In this paper we present an elegant approach to reconstruct slowly
varying gross primary production (GPP) as a function of time, based on

Accurate rate estimates of whole ecosystem metabolism are crucial for our
understanding of food web dynamics and biogeochemical cycling in aquatic
ecosystems. Starting with the seminal work of

Both approaches are based on in situ

Time domain and frequency domain methods differ on a major aspect, namely on
how GPP is separated from other processes impacting on

By focussing specifically on the diel harmonic in

In this paper we investigate an elegant approach to reconstruct the slowly
varying GPP as a function of time, based on a direct analogy with amplitude-modulated (AM) radio signals (Fig.

The analogy between sound transmission with AM radio and GPP estimation from

In

When GPP varies slowly with time, so will

Diurnal tidal constituents moving back and forth a horizontal oxygen profile
will also result in diurnal

To assess the performance of complex demodulation to estimate time-varying
GPP, we use artificial data sets generated with two numerical models. The
first model describes a water body with no appreciable lateral transport of
oxygen, representative of a lake or the surface layer of the ocean, where
vertical turbulence and air–water exchange are the dominant transport
processes. The second describes a typical riverine or estuarine situation,
characterized by substantial horizontal gradients in the

As the models are forced with observed hourly irradiance data, GPP is a function of time. This causes an overall seasonality in GPP as well as shorter-term variability due to changes in cloudiness. As forcing we used incident light recorded in 2009 on the roof of NIOZ-Yerseke (the Netherlands) using a LI-COR LI-190 SA cosine sensor. Additionally, the dynamic build-up and break-down of algal biomass add both to the seasonality and to the short-term variability.

We use a single numerical model to simulate typical riverine and estuarine
situations. To emulate the occurrence of tides, the output of the riverine
model is resampled, using simulated velocities generated with a separate 1-D
tide-resolved hydrodynamic model. We thus simulate estuarine transport with
the riverine model, assuming a reference frame moving with the tides. This
allows us to investigate the influence of tides on the GPP estimates (more
details in

To test the performance on real-world data, we used a full year (2008) of

Simulated

The results of the simulations with the open water model are shown in Fig.

Simulated

When applied to simulated

Spectral amplitudes of the velocity time series used for estuarine model calculations. Quasi-diurnal frequencies O1, Q1, P1 and K1 are marked.

Simulated GPP in the estuarine model and reconstructed GPP by complex
demodulating the

Given the dominant contribution of the first-order term, we focus on its
nature for better understanding of the impact of advective transport on

If the horizontal

How those quasi-diurnals show up in

In contrast, the impact of O1 and Q1 can be largely removed simply by
decreasing the filter width (i.e. increasing the averaging time). Indeed,
using a moving-average filter of 15 days has almost the same effect as
applying the correction term (Fig.

Result of the bookkeeping method applied to the simulated estuarine

GPP rates for Hörnum Tief (

We can not stress enough that the impact of tidal harmonics on

The small difference that is still present between the first-order-corrected
and the 15-day-filtered complex demodulation has multiple potential causes (Fig.

Applying the complex demodulation procedure to real-world

Inter-annual differences in phytoplankton bloom dynamics hamper a comparison
of the 2004 and 2008 estimates of GPP rates (see

The complex demodulation procedure presented here is an expansion of the
Fourier method. The major benefit over the approach in

A second advantage is that this theoretical framework allows to understand
and analyse the impact of different tidal harmonics on

We have shown that this spurious GPP can be largely filtered out with a
15-day
moving-average filter. This comes at the expense of the resolution of the GPP
estimate: variability on timescales smaller than 15 days can not be
resolved. Resolving variability on timescales smaller than 15 days will be
difficult, perhaps impossible, in tidal systems with significant diurnals. This
boils down to measuring or estimating at least the first-order spatial
correction term, which consists of two factors: the tidal excursion and the

The choice of Hörnum Tief as a case study was motivated by the availability
of a long-term, quasi-continuous time series. Although high-frequency

The purpose of including a real-world time series was to demonstrate that the
theoretically predicted imprint of tidal diurnals is easily identified in
real systems and can be very large. As a side result, the GPP rate estimates
seem to correspond well with reference data from bottle incubations, apart
from the period of the

A more detailed study to the applicability of Fourier methods to tidal basins
such as Hörnum would also allow for assessing another crucial assumption
underlying Eq. (

Diurnal fluctuations in

To conclude, the correspondence with results from bottle incubations in the nearby List tidal basin is encouraging but not conclusive and a more detailed numerical study is required.

In order to assess the accuracy of the GPP rate estimates, a case-by-case
approach to quantify the different types of bias is still needed. Nevertheless, the analysis
in this paper again stresses the power of analysing

The Fourier method and complex demodulation described in this paper are
implemented in an R package

TJSC contributed to theory development, numerical simulations and manuscript writing. KS and JEEvB contributed to manuscript writing.

The authors declare that they have no conflict of interest.

We are grateful to Götz Flöser and Rolf Riethmüller from HZG for providing the Hörnum Tief time series. This work received funding from the EU (grant no. MARE/2012/10). Tom J. S. Cox thanks the Belgian Science Policy Office for the Belspo Return Grant (selection 2012) he received, which enabled this research. Edited by: Gerhard Herndl Reviewed by: two anonymous referees