Parameter identifiability, constraint, and equifinality in data assimilation with ecosystem models.

One of the most desirable goals of scientific endeavor is to discover laws or principles behind ‘‘mystified’’ phenomena. A cherished example is the discovery of the law of universal gravitation by Isaac Newton, which can precisely describe falling of an apple from a tree and predict the existence of Neptune. Scientists pursue mechanistic understanding of natural phenomena in an attempt to develop relatively simple equations with a small number of parameters to describe patterns in nature and to predict changes in the future. In this context, uncertainty had been considered to be incompatible with science (Klir 2006). Not until the early 20th century was the notion gradually changed when physicists studied the behavior of matter and energy on the scale of atoms and subatomic particles in quantum mechanics. In 1927, Heisenberg observed that the electron could not be considered as in an exact location, but rather in points of probable location in its orbital, which can be described by a probability distribution (Heisenberg 1958). Quantum mechanics lets scientists realize that inherent uncertainty exists in nature and is an unavoidable and essential property of most systems. Since then, scientists have developed methods to analyze and describe uncertainty. Ecosystem ecologists have recently directed attention to studying uncertainty in ecosystem processes. The Bayesian paradigm allows ecologists to generate a posteriori probabilistic density functions (PDF) for parameters of ecosystem models by assimilating a priori PDFs and measurements (Dowd and Meyer 2003). Xu et al. (2006), for example, evaluated uncertainty in parameter estimation and projected carbon sinks by a Bayesian framework using six data sets and a terrestrial ecosystem (TECO) model. The Bayesian framework has been applied to assimilation of eddy-flux data into simplified photosynthesis and evapotranspiration model (SIPNET) to evaluate information content of the net ecosystem exchange (NEE) observations for constraints of process parameters (e.g., Braswell et al. 2005) and to partition NEE into its component fluxes (Sacks et al. 2006). Verstraeten et al. (2008) evaluate error propagation and uncertainty of evaporation, soil moisture content, and net ecosystem productivity with remotely sensed data assimilation. Nevertheless, uncertainty in data assimilation with ecosystem models has not been systematically explored. Cressie et al. (2009) proposed a general framework to account for multiple sources of uncertainty in measurements, in sampling, in specification of the process, in parameters, and in initial and boundary conditions. They proposed to separate the multiple sources of uncertainty using a conditional-probabilistic approach. With this approach, ecologists need to build a hierarchical statistical model based on the Bayesian theorem, and to use Markov chain Monte Carlos (MCMC) techniques for sampling before probability distributions of interested parameters or projected state variables can be obtained for quantification of uncertainty. It is an elegant framework for quantifying uncertainties in the parameters and processes of ecological models. At the core of uncertainty analysis is parameter identifiability. When parameters can be constrained by a set of data with a given model structure, we can identify maximum likelihood values of the parameters and then those parameters are identifiable. Conversely, there is an issue of equifinality in data assimilation (Beven 2006) that different models, or different parameter values of the same model, may fit data equally well without the ability to distinguish which models or parameter values are better than others. Thus, the issue of identifiability is reflected by parameter constraint and equifinality. This essay first reviews the current status of our knowledge on parameter identifiability and then discusses major factors that influence it. To enrich discussion, we use examples in ecosystem ecology that are different from the one on population dynamics of harbor seals in Cressie et al. (2009).