The Liouville Equation and Its Potential Usefulness for the Prediction of Forecast Skill. Part I: Theory

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Abstract
The Liouville equation provides the framework for the consistent and comprehensive treatment of the uncertainty inherent in meteorological forecasts. This equation expresses the conservation of the phase-space integral of the number density of realizations of a dynamical system originating at the same time instant from different initial conditions, in a way completely analogous to the continuity equation for mass in fluid mechanics. Its solution describes the temporal development of the probability density function of the state vector of a given dynamical model. Consideration of the Liouville equation ostensibly avoids in a natural way the problems inherent to more standard methodology for predicting forecast skill, such as the need for higher-moment closure within stochastic-dynamic prediction, or the need to generate a large number of realizations within ensemble forecasting. These benefits, however, are obtained only at the expense of considering high-dimensional problems. The purpose of this ... Abstract The Liouville equation provides the framework for the consistent and comprehensive treatment of the uncertainty inherent in meteorological forecasts. This equation expresses the conservation of the phase-space integral of the number density of realizations of a dynamical system originating at the same time instant from different initial conditions, in a way completely analogous to the continuity equation for mass in fluid mechanics. Its solution describes the temporal development of the probability density function of the state vector of a given dynamical model. Consideration of the Liouville equation ostensibly avoids in a natural way the problems inherent to more standard methodology for predicting forecast skill, such as the need for higher-moment closure within stochastic-dynamic prediction, or the need to generate a large number of realizations within ensemble forecasting. These benefits, however, are obtained only at the expense of considering high-dimensional problems. The purpose of this ...

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