OR/17/062 Magnitude

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Baptie, B, Ford, G, and Galloway, D. 2017. The Moidart earthquakes of 4 August 2017. British Geological Survey Internal Report, OR/17/062.

Local magnitude

A local magnitude of 4.0 ± 0.2 ML was computed for the mainshock as the average of estimates derived from the largest S-wave amplitudes at 24 stations in the distance range 62 km to 588 km. The amplitudes were measured on horizontal component simulated Wood–Anderson records. Amplitudes were measured on both horizontal components giving 48 readings. The ML formula derived for California by Hutton and Boore (1987)[1] is currently used for determination of earthquake magnitude in the UK.

where the amplitude A is in nanometres and the distance r is in kilometres.

A histogram determined from the individual magnitude readings is shown in Figure 6.

Figure 6    Histogram of individual magnitude readings.

Moment magnitude

We determined the seismic moment, M0, and the stress drop, Δσ, of the mainshock by modelling the source displacement spectra. We use the spectral fitting method of Ottemöller and Havskov (2003)[2], where M0 and the corner frequency, fc, are determined using a grid search. The observed spectra, A(f), are corrected for instrument response and for the effects of geometrical spreading, G, and frequency dependent attenuation, D(f). We also correct for the effect of the free surface, F, and the source radiation pattern, R, using correction factors of 2.0 and 0.6, respectively. The instrument response the observed displacement spectrum is given by

Following Herrmann and Kijko (1983)[3], we assume geometrical spreading for S- and Lg-waves

where r (km) is the hypocentral distance.

The correction for attenuation D(f) is commonly constructed in two parts. The first part accounts for attenuation along the path described by Q(f) and the second accounts for near-surface attenuation κ (sec) near the receiver (Singh et al., 1982[4])

where T (sec) is the travel time. We used the United Kingdom average attenuation model derived for Lg waves of Sargeant and Ottemöller (2009)[5]

This attenuation model was derived assuming the same geometrical spreading as used here. In this analysis we use only vertical component data, which means that correction for site amplification is not required. Following Ottemöller and Sargeant (2010)[6] we use κ=0.02 sec.

The seismic moment is then given by

where we use the density ρ=2.7 g/cm3 and the S-wave velocity at the source vs = 3.5 km/sec, and A0 is the amplitude of the flat part of the spectrum A(f).

We use the ω2 model (Aki, 1967[7]; Brune, 1970[8]) for the shape of the earthquake source spectrum S(f), where

The observed and modelled spectra are shown in Figure 7. In general the modelled spectra provide a good fit for the observations at all stations. Similarly, the observed spectra show good signal-to-noise levels at frequencies above the corner frequency. The average value for Mw determined from the 14 observations is 3.6 ± 0.1.

We compute the source radius r (km) from the corner frequency fc (Brune, 1970[8])

The stress-drop Δσ (bar) is given by

assuming a circular fault (Eshelby, 1957[9]). We find average values for the source radius and stress drop of 0.572 ± 0.277 km and 17.2 ± 14.5, respectively. The large uncertainty in the stress drop reflects the station-to-station variability of the corner frequency measurement.

Figure 7    Observed displacement spectra (black) at the stations used to determine Mw. The red line shows the modelled displacement spectrum and the grey line shows the amplitude of the noise.

References

  1. HUTTON, L K, and BOORE, D M. 1987. The ML scale in southern California. Bulletin of the Seismological Society of America, 77, 2074–2094.
  2. OTTEMÖLLER, L, and HAVSKOV, J. 2003. Moment magnitude determination for local and regional earthquakes based on source spectra. Bulletin of the Seismological Society of America, 93, 203–214.
  3. HERRMANN, R B, and KIJKO, A. 1983. Modeling some empirical vertical component Lg relations. Bulletin of the Seismological Society of America 73, 157–171.
  4. SINGH, S K, APSEL, R J, FRIED, J, and BRUNE, J N. 1982. Spectral attenuation of SH waves along the Imperial fault. Bulletin of the Seismological Society of America, 72, 2003–2016.
  5. SARGEANT, S, and OTTEMÖLLER, L. 2009. Lg wave attenuation in Britain. Geophys. J. Int. 179, no. 3, 1593–1606.
  6. OTTEMÖLLER, L, and SARGEANT, S. 2010. Ground-Motion Difference between Two Moderate-Size Intraplate Earthquakes in the United Kingdom. Bulletin of the Seismological Society of America, 100, 4, 1823–1829
  7. AKI, K. 1967. Scaling law of seismic spectrum. J. Geophys. Res., 72, 1217–1231.
  8. 8.0 8.1 BRUNE, J N. 1970. Tectonic stress and the spectra of seismic shear waves from earthquakes. J. Geophys. Res., 75, 4997–5009.
  9. ESHELBY, J. 1957. The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. R. Soc. London A 241, 376–396.