Parameter estimation using paleodata assimilation

Estimation of model parameter values is of particular interest in paleoclimate and climate change research, since it is the formulation of model parameterizations, rather than the initial conditions, which is the main source of uncertainty regarding the climate’s long-term response to natural and anthropogenic forcings. We should recognize at the outset that the question of a “correct” parameter value might in many cases be quite contentious and disputable. There is, for example, no single value to describe the speed at which ice crystals fall through the atmosphere, or the background rate of mixing in the ocean, to mention two parameters which are commonly varied in General Circulation Models (GCMs). Generally the best we can hope for is to find a set of parameter values, which perform well in a range of circumstances, and to make allowances for the model’s inadequacies, i.e. structural errors due to inadequate equations and parameterizations. However, inadequacies will always be present no matter how carefully parameter values are chosen: this should serve as a caution against over-tuning. It may not be immediately clear how one can use proxy-derived observational estimates of climatic state variables such as temperature or precipitation to estimate the values of a model’s internal parameters. However, from a sufficiently abstract perspective, the problem of parameter estimation can be considered as equivalent to state estimation, via a standard approach in which the state space of a dynamical model is augmented by the inclusion of model parameters (Jazwinski 1970; Evensen et al. 1998). To see how this works, consider a system described by a dynamical model f, which uses a set of internal parameters θ and propagates a state vector x through time through a set of differential equations: