## 1. Introduction

[2] Predictions of future climate strongly depend on the concentrations of greenhouse gases in the atmosphere with CO_{2} being the most important one. Atmospheric CO_{2} concentrations are determined by the size of the global exchange fluxes with the oceans and the land as well as the anthropogenic emissions. Terrestrial ecosystem models (TEMs) can be used to estimate the net exchange flux of CO_{2} between the land and the atmosphere and therefore play an important role in the Earth system.

[3] State-of-the-art TEMs such as the Joint UK Land Environment Simulator (JULES) [*Best et al.*, 2011; *Clark et al.*, 2011] or the Biosphere Energy Transfer and Hydrology (BETHY) scheme [*Knorr*, 2000] contain a large number of biogeochemical processes, which makes them very complex models with a large number of process parameters involved. In most cases, we do not know the exact value of the parameters, and prior parameter values are therefore based on expert knowledge. In some cases this is little more than an informed guess. The large uncertainties associated with prior parameter values also lead to large variations in the predictions of the future land-atmosphere CO_{2} fluxes [*Knorr and Heimann*, 2001], which in turn contributes to the uncertainties in future climate projections.

[4] Due to the increasing number of process parameters involved in state-of-the-art TEMs, it becomes more and more important to focus on the reduction of their uncertainties. Parameter estimation methods are very useful in this context, because they provide an objective way of constraining the model against observations and in this way are able to reduce the parameter uncertainties.

[5] Various parameter estimation methods such as adjoint, genetic algorithm, Kalman Filter, Levenberg-Marquardt and Monte Carlo inversion have been compared for example in the OptIC (Optimization InterComparison) project [*Trudinger et al.*, 2007]. The aim here was to estimate four parameters in a highly simplified representation of the carbon dynamics in a TEM with only two state variables. A forward run of the model was used to generate artificial data, which were then treated as observations after degradation through added noise, correlations, drifts and gaps. It was found that all methods were equally successful at estimating the parameters. A comparison in terms of computational efficiency was not made, due to the fact that the model was inexpensive to run. Also, the model did not have multiple minima, which therefore did not allow for a comparison in terms of the ability to find the global minimum.

[6] The REFLEX project [*Fox et al.*, 2009] compared methods based on genetic algorithm, Kalman Filter and Monte Carlo inversion using the Data Assimilation Linked Ecosystem Carbon (DALEC) model [*Williams et al.*, 2005]. DALEC is a simple box model of carbon pools used here in two versions, as a model for evergreen and a model for deciduous vegetation. The evergreen version required calibration of 11 parameters related to allocation and turnover of carbon pools, whereas the deciduous version required calibration of 17 parameters. REFLEX used both synthetic (generated from the model with added noise) and real data. It was found that estimates of confidence intervals varied among algorithms. Again, the main focus here was not on comparing the methods in terms of their computational efficiency nor their ability to find the global minimum.

[7] Many parameter estimation methods use the Bayesian approach, which has proven to provide a powerful and convenient framework for combining prior knowledge about parameters with additional information such as observations [*Rayner et al.*, 2005]. The resulting inverse problem described by Bayes' theorem can be solved in different ways. Here we focus on the comparison of two types of methods: Monte Carlo inversion [*Sambridge and Mosegaard*, 2002] and variational data assimilation [*Talagrand and Courtier*, 1987]. Monte Carlo inversion methods such as the Markov Chain Monte Carlo method (MCMC) have a better chance to converge to the global minimum than have gradient-based methods for example. In principle, the MCMC will converge to the global minimum if the number of iterations is large enough. However, the maximum number of iterations may be restricted by the computing time of the model. MCMC methods are easy to implement and they require no assumptions about the model (i.e. continuity) and the posterior probability distribution of parameters may be non-Gaussian, even if the prior distribution is assumed to be Gaussian (normally distributed).

[8] Variational data assimilation, such as the four-dimensional variational (4D-Var) scheme, is one of the most advanced approaches to assimilate observed information into a model. It uses derivative code (i.e. the adjoint of the model) for the optimization of the parameters and therefore requires the model to be differentiable with respect to all parameters. Although the 4D-Var approach is computationally very efficient in most cases, the optimization might only identify a local minimum due to the non-linearity and high dimensionality of the model. Another criticism of the 4D-Var method is that it focuses only on the optimal solution, i.e. the mode of the probability density function (PDF) without considering uncertainties. However, some 4D-Var schemes, such as the Carbon Cycle Data Assimilation System (CCDAS) [*Rayner et al.*, 2005], allow the calculation of posterior parameter uncertainties using the inverse of the Hessian (second order derivative) of the cost function at the global minimum. Unfortunately, this is only correct for linear problems. If the model is non-linear and a Gaussian distribution is assumed for the prior parameters, the model needs to be linearized around the optimum in parameter space, and the posterior distribution will only be approximated by a Gaussian [*Tarantola*, 1987]. This approximation might not always be reasonable, considering that most TEMs are highly non-linear.

[9] In this contribution we compare the 4D-Var (adjoint) approach as implemented in CCDAS, with the Metropolis algorithm (MA) [*Metropolis et al.*, 1953; *Mosegaard and Tarantola*, 1995], which is one possible MCMC method. We apply both methods in order to estimate the posterior PDF of 19 process parameters in the terrestrial ecosystem model BETHY. BETHY is a complex grid-based model, which simulates carbon assimilation and soil respiration within a full energy and water balance and phenology scheme. The main focus is on the performance of the two methods in terms of their efficiency (i.e. number of required model runs) and their ability to find the global minimum. In addition, the MA will allow us to assess the full shape of the PDF of single model parameters – as part of the evaluation of the full PDF containing the dependence on all model parameters simultaneously – and thus provides an indication of whether or not the assumption of a Gaussian posterior PDF of parameters made in CCDAS is justified.