Improvement of robustness and stability in estimating Rayleigh–Schuster’s hodograph parameters using different procedures of vector normalization

This paper deals with Rayleigh–Schuster’s hodographs, intended for the detailed investigation of changes in the phase of quasiperiodic signals in time series. The hodographs are also known as the phasor-walkout method. A procedure of conditional vector normalization is proposed: it takes into account the vector amplitude for each period under consideration. The procedure considerably improves the robustness and stability of the hodograph approach to changes in the character of the processed data distributions and to various defects in the data. For example, when analyzing the earthquake catalog, the procedure strongly diminishes the influence of the event clustering caused, in particular, by swarms of earthquakes with comparable magnitude and the aftershock sequences of strong earthquakes. At the first stage, we calculate the vector sums (resulting vectors) for each period under investigation throughout the time series duration. For example, investigating diurnal periodicity of earthquakes, we first calculate the resulting vectors for each day of the observation. The further analysis of resulting vectors for each period throughout the time series duration can be performed with different procedures. We compare three procedures for normalization of the obtained resulting vectors, which are as follows: (1) th traditional one, preserving the real signal amplitude; (2) that with reduction of the obtained resulting vectors to the unit vector (phasor); and (3) that with conditional vector normalization, taking into account the amplitude of resulting vectors for each period throughout the time series duration. The third procedure diminishes the possible instability in some special distributions of the investigated data when the resulting vector for a period is close to zero. The procedures are compared using model signals and samples from real earthquake catalogs. All the procedures used give close results when processing random time series.