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Automatic processing of seismic data is today a key element in the efforts to achieve high quality seismic systems. Automated procedures for locating seismic events with a network including arrays and single element seismometers usually incorporate back-azimuth estimates, arrival-time data, and associated uncertainties into a least-squares-inverse location algorithm. Such an algorithm is quite cumbersome and requires expanding a set of non-linear equations in a Taylor series. Second-order terms usually not included in the algorithm can be important if the initial estimate is far from the solution.