# Seismological Research Letters

- © 2005 by the Seismological Society of America

## INTRODUCTION

The most common catalog magnitude for eastern North America is the Nuttli magnitude (Nuttli, 1973), referred to as *M _{N} or M_{bLg}*. The Nuttli magnitude is based on the amplitude of the Lg phase (multiply reflected and refracted shear waves).

*M*is quickly and easily determined from the time series, making it convenient for routine catalog purposes. The moment magnitude scale (Hanks and Kanamori, 1979) is generally preferred by seismologists for many applications, as it is more reflective of the actual size of the earthquake. Moment magnitude (

_{N}**M**), however, is more difficult to routinely determine reliably, especially for small to moderate earthquakes. Thus it is useful to develop empirical relationships between

**M**and

*M*, so that

_{N}**M**can be estimated for events for which only

*M*has been determined.

_{N}Atkinson (1993) developed the empirical relationship (1) for eastern North American earthquakes within the magnitude range of 4 < *M _{N}* < 6. In this note, we extend this relationship below

*M*4, using well recorded small earthquakes in the Charlevoix seismic zone (CSZ) of southeastern Canada, where the relatively dense Charlevoix array of the Canadian National Seismographic Network (CNSN) has been active since 1994. Seven seismographic stations comprise the Charlevoix array, separated by no more than 40 km, as shown in Figure 1; events of magnitude as small as

_{N}*M*1.0 are usually well recorded (Lamontagne, 1999).

_{N}We combine the calculated moment magnitudes from the CSZ earthquakes of small magnitude and the data presented by Atkinson (1993) to form a new, more robust equation that is applicable for earthquakes of 1 ≤ *M _{N}* ≤ 6.

## DATABASE

The Geological Survey of Canada (GSC) produces a Web listing of all earthquakes occurring within Canada, which includes the larger events occurring in the northern United States (http://www.seismo.nrcan.gc.ca/). Based on this list, ∼350 earthquakes of magnitude greater than 1 (either *M _{L}* or

*M*) occurred in the CSZ between June 1999 and April 2002, as shown in Figure 1. This data set was further limited by Sonley (2004) to earthquakes within co-located clusters exhibiting similar waveforms, termed multiplets (see Lamontagne, 1999). (Note: The selection of multiplet events was made for purposes not related to this study.) Seismographic waveform data for these earthquakes were collected from the three-component, seven-station array in the CSZ (part of the Canadian National Seismographic Network). The array used short-period instruments prior to October 2000 but has used broadband instruments since that time. The instrument response curves are well determined for both instrument types.

_{N}## METHOD

Apparent source spectra were determined from the Fourier spectra of the shear-wave window based on the direct method (for full details see Sonley and Atkinson, 2001; Sonley, 2004). In this method, empirically determined regional values of attenuation are used to remove path effects from the observed spectra according to the equation (2) where *R* is hypocentral distance, β is the shear-wave velocity (3.8 km/s, after Atkinson, 1993) and *Q* is the quality factor (*Q*= 680*f*^{0.36}, after Atkinson and Boore, 1995). Site effects, *S*(*f*), are minimal, as the instruments are all sited on hard rock; the modest site amplifications are estimated by the horizontal to vertical (*H/V*) ratios (determined empirically by Siddiqqi and Atkinson, 2002) and applied as a correction to the horizontal component amplitudes (*e.g., S*(*f*) = *H/V*(*f*)). We assume that the vertical-component spectrum is approximately equal to the unamplified (*e.g.*, no site effects) horizontal component. Thus we have for each station three estimates (two corrected horizontal and one vertical) of the apparent source spectruna.

The apparent source displacement spectra (*D*) are then fit to the Brune point-source model (Brune 1970, 1971, as described by Boore, 1983): (3) where *T _{p}* is the rms value of the radiation pattern for shear waves (0.55),

*F*is the flee-surface amplification for vertical incidence (2.0),

*V*is the amplitude partition onto a randomly oriented horizontal component (0.71), is the crustal density (2.8 g/cm

^{3}, after Atkinson, 1993),β and

*R*are defined as above,

*M*

_{0}is the seismic moment (in dyne-cm), and

*f*

_{0}is the corner frequency (in Hz).

*M*

_{0}is determined from the long-period displacement level, for

*f/f*< < 1, for which

_{0}*D*(

*f*) →

*CM*

_{0}. From

*M*

_{0},

**M**is determined according to (4) (after Hanks and Kanamori, 1979). Note that Equation 2 assumes body-wave spreading (R

^{-l}), although recent studies (Atkinson, 2004; confirmed by Sonley, 2004) show empirically that

*R*

^{-1.3}may be more appropriate for distances less than 70 km. The effect of uncertainty in the geometric spreading is examined later.

## RESULTS

The moment magnitudes determined for earthquakes in this study can be used to extend the Atkinson (1993) relation over a broader *M _{N}* range, from

*M*> 1. A linear, least-squares regression was performed on a combined data set including both the Atkinson (1993) study events (

_{N}**M**> 4) and the data described above and the following empirical relationship was determined: (5) Equation 5 predicts

**M**from

*M*for 1 ≤

_{N}*M*≤6, within a standard deviation of 0.1 units. The two empirical relationships (Equations 1 and 5) are nearly identical, as shown in Figure 2. This is remarkable considering the very different magnitude ranges of the two studies and suggests that this empirical relationship is well constrained over a broad magnitude range.

_{N}

Recent studies (Atkinson, 2004; Sonley and Atkinson, 2005) have shown empirically that *R*^{-1.3} is more appropriate in Equation 2 for distances less than 70 km. For the average observation distance of about 30 km, the use of the steeper attenuation model would imply an increase in **M** by about 0.3 units. We would also expect to see a clear trend in the magnitude residuals plotted against distance, with near stations showing positive magnitude residuals and distant stations showing negative residuals. Figure 3 shows the magnitude residuals versus distance, measured at every station for each earthquake. Symbols with error bars show the mean and standard deviation of the residuals in 10-km distance bins. There is a slight tendency to positive residuals at the closest distances, suggesting underestimation of the near-source amplitude (and hence the actual magnitude). We conclude that the moment-magnitude determinations for the small events may be weakly biased (too low) by about 0.1 units. This potential bias is small relative to the overall scatter (see Figure 2). In conclusion, moment magnitude can be estimated from *M _{N}* for events of 1 ≤

*M*using Equation 5, with an uncertainty of about 0.2 units.

_{N}≤ 6## Acknowledgments

This research was supported by the Natural Sciences and Engineering Research Council of Canada. Data were provided by the Geological Survey of Canada.

## Footnotes

**Carleton University**↵* Now at Boston University