Abstract
Abstract. One characteristic of biogeochemical models is uncertainty about their formulation. Data assimilation should take this uncertainty into account. A common approach is to use an ensemble of models. We must assign probabilities not only to the parameters of the models but also to the models themselves. The method of hierarchical modelling allows us to calculate these probabilities. This paper describes the approach, develops the algebra for the most common case and then applies it to the Atmospheric Tracer Transport Model Intercomparison Project (TransCom). We see that the discrimination among models is unrealistically strong, due to optimistic assumptions inherent in the underlying inversion. The weighted ensemble means and variances from the hierarchical approach are quite similar to the conventional values because the best model in the ensemble is also quite close to the ensemble mean. The approach can also be used for cross-validation in which some data are held back to test estimates obtained with the rest. We demonstrate this with a test of the TransCom inversions holding back the airborne data. We see a slight decrease in the tropical sink and a notably different preferred order of models.
Highlights
Models of any interesting biogeochemical system are inexact
Uncertainties in model predictions arising from parametric uncertainty can be generated by semi-analytic error propagation (e.g. Scholze et al, 2007; Rayner et al, 2011) or by generating ensembles of model simulations from samples of the probability density functions (PDFs) of parameters (e.g. Murphy et al, 2007; Bodman et al, 2013)
Once we have calculated p(x, Hi|y) we can either integrate over x if we are interested in the relative probabilities of different observation operators or we can sum over the various choices of observation operators to obtain the PDF for x
Summary
Models of any interesting biogeochemical system are inexact. They cannot include all interesting processes, the governing equations of processes are not known exactly or computational resolution limits the accuracy of the solution. The clearest exception to this is the case of global-scale atmospheric inversions where the Atmospheric Tracer Transport Model Intercomparison Project (TransCom; Gurney et al, 2002, 2003, 2004; Baker et al, 2006) used an ensemble of atmospheric transport models and common inversion systems to infer regional CO2 fluxes from atmospheric concentrations All these studies faced the problem of estimating properties of the ensemble such as its mean and some measure of spread. Equal weighting was challenged by Stephens et al (2007), who compared the seasonality of vertical gradients in model simulations and observations They found that only a subset of models produced an acceptable simulation and that this subset favoured larger tropical uptake than the ensemble mean.
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