Determination of a Local Magnitude Scale: A Generalized Inverse Solution

Current techniques for the development of a local magnitude scale involve the simultaneous determination of the following: the magnitudes of a number of earthquakes, site-specific correction terms for each of the recording stations, and the two constants in the equation used to represent the variation of amplitude as a function of distance. The relation between these parameters and wave amplitudes is linear, and solving the corresponding system of equations can be achieved in one step when the number of events and stations is relatively small. When the number is so large that this approach is not practical, the solution is obtained iteratively, which requires initial estimates for the values of the parameters, but as shown here, it is possible to solve the system in two noniterative steps without starting values. To achieve this simplification the method of separation of parameters, introduced in the context of the joint hypocentral determination, is used.