Stochastic Dynamic Prediction Research Articles

Edward Epstein's Stochastic–Dynamic Approach to Ensemble Weather Prediction

In the late 1960s, well before the availability of computer power to produce ensemble weather forecasts, Edward Epstein (1931–2008) developed a stochastic–dynamic prediction (SDP) method for calculating the temporal evolution of mean value, variance, and covariance of the model variables: the statistical moments of a time-varying probability density function that define an ensemble forecast. This statistical–dynamical approach to ensemble forecasting is an alternative to the Monte Carlo formulation that is currently used in operations. The stages of Epstein's career that led to his development of this methodology are presented with the benefit of his oral history and supporting documentation that describes the retreat of strict deterministic weather forecasting. The important follow-on research by two of Epstein's proteges, Rex Fleming and Eric Pitcher, is also presented. A low-order nonlinear dynamical system is used to discuss the rudiments of SDP and Monte Carlo and to compare these approximate methods...

Bayesian statistical data assimilation for ecosystem models using Markov Chain Monte Carlo

This study considers advanced statistical approaches for sequential data assimilation. These are explored in the context of nowcasting and forecasting using nonlinear differential equation based marine ecosystem models assimilating sparse and noisy non-Gaussian multivariate observations. The statistical framework uses a state space model with the goal of estimating the time evolving probability distribution of the ecosystem state. Assimilation of observations relies on stochastic dynamic prediction and Bayesian principles. In this study, a new sequential data assimilation approach is introduced based on Markov Chain Monte Carlo (MCMC). The ecosystem state is represented by an ensemble, or sample, from which distributional properties, or summary statistical measures, can be derived. The Metropolis-Hastings based MCMC approach is compared and contrasted with two other sequential data assimilation approaches: sequential importance resampling, and the (approximate) ensemble Kalman filter (including computational comparisons). A simple illustrative application is provided based on a 0-D nonlinear plankton ecosystem model with multivariate non-Gaussian observations of the ecosystem state from a coastal ocean observatory. The MCMC approach is shown to be straightforward to implement and to effectively characterize the non-Gaussian ecosystem state in both nowcast and forecast experiments. Results are reported which illustrate how non-Gaussian information originates, and how it can be used to characterize ecosystem properties.

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

The Liouville equation represents the consistent and comprehensive framework for the treatment of the uncertainty inherent in meteorological forecasts. By its very nature, consideration of the Liouville equation avoids problems that are inherent to commonly used methods for predicting forecast skill, such as the need for higher-moment closure within stochastic-dynamic prediction, or the need for the generation of large ensemble sizes within ensemble forecasting. The general analytical solution of the Liouville equation presented in the first part of this work is used here to find the solution of the Liouville equation relevant for two low-dimensional nonlinear dynamical systems and to investigate the potential usefulness of the Liouville equation in the context of predicting forecast skill. The analytical solution of the Liouville equation in these examples, namely, the time-dependent probability density function, is discussed and compared with corresponding results obtained by stochastic-dynamic prediction and ensemble forecasting. The negative effect of the unavoidable higher-moment discard within stochastic-dynamic prediction is quantitatively demonstrated. It is also shown that a large number of ensemble members is necessary to obtain accurate estimates of statistics, such as means and (co)variances, even in the low-dimensional situations considered. It is concluded that, due to its fundamental role in dealing with initial state uncertainty in dynamical models, the Liouville equation must be considered as a highly valuable and useful guideline during the process of developing a coherent methodology for forecasting forecast skill that minimizes deficiencies of currently used methods.

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

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 ...

Dynamics of Short-Term Univariate Forecast Error Covariances

Abstract The covariance equation based on second-order closure for dynamics governed by a general scalar nonlinear partial differential equation (PDE) is studied. If the governing dynamics involve n space dimensions, then the covariance equation is a PDE in 2n space dimensions. Solving this equation for n = 3 is therefore computationally infeasible. This is a hindrance to stochastic-dynamic prediction as well as to novel methods of data assimilation based on the Kalman filter. It is shown that the covariance equation can be solved approximately, to any desired accuracy, by solving instead an auxiliary system of PDEs in just n dimensions. The first of these is a dynamical equation for the variance field. Successive equations describe, to increasingly high order, the dynamics of the shape of either the covariance function or the correlation function for points separated by small distances. The second-order equation, for instance, describes the evolution of the correlation length (turbulent microscale) field...

The dynamics of error covariances in a barotropic model

Under the hypothesis of linear error growth, the tangent linear model can be used to provide a simplified form of stochastic-dynamic prediction, in which the uncertainty on a model forecast is defined by the error covariances between the model state variables. We apply this method on a global vorticity equation model with a simple initial condition for the covariance matrix. The forecast error variances and correlations undergo rapid modifications in relation with the atmospheric flow pattern. This may have some implications for operational assimilation schemes in numerical weather prediction. A deeper understanding of the relevant processes is sought, using an adjoint form of the prediction equation, in order to assess the impact of the initial variances and correlations, and of the intrinsic dynamics of the model flow. The application on some simplified flows shows the relevance of diffusion, advection, Rossby wave dispersion and barotropic instability to forecast skill prediction. DOI: 10.1034/j.1600-0870.1993.t01-2-00006.x

A Simple Approximate Method of Stochastic-Dynamic Prediction for Small Initial Errors and Short Range

Starting with the vorticity equation for barotropic flow, we derive a system of stochastic differential equations that determines the time-evolution of the local variance of vorticity error originating in a large ensemble of initial states containing random and statistically isotropic initial errors. Those equations show that the local growth or decay of error variance depends primarily on the detailed structure of the true vorticity field; in general, the most rapid growth of error can be expected in concentrated regions of strong vorticity gradient. Those stochastic differential equations provide the basis for a simple method of stochastic-dynamic prediction. It requires only a modest increase over the total volume of computation for deterministic prediction.

Lagged average forecasting, an alternative to Monte Carlo forecasting

In order to use the information present in past observations and simultaneously to take advantage of the benefits of stochastic dynamic prediction we formulate the lagged average forecast (LAF) method. In a LAF, just as in a Monte Carlo forecast (MCF), sample statistics are calculated from an ensemble of forecasts. Each LAF ensemble member is an ordinary dynamical forecast (ODF) started from the initial conditions observed at a time lagging the start of the forecast period by a different amount. These forecasts are averaged at their proper verification times to obtain an LAF. The LAF method is operationally feasible since the LAF ensemble members are produced during the normal operational cycle. To test the LAF method, we use a two-layer, f-plane, highly truncated spectral model, forced by asymmetric Newtonian heating of the lower layer. In the experiments, a long run is generated by the primitive equation version of the model which is taken to represent nature, while forecasts are made by the quasigeostrophic version of the model. On the basis of forecast skill, the LAF and MCF are superior to the ODF; this occurs principally because ensemble averaging hedges the LAF and MCF toward the climate mean. The LAF, MCF and ODF are all improved when tempered by a simple regression filter; this procedure yields different weights for the different members of the LAF ensemble. The tempered LAF is the most skillful of the forecast methods tested. The LAF and MCF can provide a priori estimates of forecast skill because there is a strong correlation between the dispersion of the ensemble and the loss of predictability. In this way the time at which individual forecasts lose their skill can be predicted. The application of the LAF method to more realistic models and to monthly or seasonally averaged forecasts is briefly discussed. DOI: 10.1111/j.1600-0870.1983.tb00189.x

Introduction of the characteristic function in stochastic dynamic modelling

The probability characteristic function is used to generalize Epstein's stochastic dynamic prediction model. The time-differencing scheme results in an infinite system of equations which is closed by a quasi-normal procedure. This system yields an explicite approximation to the common or arbitrary marginal characteristic function, which in turn may be transformed to an estimation of the density function. The results are illustrated for Lorenz's minimum hydrodynamic equations. DOI: 10.1111/j.2153-3490.1978.tb00848.x

Comments on “Application of Stochastic Dynamic Prediction to Real Data”

Comments on “Application of Stochastic Dynamic Prediction to Real Data”

Application of Stochastic Dynamic Prediction to Real Data

Abstract The technique of stochastic dynamic prediction proposed by Epstein is applied to atmospheric data. The motivation for the approach is discussed and a review is given of the development of the stochastic dynamic equations which, subject to the third-moment discard approximation, describe the evolution of the first two moments of a probability density characterizing an ensemble of possible true states. The method of “least squares” is used to extract the moments directly from radiosonde observations of the 500 mb geopotential height field. Approaching the analysis problem from a Bayesian standpoint leads to a weighted average of the new observations and the forecast, the appropriate weighting for the latter being supplied by the stochastic forecast itself. The basic physical model employed is a spectral form of the equivalent barotropic. The effects of the simplicity of the dynamical model on the growth of error (external error growth) must be considered explicitly when making stochastic forecasts,...

Probability and Statistics in Meteorology: a Review of Some Recent Developments

Abstract Recent developments in six areas of statistical meteorology are described and the importance of interdisciplinary research in these areas is indicated. The areas are stochastic-dynamic prediction, assimilation of observed data, time-series analysis, statistical weather forecasting, probability forecasting, and precipitation modification statistics. Several problems are identified within each area and the need for further interdisciplinary research to solve these problems is emphasized.

Stochastic Analysis of Meteorological Fields

Abstract The result of a stochastic dynamic prediction is the expected values of the model parameters and the covariances among all the parameters. By adopting a Bayesian approach to the problem of analysis and making certain assumptions, one can utilize the vast amount of information in a stochastic dynamic prediction along with the information contained in observations. By making simulated observations of a pre-defined atmosphere, it is shown that the uncertainty in the analyzed values is substantially less than either the uncertainty in the forecast or in the observation. In addition, the results indicate that the effects of the limiting assumptions are minimal. Further experiments are performed in which only heights or only temperatures are actually observed, and in each case it is possible to obtain an analysis for all the parameters in the model. The method is particularly useful for assessing the value and impact of different amounts or types of data.

Depicting Stochastic Dynamic Forecasts

Abstract Stochastic dynamic prediction provides information on the variances and covariances of the predicted meteorological fields as well as the expected values of the fields themselves. It is shown that this new information can be depicted in a variety of graphical formats that illustrate various aspects of the certainty and uncertainty of the predictions and demonstrate the value of specific information on uncertainty.

Stochastic dynamic prediction

Stochastic dynamic prediction assumes the laws governing atmospheric behavior are entirely deterministic, but seeks solutions corresponding to probabilistic statements of the initial conditions, th...

Stochastic dynamic prediction<sup>1</sup>

Stochastic dynamic prediction assumes the laws governing atmospheric behavior are entirely deterministic, but seeks solutions corresponding to probabilistic statements of the initial conditions, thus recognizing the impossibility of exact or sufficiently dense observations. The equation that must be solved is the continuity equation for probability. For practical reasons only approximate solutions to this equation are possible in general. Deterministic forecasts represent a very low order of approximation. More exact methods are developed and some of the attributes and advantages of stochastic dynamic predictions are illustrated by applying them to a low order set of dynamic equations. Stochastic dynamic predictions have significantly smaller mean square errors than deterministic procedures, and also give specific information on the nature and extent of the uncertainty of the forecast. Also the range of time over which useful forecasts can be obtained is extended. However, they also require considerably more extensive calculations. The question of analysis to obtain the initial stochastic statement of the atmospheric state is considered and one finds here too a promise of significant advantages over present deterministic methods. It is shown how the stochastic method can be used to assess the value of new or improved data by considering their influence on the decrease in the uncertainty of the forecast. Comparisons among physical-numerical models are also made more effectively by applying stochastic methods. Finally the implications of stochastic dynamic prediction on the question of predictability are briefly considered, with the conclusion that some earlier estimates have been too pessimistic. DOI: 10.1111/j.2153-3490.1969.tb00483.x