Empirical dynamic modelling - a promising tool to assess causal links between water quality time series

Empirical dynamic modelling (EDM) is a relatively novel method to assess causality from time series in coupled dynamic systems. From the literature, EDM appeared to be useful in a number of applications including analysis of water quality data. It was therefore hypothesized that this technique has a potential to revisit existing long term data of solutes from catchment streams. More specifically, we proposed that causal links between concentration time series could be revealed, which were previously overlooked when only standard linear regression and correlation methods were used. We applied EDM to long term concentrations of dissolved organic carbon (DOC), total Fe and pH from the Lehstenbach stream in Germany, time series that were formerly evaluated using various other techniques in the context of DOC mobilisation from peat catchments. To assess causal links between solute time series, three steps of analysis were conducted. Firstly, the embedding dimension for each time series (that is the number of time lags required for the best possible one-time-step-ahead forecast for a particular time series) was computed. In a second step, using the computed embedding dimension, non-linearity of the system was assessed using a technique called s-mapping to explore if EDM is expected to be a beneficial tool for time series analysis. Finally, convergent cross mapping was applied to test different combinations of variables for causal links. The basic idea of convergent cross mapping is that - if X causes Y - the response variable Y contains information on its driver X but not vice versa (because X is independent of Y) and hence X can be predicted from time series of Y. So X is predicted from Y by mapping nearest neighbours in state space (so called cross mapping). With increasing length of Y, predictions shall improve to a saturation level, that is cross mapping skill converges, which is used as a criterion of causality. We applied EDM to weekly, discharge-corrected, temporal changes of concentration time series. Discharge corrections was conducted to eliminate discharge as a common driver of concentrations. Our data analysis implied a causal interaction between Fe and DOC that was proposed in earlier work. But surprisingly, DOC seems to drive pH but not the other way around, a result that needs to be investigated in more detail, but a likely explanation would be that pH of streamwater is mainly decreasing with acidic inputs of DOC from riparian wetlands. We concluded that using EDM additional insights on the links between catchment time series can be obtained.