The standard error of the magnitude-frequency<i>b</i>value

TLDR

Estimated b values in the Gutenberg–Richter relation are widely used for seismicity comparison and risk assessment, yet their uncertainties have been poorly characterized. The study presents the F probability density distribution for b and compares it with an asymptotic form for temporally varying b. Convenient tables for the standard error of b are provided, allowing statistical tests of temporal and spatial variations, with σ(b̂)=2.30 b² σ( M̄ ) for large samples and slow changes. In a central California example, stable b estimates require about 100 earthquakes; the average b over 1952–1978 was 0.95 (90 % CI 0.65–1.89), b fluctuated significantly between 0.60 and 1.39, and these variations showed no correlation with large earthquakes.

Abstract

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 ^ ) = 2.30 b 2 σ ( M _ ) , where σ 2 ( M _ ) = ∑ i − 1 n ( M i − M _ ) 2 / n ( n − 1 ) . 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 &amp;gt; 5) and b variations occurred.