Statistical Modelling in Climate Science

When it comes to modelling in atmospheric and climate science, the two main types of models are taken into account – dynamical and statistical models. The former ones have a physical basis: they utilize discretized differential equations with a set of conditions (boundary conditions + present state as an initial condition) and model the system’s state by integrating the equations forward in time. Models of this type are currently used e.g. as a numerical weather prediction models. The statistical models are considerably different: they are not based on physical mechanisms underlying the dynamics of the modelled system, but rather derived from the analysis of past weather patterns. An example of such a statistical model based on the idea of linear inverse modelling, is examined for modelling the El Niño – Southern Oscillation phenomenon with a focus on modelling cross-scale interactions in the temporal sense. Various noise parameterizations and the possibility of using a multi-variable model is discussed among other characteristics of the statistical model. The prospect of using statistical models with low complexity as a surrogate model for statistical testing of null hypotheses is also discussed. 1 Modelling in climate science Climate models, which rely on the use of quantitative methods to simulate interactions in the climate system, are one of the most important tools to predict and asses future climate projections or to study the climate of the past. In general, two types of models are mainly used: dynamical models and statistical models. The base for a dynamical model is a set of discretized differential equations which are integrated forward in time from the present state, posing as an initial condition. The most prominent example of the usage of dynamical models is without doubt a general circulation model (GCM hereafter). It employs a mathematical model of circulation of the planetary atmosphere and oceans, therefore it uses the Navier-Stokes equations on a rotating sphere (describing a motion of viscous fluid) with thermodynamic terms for energy sources and sinks. The above described model is used in numerical weather prediction, to infer the reanalysis datasets of the past climate and for future climate projections in climate model intercomparison projects CMIP3 [1] and CMIP5 [2]. The uncertainties of the forecast arisen from the GCM models are usually classified into two types: the first one is related to the initial errors (errors in determining the “true” present state of the climate), while the second one is due to the model errors [3] and these are intrinsic. The problem with initial errors is usually tackled by considering an ensemble of model forecasts (instead of just one realization integration from single initial state), starting with slightly different initial conditions. The model errors are intrinsically connected with the exponential error growth emerging from the chaotic behaviour related to nonlinearities in discretized equations [4]. This limits the predictability of such GCMs to 6-10 days maximum (e.g. [5]). 1.1 Statistical models The second kind of models used in climate science are statistical models. In their design, they are considerably different than the dynamical models in the sense that they are not based on physical mechanism underlying the dynamics of the modelled system, but rather derived from the analysis of past weather patterns. Probably the most used concept is that of inverse stochastic model [6], where the model is designed, then estimated using past data and, finally, stochastically integrated forward in time to obtain the prediction. The disadvantages connected to this type of models consist of the selection of variables that capture the system we are trying to model. Other possible issue could be the non-stationarity of the modelled system since the statistical model does not involve the underlying physical mechanisms, just the interaction between subsystems (ignoring hidden variables), the model estimated on some subset of the past data may not correctly capture all possible states of the system. In other words, the training period of the past data used to estimate the statistical model may not capture the full phase space of the modelled system. The motivation for building a statistical model for particular phenomenon, apart from its forecasting, would be to scale down the complexity of the problem. When we find some e.g. nonlinear interactions in the observed data, and we are interested in uncovering the mechanisms, constructing a models of different complexity and seeking such interactions in them would help to expose the mechanisms and shed some light on the problem. In the following sections, the inverse stochastic model for forecasting the El Niño Southern Oscillation (ENSO hereafter) phenomenon is built following [7], with the focus on various noise parametrizations and possible use of multiple variables. ITAT 2016 Proceedings, CEUR Workshop Proceedings Vol. 1649, pp. 102–109 http://ceur-ws.org/Vol-1649, Series ISSN 1613-0073, c © 2016 N. Jajcay, M. Paluš