Earthquake location based on the gradient method and on minima of beams of directions

Abstract

abstract Earthquake location is usually made by means of the Gauss-Newton iterative method, known in seismology as the Geiger method. As this method may fail to be efficient in some cases, attention is turned to the possibilities offered to the problem by the gradient method, which is always convergent, and, when properly used, may lead to the vicinity of the solution after a few iterations. But in the vicinity of the solution, the gradient method becomes slowly convergent. Therefore, in addition, another convergent method based on the minimum value, g, of the sum of the squared residuals taken on a beam of directions, is presented. In it the sum g is approximated by a polynomial of Laplace spherical harmonics. The new method includes the gradient method as a particular case; it is well suited in the vicinity of the solution and may lead quickly to it.