On seasonal signals in geodetic time series

We explore implications for modeling and noise analysis of stochastic seasonal processes of climatic origin in geodetic time series. Seasonal signals are generally modeled as sinusoids with annual periods (and harmonics thereof), each with constant amplitude and phase. However, environmental noise that underlies the seasonal signal in geodetic time series has a reddened power spectral density (PSD). We investigate the form of the PSD of a time series having a stochastic seasonal component and find that for frequencies greater than the nominal seasonal frequency, the PSD of the time series reflects the PSD of the seasonal amplitudes. For example, if the PSD of the seasonal amplitudes can be expressed as an inverse power law, then the PSD of the time series will behave as an inverse power law for high frequencies. Stochastic seasonal variability will also induce a peak near the nominal seasonal frequency in addition to that of the mean seasonal signal and will be relatively flat below this frequency. It is therefore possible that some of the noise in Global Navigation Satellite Systems (GNSS) time series reported by others may be associated with neglecting the stochastic component of the seasonal signal. We use a GNSS time series from site ZIMM as an example to demonstrate the existence of a variable seasonal signal (without attributing its cause), and we use an example Gravity Recovery and Climate Experiment (GRACE) time series from Alaska to demonstrate that use of a nonstochastic seasonal model can have a significant impact on the value and uncertainty of time-variable rates estimated from the time series.