Nonstationary analysis of tides using higher order stochastic cycles

Abstract

Estuarine tides, currents and shelf phenomena are often non-stationary, with a frequency composition that evolves with time and forcing such as river flow and density structure. Such tides are difficult to characterize quantitatively, particularly against a background of significant, multi-scale sub-tidal and cusp energy. One challenge identified in past work is the separation of diurnal and subtidal components of the signal. The difficulty is exacerbated in basins such as the San Francisco Bay-Delta, where the diurnal species is subject to fortnightly fluctuation due to O1-K1 beating (the tropical spring-neap cycle) and the subsequent interaction with the semidiurnal species leads to the production of modulated terdiurnal energy. This paper extends the method of higher order stochastic cycles, a construct from the structural time series literature, for use in analyzing these nonstationary tides. Interpretable as a multi-species generalization of Butterworth low pass and bandpass filters, Stochastic Cycle Harmonic Analysis (SCHA) possesses an interpretable stationary frequency response and can accurately extract terdiurnal frequencies. Represented by a state space formulation and fit using a Markov Chain Monte Carlo (MCMC) method, the method can produce Bayesian confidence intervals for derived quantities such as amplitudes, is robust to patches of missing data, and is energy conserving. We evaluate the method on benchmark problems that test its time and frequency resolution. In a field application on the Sacramento River in the San Francisco Bay-Delta SCHA confirms that the declinational beating of O1 and K1, heavily modulated by river flow, dominates fortnightly variation in the tide and its terdiurnal response.