Detection of a sinusoidal oscillation of unknown frequency in a time series – a geodetic approach

Abstract

Geodetic and geophysical time series may contain sinusoidal oscillations of unknown angular frequency. Often it is required to decide if such sinusoidal oscillations are truly present in a given time series. Here we pose the decision problem as a statistical hypothesis test, an approach very popular in geodesy and other scientific disciplines. In the case of unknown angular frequencies such a test has not yet been proposed.We restrict ourselves to the detection of a single sinusoidal oscillation in a one-dimensional time series, sampled at non-uniform time intervals.We compare two solution methods: the likelihood ratio test for parameters in a Gauss-Markov model and the analysis of the Lomb-Scargle periodogram. Whenever needed, critical values of these tests are computed using the Monte Carlo method. We analyze an exemplary time series from an absolute gravimetric observation by various tests. Finally, we compare their statistical power. It is found that the results for the exemplary time series are comparable. The LR test is more flexible, but always requires the Monte Carlo method for the computation of critical values. The periodogram analysis is computationally faster, because critical values can be approximately deduced from the exponential distribution, at least if the sampling is nearly uniform.

Keywords

• data analysis
• least squares spectral analysis
• likelihood ratio test
• Lomb-Scargle periodogram
• significance