FITTING THE DYNAMIC-MODEL PCLAKE TO A MULTI-LAKE SURVEY THROUGH BAYESIAN STATISTICS

A method is presented for the regression of dynamic lake ecosystem models on multi-lake data. The method draws upon Bayesian Statistics as the main inference engine, as outlined by Box and Draper (1965), M.J. Box (1971) and Box and Tiao (1973/1992). The Bayesian approach allows the calculation of the uncertainty of parameters and predictions both before and after the model is confronted with data. There are several modelling objectives that can be dealt with through this technique in a unifying way: calibration of parameters, on the basis of prior knowledge and the available data; estimation of parameter uncertainty and correlation structure; estimation of predictive uncertainty for the assessment of trends and scenario analyses; validation of model structure in relation to residual errors. Moreover the method allows an iterative approach between hypothesis generation and data analysis. The method is applied to perform a regression and uncertainty analysis of the model PCLake on data from an 18-lake Dutch survey. PCLake is a dynamic nutrient-ecosystem model with a closed nutrient budget, comprising one water and one sediment layer. The model was run until steady state was reached and results were compared to summer-averaged field data from the survey. The output variables selected are chlorophyll-a and total phosphorus concentrations. The analysis was done for three selected parameters, considered uniformly distributed within predefined ranges. Posterior distributions were calculated for each lake on the basis of 125 three-parameter combinations. The residual error of the prediction of chlorophyll-a was reduced from a factor of 3.8 on the basis of the prior uncertainty analysis down to a factor of 2.3 after regression on both variables. For total P concentrations these factors were 1.4 before regression and 1.5 afterwards, hence a small trade-off to match chlorophyll levels. The prior uncertainty factor of mean chlorophyll predictions was reduced from 1.9 before regression to 1.1 after regression on both outputs, while for mean total P predictions these factors were 1.1 and 1.05, respectively. The conclusion can be drawn that chlorophyll-a is particularly sensitive to the three parameters, while total P is determined to a large extent by the lake-specific input parameters. The posterior parameter distributions reflected these differences in sensitivity. Painwise correlations between parameters were low. Analysis of systematic and case-specific deviations between model regression and data helps to identify other critical parameters and possible structural modifications.