Alternative representation of the Gutenberg–Richter relation in terms of the logarithmic mean annual seismicity rate and its standard deviation

Gutenberg and Richter developed an empirical relation, $$\log_{10} N(M) = a - bM$$ log 10 N ( M ) = a - b M , to quantify the seismicity rate of various magnitudes in a given region and time period. They found the equation fit observed data well both globally and for particular regions. In conventional G–R relation, N(M) represents an arithmetic mean. As a result, the arithmetic standard deviation cannot be explicitly incorporated in the log-linear G–R relation. Moreover, this representation is susceptible to influence of spuriously large numbers of aftershocks of major earthquake sequences. To overcome these shortcomings, we propose an alternative representation of the G–R relation in terms of the logarithmic mean annual seismicity rate and its standard deviation. We select the crustal earthquake data from 1973 to 2011, as listed in the National Earthquake Information Center (NEIC) global catalog and the Central Weather Bureau (CWB) Taiwan regional catalog, to illustrate our methodology. We first show that by using the logarithmic annual seismicity rates we can significantly suppress the influences of spuriously large numbers of aftershocks following major earthquake sequences contained in the Taiwan regional catalog. More significantly, both the logarithmic mean annual seismicity rate and its standard deviation can be explicitly represented in the Gutenberg–Richter relation as follows: $${\text{For}}\,{\text{global}}\,{\text{crustal}}\,{\text{seismicity}}{:}\;\log_{10} N = 8.14 - 1.03M \pm (0.04M - 0.13);$$ For global crustal seismicity : log 10 N = 8.14 - 1.03 M ± ( 0.04 M - 0.13 ) ; $${\text{For}}\,{\text{Taiwan}}\;{\text{crustal}}\,{\text{seismicity}}{:}\;\log_{10} N = 5.62 - 0.90M \pm (0.02M + 0.17)$$ For Taiwan crustal seismicity : log 10 N = 5.62 - 0.90 M ± ( 0.02 M + 0.17 ) where log10 N represents the logarithmic annual seismicity rate. Above analytical equations are very well constrained by observed global seismicity data with $$5.0 \le M \le 7.0$$ 5.0 ≤ M ≤ 7.0 and by Taiwan seismicity data with $$3.0 \le M \le 5.0$$ 3.0 ≤ M ≤ 5.0 . Both equations can be extrapolated with confidence to simultaneously estimate not only the median annual seismicity rates but also their uncertainties for large earthquakes for the first time since inception of the G–R relation. These equations can be used to improve the conventional probabilistic seismic hazard assessment by including the dispersion of the annual seismicity rate. Finally, the corresponding numerical median annual seismicity rate with its upper and lower bounds obtained from above equations for $$5.0 \le M \le 9.0$$ 5.0 ≤ M ≤ 9.0 is listed in Table 1. Table 1 Observed and estimated median annual seismicity rate and return period with their dispersions for Taiwan and global crustal earthquakes Magnitude Catalog Taiwan catalog (CWB) Taiwan catalog (CWB) Global catalog (NEIC) Global catalog (NEIC) Annual rate (event/year) Return period (year) Annual rate (event/year) Return period (year) M ≥ 5.0 24.55 13.18 7.08 0.041 0.076 0.14 1148.16 977.24 831.76 0.0009 0.001 0.0012 M ≥ 5.5 8.91 4.68 2.45 0.11 0.21 0.41 367.28 298.54 242.66 0.0027 0.0033 0.0041 M ≥ 6.0 3.24 1.66 0.85 0.31 0.60 1.18 117.49 91.20 70.79 0.0085 0.011 0.014 M ≥ 6.5 1.17 0.59 0.30 0.85 1.69 3.33 37.58 27.86 20.65 0.027 0.036 0.048 M ≥ 7.0 0.43 0.21 0.10 2.33 4.76 10.0 12.02 8.51 6.03 0.083 0.12 0.17 M ≥ 7.5 0.15 0.074 0.036 6.67 13.51 27.78 3.85 2.60 1.76 0.26 0.38 0.57 M ≥ 8.0 0.056 0.026 0.012 17.86 38.46 83.33 1.23 0.79 0.51 0.81 1.27 1.96 M ≥ 8.5 0.020 0.009 0.004 50.00 111.11 250.0 0.39 0.24 0.15 2.56 4.17 6.67 M ≥ 9.0 0.0074 0.0033 0.0015 135.14 303.03 666.67 0.13 0.074 0.04 7.69 13.51 25.00 Observed value is shown in bold number, estimated value in regular number $$\log_{10} N = 5.62 - 0.90M \pm (0.02M + 0.17)$$ log 10 N = 5.62 - 0.90 M ± ( 0.02 M + 0.17 ) for Taiwan crustal earthquakes $$\log_{10} N = 8.14 - 1.03M \pm (0.04M - 0.13)$$ log 10 N = 8.14 - 1.03 M ± ( 0.04 M - 0.13 ) for global crustal earthquakes