Comments on the joint determination of hypocenters and station corrections

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Abstract
Some aspects of the joint determination of hypocenters and station corrections (JHD) are discussed. An alternative solution to one of the methods currently in use (Pavlis and Booker, 1983) is proposed. It requires only a modest computational effort and is very fast. This simplification is not achieved at the expense of numerical stability and is backed by a theoretical study of the properties of the matrices involved. These matrices are positive semidefinite, and this allows a general analysis of the JHD solution. Several facts, some of them known and some new, are formally proved. It is possible, for example, to carry out a detailed study of the eigenvalues of the matrix from which the station corrections are derived. It is also shown that the mean value of the initial estimate of station corrections are retained as a bias in the final solution. The relationship between the approximate solution of Frohlich (1979) and the exact JHD solution can also be studied, and the reason for its success in certain cases is explained.

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