Tune in on 11.57 µHz and listen to primary production

In this paper we present an elegant approach to reconstruct slowly varying gross primary production (GPP) as a function of time, based on O2 time series. The approach, called complex demodulation, is based on a direct analogy with amplitude-modulated (AM) radio signals. The O2 concentrations oscillating at the diel frequency (or 11.57 µHz) can be seen as a carrier wave, while the time variation in the amplitude of this carrier wave is related to the time-varying GPP. The relation follows from an analysis in the frequency domain of the governing equations of O2 dynamics. After the theoretical derivation, we assess the performance of the approach by applying it to three artificial O2 time series, generated with models representative of a well-mixed vertical water column, a river and an estuary. These models are forced with hourly observed incident irradiance, resulting in a variability of GPP on scales from hours to months. The dynamic build-up of algal biomass further increases the seasonality. Complex demodulation allows for reconstruction, with great precision, of time-varying GPP of the vertical water column and the river model. Surprisingly, it is possible to derive daily averaged GPP – complex demodulation thus reconstructs the amplitude of every single diel cycle. Also, in estuaries time-varying GPP can be reconstructed to a great extent. But there, the influence of the tides prevent achieving the same temporal resolution. In particular, the combination of horizontal O2 gradients with quasi-diurnal harmonics in the tides interferes with the complex demodulation procedure and introduces spurious amplitude variation that can not be attributed to GPP. We demonstrate that these spurious effects also occur in real-world time series (Hörnum Tief, Germany). The spurious effects due to K1 and P1 quasi-diurnals can not be distinguished from GPP. However, the spurious fluctuations introduced by O1 and Q1 can be removed to a large extent by increasing the averaging time to 15 days. As such, we demonstrate that a good estimate of the running 15-day average of GPP can be obtained in tidal systems. Apart from the direct merits of estimating GPP from O2 time series, the analysis in the frequency domain enhances our insights into O2 dynamics in tidal systems in general, and into the performance of O2 methods to estimate GPP in particular.

COMPLEX DEMODULATION: A NOVEL TIME SERIES METHOD FOR ANALYSING SEASONAL INFECTIOUS DISEASES

Understanding how seasonal patterns change from year to year is important for the management of infectious disease epidemics. Here, we present a mathematical formalization of the application of complex demodulation, which has previously only been applied in an exploratory manner in the context of infectious diseases. This method extracts the changing amplitude and phase from seasonal data, allowing comparisons between the size and timing of yearly epidemics. We first validate the method using synthetic data that displays the key features of epidemic data. In particular, we analyse both annual and biennial synthetic data, and explore the effect of delayed epidemics on the extracted amplitude and phase. We then demonstrate the usefulness of complex demodulation using national notification data for influenza in Australia. This method clearly highlights the higher number of notifications and the early peak of the influenza pandemic in 2009. We also identify that epidemics that peaked later than usual generally followed larger epidemics and involved fewer overall notifications. Our analysis establishes a role for complex demodulation in the study of seasonal epidemiological events.

Tune in on 11.57 µHz and listen to primary production

In this manuscript we present an an elegant approach to reconstruct slowly varying GPP as a function of time, based on O2 time series. The approach, called complex demodulation, is based on on a direct analogy with amplitude modulated (AM) radio signals. The O2 concentrations oscillating at the diel frequency (or 11.57 μHz) can be seen as a carrier wave, while the time variation in the amplitude of this carrier wave is related to the time varying GPP. The relation follows from an analysis in the frequency domain of the governing equation of O2 dynamics. After the theoretical derivation, we assess the performance of the approach by applying it to 3 artificial O2 time series, generated with models representative for a well mixed vertical water column, a river and an estuary. These models are forced with hourly observed incident irradiance, resulting in a variblity of GPP on scales from hours to months. The dynamic build-up of algal biomass further increases the seasonality. Complex demodulation allows to reconstruct with great precision time varying GPP of the vertical water column and the river model. Surprisingly, it is possible to derive daily averaged GPP – complex demodulation thus reconstructs the amplitude of every single diel cycle. Also in estuaries time varying GPP can be reconstructed to a great extent. But there, the influence of the tides prevent achieving the same temporal resolution. In particular, the combination of horizontal O2 gradients with the O1 and Q1 harmonics in the tides, interferes with the complex demodulation procedure, and introduces spurious amplitude variation that can not be attributed to GPP. But also other tidal harmonics, in casu K1 and P1, introduce diel fluctuations that can not be distinguished from GPP. We demonstrate that these spurious effects also occur in real-world time series (Hörnum Tief, De). The spurious fluctuations introduced by O1 and Q1 can be removed to a large extent by increasing the averaging time to 15 days. As such, we demonstrate that a good estimate of the running 15 day average of GPP can be obtained in tidal systems. Apart from the direct merits to estimating GPP from O2 time series, the analysis in the frequency domain enhances our insights in O2 dynamics in tidal systems in general, and in the performance of O2 methods to estimate GPP in particular.