Detection of Large-Scale Noisy Multi-Periodic Patterns with Discrete Double Fourier Transform. II. Study of Correlations Between Patterns

Abstract

The discrete double Fourier transform (DDFT) was developed to search for large-scale multi-periodic patterns in the presence of noise and is based on detection of the equidistant series of harmonics generated by the periodic patterns in the discrete Fourier transform (DFT) spectra. As DDFT retains all generic features of the Fourier transform, the corresponding pattern correlation function (PCF) related to DDFT can be introduced similarly to the data correlation function (DCF) related to DFT on the basis of the Wiener–Khinchin relationship. Peaks in PCF indicate the number of periodic patterns in a dataset under analysis and have direct correspondence with the counterpart peaks in the DFT spectrum. The close correspondence between positions of the peaks in the PCF and DFT spectra strongly enhances statistical significance of detected periodicities. Similar PCFs can also be defined for the cepstrum transform. The combined DFT–DCF and DDFT–PCF technique was applied to the detection of cycles in geomagnetic activity using disturbance storm-time (Dst) index. In addition to the known 27-day, semiannual and 11-year cycles of geomagnetic activity, we have also found the annual cycle of activity. The results were compared with those obtained by the cepstrum transform. A multiple cross-check makes the combined technique much more efficient and robust in comparison with the detection based on a unique particular method.