Optimal nonlinear models from empirical time series: An application to climate

In this work we propose a method that exploits the feedback between empirical and theoretical knowledge of a complex macroscopic system in order to build a nonlinear model. We apply the method to the monthly earth's mean surface temperature time series. The problems of contamination and stationarity are considered noting the importance of observation and modeling scales. We construct a dynamical system of ordinary differential equations where the vector field relating the relevant degrees of freedom and their variations in time is expressed in terms of a polynomial base orthonormal to the measure associated to the time series under study. The optimal size of the model and the values of its parameters are estimated with the principle of minimum description length and the Adams-Molton predictor-corrector method. This procedure is self-consistent because it does not use any external parameter or assumption. We then present a first approach to find the closest chaotic dynamical system corresponding to the earth's mean surface temperature and compare it with scale consistent theoretical or phenomenological models of the lower atmosphere. This comparison allows us to obtain an explicit functional form of the heat capacity of the earth's surface as a function of the earth's mean surface temperature.