On the overparameterization of earthquake location algorithms

The standard regression model for hypocenter determination contains four parameters: latitude correction, longitude correction, focal depth correction, and origin time correction. Similarity between the function ∂t/∂Δ and ∂t/∂H for values of Δ < 20° and again for Δ > 25° causes linear dependence between the coefficient vectors, provided that the observations are contained within a relatively narrow azimuthal range. The regression is overparameterized, and nothing can be inferred about the solution even when the procedure does not diverge. A solution can always be obtained in such cases by reducing the number of parameters (e.g., by restricting the focal depth); but such a solution remains dependent on the error in focal depth.Because of the geometry of the Earth's subduction zones largely surrounding the continental masses, many earthquake observations are clustered within an azimuthal range of the order of 90°. Thus, the problem of overparameterization is inherent in the location model as set up originally by Geiger. For example, the data set for LONGSHOT is shown to contain 150 observations in the azimuth range 350° to 90°, including all European and North American stations. Hence, the LONGSHOT least-squares location is poorly conditioned. In order to obtain well-conditioned solutions, the hypocentral location model must be reformulated in such a way that the epicenter is determined independently of the focal depth, thus avoiding overparameterization.