at an open ore pass and the ones in C2 are in the most active mining area, the events might be related to a direct opening of the rock matrix toward the cavities. This could in turn lead to a more tensile character of the earthquake’s source mechanisms, thereby leading to a smaller part of the energy being transferred to shear-wave energy as compared with a pure shear-source mechanism. The seismic efficiency g is defined as the ratio of the radiated seismic wave energy Erad to the change in strain
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Figure 15. Energy-to-moment ratios for several different data sets, showing a variation of about 3 orders of magnitude over a range of 11 moment magnitudes. The results from the present study include events from clusters C1 and C2, assuming Q(f) based on the MEGF method (gray diamonds). Black lines indicate trends for Erad /M0 for the scaling relation M0 fcⳮ(3Ⳮe), with e ⳱ 0, 0.5, and 1. Results from other studies include deep South African gold mines (Richardson and Jordan, 2002), the Long Valley Caldera, California (Ide et al., 2003), the Hi-net, Japan (Ide et al., 2004), 18 events that are representative for the Underground Research Laboratory data (Gibowicz et al., 1991), and shallow crustal earthquakes in California and Japan (compiled by Kanamori and Rivera [2004]).
energy DW at the source. Because the absolute values of stresses at the fault before (r1) and after the rupture (r2) are practically impossible to determine by seismological methods alone, we follow the approach of Savage and Wood (1971). The seismic efficiency can be expressed by the ratio g ⳱ rapp /r¯ , with r¯ ⳱ (r1 Ⳮ r2)/2 (Kasahara, 1981). We calculate the seismic efficiency using an average stress of r¯ 40 MPa (Ledger, 1999) and receive values far below the proposed maximum seismic efficiency of 0.06 (McGarr, 1999); in fact, we find g ⳱ 0.0014 Ⳳ 0.0012 with a maximum of 0.0052, using results from the MEGF method. For the full data set of 1500 events using constant Q ⳱ 350 and the x2 fit, we get a mean value for g of 0.0021 Ⳳ 0.0091, with only 10 outliers above the 6% limit. Assuming overshooting, in other words the slipped fault locks again at a stress r2 somewhat less than the dynamicfriction stress, the Savage–Wood inequality is valid rapp 0.5 DrB (Savage and Wood, 1971; McGarr, 1999; Beeler et al., 2003). All our estimated data indicate overshooting, with rapp /DrB 0.27 Ⳳ 0.12 for the spectral ratio events and rapp /DrB 0.19 Ⳳ 0.02 for all 1500 events. Beeler et al. (2003) assume constant stress drop and show that apparent stress then increases linearly with stress drop. We have demonstrated in this study that constant Q
models are not adequate to describe the attenuation in the Pyha¨salmi mine. Source parameters estimated on the basis of this assumption will be biased and may affect scaling relations. The MEGF method used in this study accounts in a better way for the actual attenuation and is therefore more reliable for determination of source parameters and scaling relations. Energy-to-moment ratios based on constant Q models indicate an increase in apparent stress with magnitude. In contrast, the results based on the MEGF method seem to yield constant apparent stresses. Because the magnitude range of this study’s Erad /M0 ratios is limited, we compare them with other studies over a larger range of magnitudes. We find that our results are 1 to 2 orders of magnitude smaller than the ratios resulting from large earthquakes and from other MEGF studies (Fig. 15). However, because of the large scatter in the compiled results our conclusion is that it is not yet possible to choose clearly between a constant stress drop or an increasing stress drop model.
Acknowledgments We thank Katja Sahala, Ilpo Ma¨kinen, and Timo Ma¨ki, INMET Mining, and Errol de Kock, Integrated Seismic Systems (ISS), for access to the
Source Parameters and Scaling Relations for Mining-Related Seismicity within the Pyha¨salmi Ore Mine, Finland Pyha¨salmi mine data and for generous support and assistance in this connection. We also thank Hiroo Kanamori for providing us with the data from Kanamori and Rivera (2004), used in Figure 15, and Tormod Kværna, Lars Lind and Steven Gibbons for discussions and help during processing. We thank G. Prieto, S. Ide, and A. McGarr for constructive comments that clearly increased the quality of the article. V.O. was supported by a grant from the Research Council of Norway, and this work has also received support from U.S. Department of Energy Award DE-FC03-02SF22636.
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Manuscript received 23 August 2004.
