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Effects of observational errors in relating magnitude scales and fitting the gutenberg-richter parameter ß

U.S. GEOLOGICAL SURVEY, BOX 25046, M.S. 966
DENVER FEDERAL CENTER, DENVER, COLORADO 80225

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., m_b, M_S, M_L, and I₀) to a common scale (e.g., m_b), by fitting the coefficients a_i and b_i in an assumed linear relationship m_b = a_i + b_im_i between m_b and each of the other size measures m_i. He then combines m_b values (either recorded m_b or fitted ) for all earthquakes to estimate a Gutenberg-Richter rate parameter, _mb, and distribution parameter, ß_mb. 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 m_max, estimates â_i, may be highly biased and uncertain. Less biased estimates of can be obtained if only magnitude values in the interior of the range are used in the fitting. Errors in carry over into estimates ß_mb(i) obtained using magnitudes converted to from m_i. This means that even if a large number, N, of m_i values are converted to m_b values, the variance of using these magnitudes may depend primarily on the sizes of the observational errors and on n the number, of pairs (m_b and m_i) used to fit , rather than on N. An estimate ß_mb(i) may be obtained without converting magnitudes m_i to m_b by estimating (the distribution parameter for m_i) using the N m_i values and then setting . Individual estimates 1/ß_mb(i) for various scales, m_i, may be weighted and combined, and then inverted to obtain an estimate of ß_mb 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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