Assimilating modelled dissolved inorganic nitrogen loads to monitored data using component-wise iterative ensemble Kalman inversion

Abstract

Understanding the fate of dissolved inorganic nitrogen (DIN) sourced from land-based runoff is an important element in the development of strategies for managing water quality on the Great Barrier Reef (GBR). Direct measurement of the generation and transport of DIN from the landscape is impractical at large scale so most of the burden is deferred to water quality models (WQMs) that simulate catchment processes. Although WQMs are subject to uncertainty as most environmental models are, until recently WQMs designed to quantify nutrient, sediment and pesticide loads being discharged into the GBR lagoon have been deterministically calibrated with no reference to uncertainty. To properly assess the reliability and performance of WQMs they must be able to produce probabilistic predictions that correctly represent model uncertainty and Bayesian inference is an appealing framework for achieving this.Although the principles of Bayesian inference can be succinctly expressed as Bayes' theorem, 𝑓(M, |d) ∝ 𝑓(d|M, )𝑓(M, ), where M represents the model parameters,  are parameters of the likelihood probability density function (pdf) and d some measured data, its application for WQMs can be challenging. In all but the most trivial cases, there is no practical way of symbolically deriving the posterior 𝑓(M, |d). In the absence of an analytical representation of the posterior, one may resort to Markov chain Monte Carlo (MCMC) and sequential Monte Carlo (SMC) methods for sampling from the posterior. Although these stochastic processes are a tour de force that can yield high quality results, they require a large number of model evaluations to converge to an equilibrated solution.Iterative ensemble Kalman inversion (IEKI) encompasses a family of algorithms, such as the ensemble randomized maximum likelihood method and ensemble smoother with multiple data assimilations that offer an efficient method for gradually propagating error statistics via an ensemble of model outputs to connect samples from the prior to an ensemble of posterior samples. The major advantage of these schemes is that they consume a fraction of the model evaluations that would be required otherwise and scale favorably with the dimensionality of the inverse problem. A major drawback of the ensemble methods in the current context is that in their formulation, they require that the likelihood 𝑓(d|M, ) is Gaussian and that the covariance, , is known. Unfortunately, measurement error statistics for water quality measurements used to inform the WQM calibration are difficult to derive and quite often unavailable. On the other hand, MCMC and SMC methods can infer the parametric likelihood given the data.The authors recently introduced the component-wise IEKI (CW-IEKI) which hybridizes MCMC and IEKI by iteratively updating the model parameters, M, using an ensemble step followed by an MCMC step to adjust the likelihood parameters, . Since the MCMC step requires no further model evaluations, the algorithm offers efficiency comparable with conventional ensemble method...