Improved methods for single-and multiple-event hypocenter determinations are developed and applied to the problem of locating earthquake clusters in the South-Central Pacific Ocean. Bayesian statistical methods are used to incorporate a priori information about arrival-time variance into the derivation of hypocenter confidence ellipsoids, permitting a more realistic calculation of critical parameters in the case where the number of stations is small. The diagonal elements of certain projection operators, called “data importances” by Minster et al. (1974), are used to evaluate network balance. The hypocentroid of an event cluster is defined to be the average location of events within the cluster, and the deviations of individual hypocenters from the hypocentroid are called cluster vectors. The problem of estimating the cluster vectors can be decoupled from the problem of estimating the hypocentroid by a simple but fundamental mathematical result, here termed the hypocentroidal decomposition theorem. The algorithm based on this analysis appears to have many advantages over other published methods for multiple-event location, both in its efficient use of available information and its computational speed. The application of this method to three clusters of shallow intraplate seismicity in the South-Central Pacific, designated Regions A, B, and C, demonstrates that the seismicity within each cluster is very localized; the rms lengths of the cluster vectors for each group of epicenters are estimated to be only 9, 6, and 12 km, respectively. Estimates of the epicentroids are

 Region A (4 events) 7.40 ± 0.023°S 148.21 ± 0.023°W Region B (3 events) 18.40 ± 0.018°S 132.87 ± 0.018°W Region C (7 events) 20.76 ± 0.015°S 126.95 ± 0.019°W
 Region A (4 events) 7.40 ± 0.023°S 148.21 ± 0.023°W Region B (3 events) 18.40 ± 0.018°S 132.87 ± 0.018°W Region C (7 events) 20.76 ± 0.015°S 126.95 ± 0.019°W

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