Ensemble Transformation Research Articles

Implications of the Form of the Ensemble Transformation in the Ensemble Square Root Filters

Abstract This paper considers implications of different forms of the ensemble transformation in the ensemble square root filters (ESRFs) for the performance of ESRF-based data assimilation systems. It highlights the importance of using mean-preserving solutions for the ensemble transform matrix (ETM). The paper shows that an arbitrary mean-preserving ETM can be represented as a product of the symmetric solution and an orthonormal mean-preserving matrix. The paper also introduces a new flavor of ESRF, referred to as ESRF with mean-preserving random rotations. To investigate the performance of different solutions for the ETM in ESRFs, experiments with two small models are conducted. In these experiments, the performances of two mean-preserving solutions, two non-mean-preserving solutions, and a traditional ensemble Kalman filter with perturbed observations are compared. The experiments show a significantly better performance of the mean-preserving solutions for the ETM in ESRFs compared to non-mean-preserving solutions. They also show that applying the mean-preserving random rotations prevents the buildup of ensemble outliers in ESRF-based data assimilation systems.

Development of bead-spring polymer models using the constant extension ensemble

We have examined a new method for generating coarse-grained models of polymers. The resulting models consist of bead-spring chains with the spring force law taken from the force-extension behavior in the constant extension ensemble. This method, called the polymer ensemble transformation method, is applied to the freely jointed chain. The resulting model illustrates why current bead-spring chain models are insufficient in describing polymer behavior at high discretization. Applying the method to the freely jointed chain with unequal rod lengths showed the effect of varying flexibility in the chain. The method was also used to generate a bead-spring model of F-actin, which shows how the method is not restricted to one molecular model and can even be applied to experimental data. The current limitations of the method are discussed, including the need for approximate bending potentials to model the worm-like chain with a bead-spring chain. We discuss practical issues such as using the bead-spring models in Brownian dynamics simulations and develop a simple spring force law that can accurately represent a freely jointed chain with only a few rods per spring. Because of the functional form of this new force law, existing computer simulations can be easily modified.

On the coarse-graining of polymers into bead-spring chains

We present a study of the coarse-graining of polymers into bead-spring chains using statistical mechanics. The force–extension behavior is examined at different levels of coarse-graining. A direct result of the springs being decoupled is that the force–extension behavior depends only on the number of flexibility lengths (e.g. persistence or Kuhn lengths) represented by each spring. This dimensionless parameter is found to govern the fluctuations around the mean extension, analogous to the conventional role of temperature. The use of an effective flexibility length to correct the model behavior is analyzed, and we have calculated bounds on the choices of this correction factor. The analytic nature of the statistical mechanical framework has also allowed for the calculation of asymptotic and universal behavior. The zero Weissenberg number rheological behavior is examined using the retarded-motion expansion coefficients of bead-spring chains at different levels of coarse-graining. The results show the trade-off between using too few or too many springs. The general framework to analyze the force–extension and rheological behavior is applied to the worm-like chain, FENE, and Fraenkel models. We introduce a new method for coarse-graining a polymer into a bead-spring chain called the Polymer Ensemble Transformation (PET) method. Application to the freely jointed chain polymer yields a set of spring force-laws called the Random Walk Spring (RWS) model. This new method illustrates why the previous spring force-laws cannot be used to finely discretize polymers and also provides new insight into how to rationally proceed in the coarse-graining of polymers into bead-spring chains.

Adaptive Sampling with the Ensemble Transform Kalman Filter. Part I: Theoretical Aspects

A suboptimal Kalman filter called the ensemble transform Kalman filter (ET KF) is introduced. Like other Kalman filters, it provides a framework for assimilating observations and also for estimating the effect of observations on forecast error covariance. It differs from other ensemble Kalman filters in that it uses ensemble transformation and a normalization to rapidly obtain the prediction error covariance matrix associated with a particular deployment of observational resources. This rapidity enables it to quickly assess the ability of a large number of future feasible sequences of observational networks to reduce forecast error variance. The ET KF was used by the National Centers for Environmental Prediction in the Winter Storm Reconnaissance missions of 1999 and 2000 to determine where aircraft should deploy dropwindsondes in order to improve 24‐72-h forecasts over the continental United States. The ET KF may be applied to any well-constructed set of ensemble perturbations. The ET KF technique supercedes the ensemble transform (ET) targeting technique of Bishop and Toth. In the ET targeting formulation, the means by which observations reduced forecast error variance was not expressed mathematically. The mathematical representation of this process provided by the ET KF enables such things as the evaluation of the reduction in forecast error variance associated with individual flight tracks and assessments of the value of targeted observations that are distributed over significant time intervals. It also enables a serial targeting methodology whereby one can identify optimal observing sites given the location and error statistics of other observations. This allows the network designer to nonredundantly position targeted observations. Serial targeting can also be used to greatly reduce the computations required to identify optimal target sites. For these theoretical and practical reasons, the ET KF technique is more useful than the ET technique. The methodology is illustrated with observation system simulation experiments involving a barotropic numerical model of tropical cyclonelike vortices. These include preliminary empirical tests of ET KF predictions using ET KF, 3DVAR, and hybrid data assimilation schemes—the results of which look promising. To concisely describe the future feasible sequences of observations considered in adaptive sampling problems, an extension to Ide et al.’s unified notation for data assimilation is suggested.

Ensemble Transformation and Adaptive Observations

Abstract Suppose that the geographical and temporal resolution of the observational network could be changed on a daily basis. Of all the possible deployments of observational resources, which particular deployment would minimize expected forecast error? The ensemble transform technique answers such questions by using nonlinear ensemble forecasts to rapidly construct ensemble-based approximations to the prediction error covariance matrices associated with a wide range of different possible deployments of observational resources. From these matrices, estimates of the expected forecast error associated with each distinct deployment of observational resources are obtained. The deployment that minimizes the chosen measure of forecast error is deemed optimal. The technique may also be used to find the perturbation that evolves into the leading eigenvector or singular vector of an ensemble-based prediction error covariance matrix. This time-evolving perturbation “explains” more of the ensemble-based prediction ...

Limitations of Gaussian assumptions for the irradiance distribution in digital imagery: nonstationary image ensemble considerations

Gaussian assumptions for the irradiance probability density in a digital image are often employed but rarely justified. We provide a mathematical justification for these assumptions and indicate the limitations of their use. Beginning with the context-dependent image ensemble considerations introduced by Hunt Cannon [IEEE Trans. Syst. Man Cybern.SMC-6, 876 (1976)], such an ensemble is found to be accurately modeled as a Gaussian random process with nonstationary mean and nonstationary variance. An ensemble transformation is deduced and confirmed empirically that yields a Gaussian, zero-mean, unit-variance, ergodic random process. This conclusion leads to an image model that predicts that the distribution of image irradiance values after local mean removal is determined only by the distribution of local standard deviation values in the image. An analytic expression is derived for the probability density function of these irradiance values and is validated experimentally. This expression indicates that the distribution of image irradiance values after local mean removal may be assumed to be Gaussian only when the local standard deviation in an image is a nearly stationary quantity.

Synchrotron radiation studies of defect structure evolution in aluminium during creep

Results are reported of in situ observations (using synchroton radiation) of the nonmonotonic evolution of the structural parameters of polycrystalline aluminium deformed under creep at intermediate temperatures. As an explanation of the experimental data, a theoretical model is proposed of dislocation ensemble transformation: chaotically distributed (in subgrain interiors) dislocations → sub-boundary dislocations.

On the Equation of State of Small Systems

Lebowitz and Percus recently described a method by which a thermodynamic function F in a canonical ensemble could be expressed in terms of a differential operation on F as described in a grand canonical ensemble. The result of such a transformation was to derive an explicit expression for the dependence of the virial coefficients in the expansion of the canonical pressure on the number of molecules in the periodic system. The purpose of this article is to elucidate the methods and assumptions used in this transformation and to derive the expression for the canonical pressure by applying this method to the (N–P–T) ensemble → (N–V–T) ensemble transformation.