abstract

Estimated b values in log N = a − bM are widely used in seismicity comparisons and risk analysis, but uncertainties have been little explored. In this paper, the usual F probability density distribution for b is given and compared with an asymptotic form for temporally varying b. Convenient tables for the standard error of b are given that allow statistical tests to accompany investigations of both temporal and spatial variations of b. With large samples and slow temporal changes in b, the standard error of b is

 
σ(b^)=2.30b2σ(M_),

where

 
σ2(M_)=i1n(MiM_)2/n(n1).

In an example from central California, stable estimates of b require a space-time window containing about 100 earthquakes. From 1952 to 1978, the average b and 90 per cent confidence limits are 0.95 (+0.94, −0.30). Some fluctuations of b are statistically significant but some are not. Within 90 per cent confidence limits, b changes from a low of 0.60 (+0.11, −0.09) in 1955 to a high of 1.39 (+0.25, −0.21) in 1967 and drops to 0.72 (+0.13, −0.10) in 1975. In this example, no correlation between large earthquakes (M > 5) and b variations occurred.

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