Application of a weighted likelihood method to hypocenter determination

Abstract

The method of least squares is a standard approach to hypocenter determination in seismology. However, this method is not useful for data contaminated by systematic errors. To address this problem, we propose a weighted likelihood method (WLL) rather than a weighted least-squares method (WLSQ). Assuming a normally distributed random error and systematic errors, both methods give the same solution; however, variances of random errors estimated by WLSQ are much smaller than those estimated by WLL. Examining reasonable random errors, we simulate a case of systematic errors varying linearly with a given parameter, where the number of unknown parameters is reduced to one for simplification. We assume that a systematic error, two different arrays of stations, and three different weights are functions of distance. In the cases where biases affected by systematic errors are adequately reduced, the variances of random errors estimated by WLL become roughly equal to that assumed, but those estimated by WLSQ are much smaller than that assumed. This result implies that WLL is a better approach than WLSQ for data contaminated by systematic errors.