A Bayesian approach to quantifying uncertainties in surface fluxes from the eddy covariance method

Eddy covariance is the state-of-the-art technique for measuring fluxes of energy and gases between the land surface and the atmosphere. However, many processing steps sit between the data that are actually measured (typically a 10-Hz time series of wind speed components and gas mixing ratios) and the calculated flux. In practice, the data processing constitutes a modelling exercise, requiring a model of the measurement system and of the surface layer of the atmosphere. In conventional data processing, we treat the parameters of this model as known constants, and allow only for a random uncertainty term which diminishes as the duration of the observations increases.  This inevitably fails to comprehensively propagate the true uncertainties.

In a Bayesian approach, we treat the parameters of the model (of the measurement system and atmosphere) as uncertain parameters, which we characterise with probability distributions. We use our knowledge of the physics of the system and previous data to specify prior probability distributions for these parameters. We then update these with the data we actually observe (the high-frequency time series) to yield the posterior distributions for these parameters. By a simple extension, we can include a model of the biological processes governing uptake and emissions of gases (e.g. in the case of carbon dioxide, photosynthesis and respiration). In this way, we can produce estimates of the fluxes of interest, such as the long-term net carbon balance of an ecosystem, in the form of posterior probability distributions, in which the uncertainty is correctly represented following the axioms of conditional probability. We thereby correctly propagate all of the systematic uncertainties, giving a much more accurate representation of the true uncertainty.
  
For example, in most eddy covariance set-ups, the time lag between the sonic anemometer and the gas analyser is  estimated from the cross-covariance function, where it shows the maximum covariance. However, whenever there is noise in the data, this time lag can be misidentified, and results in systematic over-estimates in the calculated flux. The Bayesian approach represents this more correctly, as a system with two cross-correlated random variables where the time lag between them is itself a temporally-autocorrelated random variable with uncertainty. By estimating the distribution of plausible ("posterior") time lags to form a distribution of calculated fluxes, we incorporate this uncertainty in the results. We can apply similar principles to detrending the data, coordinate rotation, frequency-response corrections, and separating advection from the eddy flux.