A mollified Ensemble Kalman filter

It is well recognized that discontinuous analysis increments of sequential data assimilation systems, such as ensemble Kalman filters, might lead to spurious high frequency adjustment processes in the model dynamics. Various methods have been devised to continuously spread out the analysis increments over a fixed time interval centered about analysis time. Among these techniques are nudging and incremental analysis updates (IAU). Here we propose another alternative, which may be viewed as a hybrid of nudging and IAU and which arises naturally from a recently proposed continuous formulation of the ensemble Kalman analysis step. A new slow-fast extension of the popular Lorenz-96 model is introduced to demonstrate the properties of the proposed mollified ensemble Kalman filter.Comment: 16 pages, 6 figures. Minor revisions, added algorithmic summary and extended appendi

On the propagation of information and the use of localization in ensemble Kalman filtering

Several localized versions of the ensemble Kalman filter have been proposed. Although tests applying such schemes have proven them to be extremely promising, a full basic understanding of the rationale and limitations of localization is currently lacking. It is one of the goals of this paper to contribute toward addressing this issue. The second goal is to elucidate the role played by chaotic wave dynamics in the propagation of information and the resulting impact on forecasts. To accomplish these goals, the principal tool used here will be analysis and interpretation of numerical experiments on a toy atmospheric model introduced by Lorenz in 2005. Propagation of the wave packets of this model is shown. It is found that, when an ensemble Kalman filter scheme is employed, the spatial correlation function obtained at each forecast cycle by averaging over the background ensemble members is short ranged, and this is in strong contrast to the much longer range correlation function obtained by averaging over states from free evolution of the model. Propagation of the effects of observations made in one region on forecasts in other regions is studied. The error covariance matrices from the analyses with localization and without localization are compared. From this study, major characteristics of the localization process and information propagation are extracted and summarized.Comment: 13 pages, 18 figures, uses ametsoc.bst and ametsoc2col.st

Predictability in models of the atmospheric circulation

It will be clear from the above discussions that skill forecasts are still in their infancy. Operational skill predictions do not exist. One is still struggling to prove that skill predictions, at any range, have any quality at all. It is not clear what the statistics of the analysis error are. The statistics of the model errors are not known and finally it is not clear how to efficiently evolve the error statistics to the time of the forecast.In chapter 2 methods are developed to determine the variability of the predictability. The study is similar to the one by Lorenz (1965). The present atmospheric model, with 30 variables rather than 28, is only slightly larger than Lorenz's model. The main difference is in the use of methods. Adjoint models are used to find the most important error structures. These methods can be transported to state of the art models. Chapter 2 has appeared as a paper in Tellus (Houtekamer 1991).In chapter 3, the method is extended. A simple inhomogeneous observing network is used to obtain an inhomogeneous distribution for the analysis error. It is shown that ignoring this inhomogeneity will lead to a skill forecast of low quality. Thus skill forecasters have to use the error statistics which are obtained during the data assimilation process. If one uses an average distribution to describe the analysis error one may already obtain a reasonable skill forecast. Chapter 3 will appear in Monthly Weather Review (Houtekamer 1992).In chapter 4 a much more advanced model is used. It has 1449 variables. It is used in conjunction with the state of the art ECMWF model. The usefulness of the methods developed in chapter 2 and 3 is tested in a realistic context. It appears that the global forecast error cannot efficiently be described with adjoint methods. Global forecast errors can better be predicted with a Monte Carlo method. Weather forecasts usually have a local nature. For the description of local forecast errors adjoint methods are feasible. It appears that the distribution of the analysis error is less variable as expected from chapter 3. The observing network, which is almost time independent, determines the main structures of the distribution of the analysis error. Because the properties of the analysis error are almost constant they need to be determined only once. This reduces the computational cost of a skill forecast enormously., This chapter is concluded with a discussion of the possible impact of a high quality skill forecast. It may increase or decrease the length of a forecast with about one day. This is significant compared to the effect of other possible improvements to the forecasting system. Chapter 4 has been submitted to Monthly Weather Review

Discrete Data Assimilation in the Lorenz and 2D Navier--Stokes Equations

Consider a continuous dynamical system for which partial information about its current state is observed at a sequence of discrete times. Discrete data assimilation inserts these observational measurements of the reference dynamical system into an approximate solution by means of an impulsive forcing. In this way the approximating solution is coupled to the reference solution at a discrete sequence of points in time. This paper studies discrete data assimilation for the Lorenz equations and the incompressible two-dimensional Navier--Stokes equations. In both cases we obtain bounds on the time interval h between subsequent observations which guarantee the convergence of the approximating solution obtained by discrete data assimilation to the reference solution

Information-based data selection for ensemble data assimilation

Ensemble-based data assimilation is rapidly proving itself as a computationally-efficient and skilful assimilation method for numerical weather prediction, which can provide a viable alternative to more established variational assimilation techniques. However, a fundamental shortcoming of ensemble techniques is that the resulting analysis increments can only span a limited subspace of the state space, whose dimension is less than the ensemble size. This limits the amount of observational information that can effectively constrain the analysis. In this paper, a data selection strategy that aims to assimilate only the observational components that matter most and that can be used with both stochastic and deterministic ensemble filters is presented. This avoids unnecessary computations, reduces round-off errors and minimizes the risk of importing observation bias in the analysis. When an ensemble-based assimilation technique is used to assimilate high-density observations, the data-selection procedure allows the use of larger localization domains that may lead to a more balanced analysis. Results from the use of this data selection technique with a two-dimensional linear and a nonlinear advection model using both in situ and remote sounding observations are discussed

Ensemble-based data assimilation and the localisation problem

The “butterfly effect” is a popularly known paradigm; commonly it is said that when a butterfly flaps its wings in Brazil, it may cause a tornado in Texas. This essentially describes how weather forecasts can be extremely senstive to small changes in the given atmospheric data, or initial conditions, used in computer model simulations. In 1961 Edward Lorenz found, when running a weather model, that small changes in the initial conditions given to the model can, over time, lead to entriely different forecasts (Lorenz, 1963). This discovery highlights one of the major challenges in modern weather forecasting; that is to provide the computer model with the most accurately specified initial conditions possible. A process known as data assimilation seeks to minimize the errors in the given initial conditions and was, in 1911, described by Bjerkness as “the ultimate problem in meteorology” (Bjerkness, 1911)

Representativity error for temperature and humidity using the Met Office high-resolution model

The observation-error covariance matrix used in data assimilation contains contributions from instrument errors, representativity errors and errors introduced by the approximated observation operator. Forward model errors arise when the observation operator does not correctly model the observations or when observations can resolve spatial scales that the model cannot. Previous work to estimate the observation-error covariance matrix for particular observing instruments has shown that it contains signifcant correlations. In particular, correlations for humidity data are more significant than those for temperature. However it is not known what proportion of these correlations can be attributed to the representativity errors. In this article we apply an existing method for calculating representativity error, previously applied to an idealised system, to NWP data. We calculate horizontal errors of representativity for temperature and humidity using data from the Met Office high-resolution UK variable resolution model. Our results show that errors of representativity are correlated and more significant for specific humidity than temperature. We also find that representativity error varies with height. This suggests that the assimilation scheme may be improved if these errors are explicitly included in a data assimilation scheme. This article is published with the permission of the Controller of HMSO and the Queen's Printer for Scotland

Morphing Ensemble Kalman Filters

A new type of ensemble filter is proposed, which combines an ensemble Kalman filter (EnKF) with the ideas of morphing and registration from image processing. This results in filters suitable for nonlinear problems whose solutions exhibit moving coherent features, such as thin interfaces in wildfire modeling. The ensemble members are represented as the composition of one common state with a spatial transformation, called registration mapping, plus a residual. A fully automatic registration method is used that requires only gridded data, so the features in the model state do not need to be identified by the user. The morphing EnKF operates on a transformed state consisting of the registration mapping and the residual. Essentially, the morphing EnKF uses intermediate states obtained by morphing instead of linear combinations of the states.Comment: 17 pages, 7 figures. Added DDDAS references to the introductio

On diagnosing observation error statistics with local ensemble data assimilation

Recent research has shown that the use of correlated observation errors in data assimilation can lead to improvements in analysis accuracy and forecast skill. As a result there is increased interest in characterizing, understanding and making better use of correlated observation errors. A simple diagnostic for estimating observation error statistics makes use of statistical averages of observation-minus-background and observation-minus-analysis residuals. This diagnostic is derived assuming that the analysis is calculated using a best linear unbiased estimator. In this work, we consider if the diagnostic is still applicable when the analysis is calculated using ensemble assimilation schemes with domain localization. We show that the diagnostic equations no longer hold: the statistical averages of observation-minus-background and observation-minus-analysis residuals no longer result in an estimate of the observation error covariance matrix. Nevertheless, we are able to show that, under certain circumstances, some elements of the observation error covariance matrix can be recovered. Furthermore, we provide a method to determine which elements of the observation error covariance matrix can be correctly estimated. In particular, the correct estimation of correlations is dependent both on the localization radius and the observation operator. We provide numerical examples that illustrate these mathematical results

Re-membering meaning in the spaces

viii, 138 leaves ; 28 cmThrough stories of elementary school children she sees as a counsellor and the tensions existing in those stories, the writer enters into an examination of stories from her own life. The main question the author attempts to explore is the importance of narrative and story in her own life and how they function as a site for resistance, reflection, interpretation and meaning making. The writing itself is the process as the author attempts a qualitative, phenomenological inquiry into her own complicity in maintaining existing structures of class, race, gender, morality and education