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Appendix S1 Methodological development of data assimilation has gone through four primary phases: (1) simple analysis (e.g., Cressman algorithm), (2) statistical or optimum interpolation, (3) variational data assimilation (VDA) and (4) sequential data assimilation.

Appendix S2 The Kalman filter, named after Rudolf E. Kalman, is a mathematical method that uses measurements observed over time.

Appendix S3 Markov Chain Monte Carlo (MCMC) techniques used to generate simulations from a probability distribution are a class of algorithms used in sampling from probability distributions based on constructing a Markov chain that has the desired distribution that it reflects its equilibrium distribution.

Appendix S4 BIOME3 (Haxeltine and Prentice 1996) is a process-based terrestrial biosphere model that includes a photosynthetic scheme that simulates the acclimation of plants to an altered state of atmospheric CO2 by the optimisation of nitrogen allocation to foliage and by accounting for the effects of CO2 on net assimilation, stomatal conductance, Leaf Area Index, and the ecosystem water balance.

Appendix S5 The most important advantage of sequential methods is the ability of the optimal sate to differ from that embodied in the model equation.

Appendix S6 The study by Chen et al. (2008) is, in effect, a joint state-parameter approach that can integrate a kernel-smoothing algorithm into an ensemble Kalman filter to overcome the dramatic, sudden changes in parameter values through time and the loss of information between two consecutive points in time.

Appendix S7 Systematic errors are cumulative in nature.

Table S1 Comparison of main data assimilation (DA) methods used in numerical weather prediction (NWP)*.

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