Process-based forest models generally have many parameters, multiple outputs of interest and a small underlying empirical database. These characteristics hamper parameterization. Bayesian calibration offers a solution to the calibration problem because it applies to models of any type or size. It provides parameter estimates, with measures of uncertainty and correlation among the parameters. The procedure begins by quantifying the uncertainty about parameter values in the form of a prior probability distribution. Then data on the output variables are used to update the parameter distribution by means of Bayes' Theorem. This yields a posterior calibrated distribution for the parameters, which can be summarized in the form of a mean vector and variance matrix. The predictive uncertainty of the model can be quantified by running it with different parameter settings, sampled from the posterior distribution. In a further step, one may evaluate the posterior probability of the model itself (rather than that of the parameters) and compare that against the probability of other models, to aid in model selection or improvement. Bayesian calibration of process-based models cannot be performed analytically, so the posterior parameter distribution must be approximated in the form of a representative sample of parameter values. This can be achieved by means of Markov Chain Monte Carlo simulation, which is suitable for process-based models because of its simplicity and because it does not require advance knowledge of the shape of the posterior distribution. Despite the suitability of Bayesian calibration, the technique has rarely been used in forestry research. We introduce the method, using the example of a typical forest model. Further, we show that reductions in parameter uncertainty, and thus in output uncertainty, can be effected by increasing the variety of data, increasing the accuracy of measurements and increasing the length of time series.

Forests are experiencing an environment that changes much faster than in at least the past several hundred years. In addition, the abiotic factors determining forest dynamics vary depending on its location. Forest modeling thus faces the new challenge of supporting forest management in the context of environmental change. This review has focused on three types of models that are used in forest management: empirical, process-based and hybrid models. Recent approaches might lead to the applicability of empirical models under changing environmental conditions, such as (i) the dynamic state-space approach, or (ii) the development of productivity-environment relationships. 25 process-based models in use in Europe were analyzed in terms of its structure, inputs and outputs having in mind a forest management perspective. Two paths for hybrid modeling were distinguished: (i) coupling of EMs and PBMs by developing signal-transfer environment-productivity functions; (ii) hybrid models with causal structure including both empirical and mechanistic components. Several gaps of knowledge were identified for the three types of models reviewed. The strengths and weaknesses of the three model types differ and all are likely to remain in use. There is a trade-off between how little data the models need for calibration and simulation purposes, and the variety of input-output relationships that they can quantify. PBMs are the most versatile, with a wide range of environmental conditions and output variables they can account for. However, PBMs require more data making them less applicable whenever data for calibration are scarce. The EMs, on the other hand, are easier to run as they require much less prior information, but their simplicity makes them less reliable in the context of environmental changes. These different deficiencies of PBMs and EMs suggest that hybrid models may be a good compromise, but a more extensive testing of these models is required in practice.

The development of vertical canopy gradients of leaf N has been regarded as an adaptation to the light gradient that helps to maximize canopy photos