Calibration of computationally demanding and structurally uncertain models with an application to a lake water quality model

Models of environmental systems are simplified representations of the reality. For this reason, their results are affected by systematic errors. This bias makes it difficult to get reliable uncertainty estimates of model parameters and predictions. A relatively simple way of considering this bias when using deterministic models is to add a statistical representation of the bias to the model output in addition to observation error and to jointly estimate model parameters, bias and observation error. When assuming Normal distributions for bias and observation error, this leads to a relatively simple likelihood function that can easily be evaluated. Nevertheless, the sampling from the posterior distribution still requires long Markov chains to be calculated which can be prohibitive for computationally demanding models. In order to extend the range of applicability of this technique to computationally demanding models, we suggest to replace Markov chain sampling by a Normal approximation to the posterior of the parameters and to estimate prediction uncertainty by linearized error propagation. We tested this procedure for a didactical example and for an application of the biogeochemical-ecological lake model BELAMO to long-term data from Lake Zurich. This is a good test application because the strong coupling of output variables makes it difficult to avoid bias in the results of this model. These tests demonstrate the applicability of the suggested procedure, the approximate reproduction of the results of the full procedure for the didactical example, and meaningful results for the lake model. For the latter, the results demonstrate that the assumption of a realistic likelihood function leads to the conclusion that prediction uncertainty may be high.