Maximum likelihood estimation of b values for magnitude grouped data

abstract The b value estimated by fitting a set of observed earthquake magnitudes to the magnitude-frequency relationship, log N(m) = a - bm, where N(m) = number of earthquakes exceeding magnitude m, is correlated with the fitting technique used. Both so-called interval and cumulative least-squares fits to the formula log N(m) = a - bm tend statistically to estimate too low a b value, because they cannot include magnitudes above the maximum observed. Maximum likelihood formulas (Aki, Utsu, and Page) for exact or continuous magnitudes give biased results if they are applied to interval data, with the bias increasing as interval size increases. The bias is small at magnitude intervals Δm = 0.1, but significant if the formulas are applied to magnitudes which have been recovered from historic intensity data at intervals of 0.6 magnitude unit. Corrections for interval size can be applied to the continuous data formulas to make them equivalent to the formula derived specifically for grouped data (e.g., Karnik, 1971). A simpler form of the grouped data formula is derived here and shows the role of interval size and maximum magnitude on the b value obtained. This paper also shows how, given a population value of b, to calculate the distribution of the estimated b values. Conversely, this paper derives an a posteriori distribution for the population b value, given the magnitudes of an observed set of earthquakes. The distribution of b values fitted by various techniques is illustrated for a number of cases. Several illustrations of probabilistic ground motions calculated for a range of b values show that a small fractional change in the assumed b value can have a substantially larger fractional effect on the ground motion calculated.