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{{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)<ref name="Hutton 1987">HUTTON, L K, and BOORE, D M. 1987. The ML scale in southern California. ''Bulletin of the Seismological Society of America'', 77, 2074–2094.
</ref> is currently used for determination of earthquake magnitude in the UK.

[[Image:OR17062equation1.jpg|frameless|center|300px| ]]

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.

[[Image:OR17062fig6.jpg|thumb|center|300px| '''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)<ref name="Ottemöller 2003">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.</ref>, 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

[[Image:OR17062equation2.jpg|frameless|center|200px| ]]

Following Herrmann and Kijko (1983)<ref name="Herrmann 1983">HERRMANN, R B, and KIJKO, A. 1983. Modeling some empirical vertical component Lg relations. ''Bulletin of the Seismological Society of America'' 73, 157–171.</ref>, we assume geometrical spreading for S- and Lg-waves

[[Image:OR17062equation3.jpg|frameless|center|300px| ]]

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<ref name="Singh 1982">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.</ref>)

[[Image:OR17062equation4.jpg|frameless|center|225px| ]]

where ''T'' (sec) is the travel time. We used the United Kingdom average attenuation model derived for ''Lg'' waves of Sargeant and Ottemöller (2009)<ref name="Sargeant 2009">SARGEANT, S, and OTTEMÖLLER, L. 2009. Lg wave attenuation in Britain. ''Geophys. J. Int''. 179, no. 3, 1593–1606.</ref>

[[Image:OR17062equation5.jpg|frameless|center|150px| ]]

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)<ref name="Ottemöller 2010">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</ref> we use κ=0.02 sec.

The seismic moment is then given by

[[Image:OR17062equation6.jpg|frameless|center|165px| ]]

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<ref name="Aki 1967">AKI, K. 1967. Scaling law of seismic spectrum. ''J. Geophys. Res.,'' 72, 1217–1231.</ref>; Brune, 1970<ref name="Brune 1970">BRUNE, J N. 1970. Tectonic stress and the spectra of seismic shear waves from earthquakes. ''J. Geophys. Res.,'' 75, 4997–5009. </ref>) for the shape of the earthquake source spectrum ''S(f)'', where

[[Image:OR17062equation7.jpg|frameless|center|165px| ]]

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<ref name="Brune 1970"></ref>)

[[Image:OR17062equation8.jpg|frameless|center|150px| ]]

The stress-drop Δσ (bar) is given by

[[Image:OR17062equation9.jpg|frameless|center|150px| ]]

assuming a circular fault (Eshelby, 1957<ref name="Eshelby 1957">ESHELBY, J. 1957. The determination of the elastic field of an ellipsoidal inclusion and related problems. ''Proc. R. Soc. London A'' 241, 376–396.</ref>). 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.

[[Image:OR17062fig7.jpg|thumb|center|500px| '''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==
{{reflist}}
[[Category: OR/17/062 The Moidart earthquakes of 4 August 2017 | 04]]

OR/17/062 Magnitude - Revision history

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Ajhil at 13:40, 21 March 2018