The Theory of Multi-Earthquake Location by Least Squares and Applications to Groups of North Island, New Zealand Mantle Earthquakes

Abstract

Aspects of the standard least squares method of locating earthquakes and its extensions are discussed. It is shown that there is a need to carefully separate and distinguish between the statistical and deterministic properties of the least squares solution and the algorithm used to obtain it. Standard linear statistical analysis gives reasonable confidence regions for the hypocentre provided that the errors in the model travel time to pairs of stations are not correlated. The travel time residuals which result from the overdetermined system are unreliable estimates of the model errors, as are the pooled residuals from groups of events whether or not the data are homogeneous. The concepts of Absolute and Relative hypocentre determination are clarified and the Homogeneous Station method is developed and demonstrated to be a good relative location method. The application of the method to a group of North Island, New Zealand subcrustal earthquakes chosen for homogeneity revealed that the earthquakes occurred in a thin, fairly that dipping zone that could be as thin as 9 km and is not thicker than 18 km. The result is a significant refinement of previous estimates for New Zealand. The method of Joint Hypocentre Determination first described by Douglas (1967) is examined. The advantage of the method is that the error in the travel time model is estimated as well as allowing for and estimating the effect of an interaction of this error with the hypocentre parameters of the earthquakes. The application of this method to groups of, North Island, New Zealand earthquakes allows very significant improvements to the travel time model to be made and confirms the result that there is a velocity contrast for both P and S of between six and ten percent between paths in and entirely out of the downgoing Pacific plate. Estimates of the velocities in the plate are 8.6±.1 km/sec. for P and 4.73± km/sec. for S. In addition, station terms are calculated which describe the average departure from the new model of travel times to the stations contributing data to the study. These terms may be interpreted as arising from crustal structure local to the station which is different from that of the average crustal model used. The conclusion is reached that apart from providing better absolute hypocentre estimates, the method of Joint Hypocentre Determination can be made to yield worthwhile information about structure on the scale considered here.