Backward Fokker-Planck equation for determination of Model Predictability with uncertain initial errors

Forecast skill analysis is of considerable interest in ocean modeling and prediction (Robinson et al, 1996; Lozano et al, 1996; Aikman et al, 1997; O'Brien et al, 1998 and others). Major difference of predictability analysis is found between atmospheric and oceanic models. For atmospheric models, the Lyapunov exponents and singular vectors are widely used. For ocean models the classical correlation analysis is employed (e.g., Robinson et al., 1996, Chu et al., 1999). The forecast skill of an atmospheric model is usually studied as a flow stability relative to uncertain initial conditions. The error growth rate is determined by either the leading (largest) Lyapunov exponent or the so-called amplification factors calculated from the leading singular vectors. The two approaches were developed from the stability theory of dynamical systems and were generalized to account the atmospheric predictability.

Forecast skill can be quantified by the predictability time (PT) (Kravtsov, 1989), which is the time when the uncertainty in the forecast exceeds some criterion or in other words, the time when the information of the initial condition is lost. Obviously, the PT can be calculated using different methods. We present several analytic measures for model predictability using Pontryagin-Kolmogorov theory.