Abstract

In order to use historic earthquakes to estimate future seismicity of a region, the analyst usually first converts earthquake sizes that have been recorded in various ways (e.g., mb, MS, ML, and I0) to a common scale (e.g., mb), by fitting the coefficients ai and bi in an assumed linear relationship mb = ai + bimi between mb and each of the other size measures mi. He then combines mb values (either recorded mb or fitted m^b) for all earthquakes to estimate a Gutenberg-Richter rate parameter, αmb, and distribution parameter, βmb. However, even if magnitude scales are linearly related over some range, when observational errors are present and earthquakes have been incompletely recorded at lower magnitudes or terminate abruptly at some finite mmax, estimates âi, b^i may be highly biased and uncertain. Less biased estimates of b^i can be obtained if only magnitude values in the interior of the range are used in the fitting. Errors in b^i carry over into estimates βmb(i) obtained using magnitudes converted to m^b from mi. This means that even if a large number, N, of mi values are converted to mb values, the variance of β^mb(i) using these magnitudes may depend primarily on the sizes of the observational errors and on n the number, of pairs (mb and mi) used to fit b^i, rather than on N. An estimate βmb(i) may be obtained without converting magnitudes mi to mb by estimating β^i (the distribution parameter for mi) using the N mi values and then setting β^mb(i)=β^i/b^i. Individual estimates 1/βmb(i) for various scales, mi, may be weighted and combined, and then inverted to obtain an estimate of βmb that has a lower variance than does the estimate obtained using all the earthquakes simultaneously. This study particularly considers estimates involving I0, epicentral intensity, because often I0 is the only size recorded for an earthquake. A relatively large range of magnitudes corresponds to a single intensity, making the analysis more complicated for this case.

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