Abstract: Crop models are important tools for impact assessment of climate change, as well as for  exploring management options under current climate. It is essential to evaluate the  uncertainty associated with predictions of these models. Several ways of quantifying  prediction uncertainty have been explored in the literature, but there have been no  studies of how the different approaches are related to one another, and how they are  related to some overall measure of prediction uncertainty. Here we show that all the  different approaches can be related to two different viewpoints about the model; either  the model is treated as a fixed predictor with some average error, or the model can be  treated as a random variable with uncertainty in one or more of model structure, model  inputs and model parameters. We discuss the differences, and show how mean squared  error of prediction can be estimated in both cases. The results can be used to put  uncertainty estimates into a more general framework and to relate different uncertainty  estimates to one another and to overall prediction uncertainty. This should lead to a  better understanding of crop model prediction uncertainty and the underlying causes of  that uncertainty. This study was published as (Wallach et al. 2016)

Abstract: Crop models are important tools for impact assessment of climate change, as well as for exploring management options under current climate. It is essential to evaluate the uncertainty associated with predictions of these models. We compare two criteria of prediction error; MSEPfixed, which evaluates mean squared error of prediction for a model with fixed structure, parameters and inputs, and MSEPuncertain(X), which evaluates mean squared error averaged over the distributions of model structure, inputs and parameters. Comparison of model outputs with data can be used to estimate the former. The latter has a squared bias term, which can be estimated using hindcasts, and a model variance term, which can be estimated from a simulation experiment. The separate contributions to MSEPuncertain(X) can be estimated using a random effects ANOVA. It is argued that MSEPuncertain(X) is the more informative uncertainty criterion, because it is specific to each prediction situation.

Abstract: Ensembles of process-based crop models are increasingly used to simulate crop growth for scenarios of temperature and/or precipitation changes corresponding to different projections of atmospheric CO2 concentrations. This approach generates large datasets with thousands of simulated crop yield data. Such datasets potentially provide new information but it is difficult to summarize them in a useful way due to their structural complexities. An associated issue is that it is not straightforward to compare crops and to interpolate the results to alternative climate scenarios not initially included in the simulation protocols. Here we demonstrate that statistical models based on random-coefficient regressions are able to emulate ensembles of process-based crop models. An important advantage of the proposed statistical models is that they can interpolate between temperature levels and between CO2 concentration levels, and can thus be used to calculate temperature and [CO2] thresholds leading to yield loss or yield gain, without rerunning the original complex crop models. Our approach is illustrated with three yield datasets simulated by 19 maize models, 26 wheat models, and 13 rice models. Several statistical models are fitted to these datasets, and are then used to analyze the variability of the yield response to [CO2] and temperature. Based on our results, we show that, for wheat, a [CO2] increase is likely to outweigh the negative effect of a temperature increase of +2 degrees C in the considered sites. Compared to wheat, required levels of [CO2] increase are much higher for maize, and intermediate for rice. For all crops, uncertainties in simulating climate change impacts increase more with temperature than with elevated [CO2].