Advances in sequential data assimilation and numerical weather forecasting: an Ensemble Transform Kalman-Bucy Filter, a study on clustering in deterministic Ensemble Square Root Filters, and a test of a new time stepping scheme in an atmospheric model

This dissertation deals with aspects of sequential data assimilation (in particular ensemble Kalman filtering) and numerical weather forecasting. In the first part, the recently formulated Ensemble Kalman-Bucy (EnKBF) filter is revisited. It is shown that the previously used numerical integration scheme fails when the magnitude of the background error covariance grows beyond that of the observational error covariance in the forecast window. Therefore, we present a suitable integration scheme that handles the stiffening of the differential equations involved and doesn’t represent further computational expense. Moreover, a transform-based alternative to the EnKBF is developed: under this scheme, the operations are performed in the ensemble space instead of in the state space. Advantages of this formulation are explained. For the first time, the EnKBF is implemented in an atmospheric model. The second part of this work deals with ensemble clustering, a phenomenon that arises when performing data assimilation using of deterministic ensemble square root filters in highly nonlinear forecast models. Namely, an M-member ensemble detaches into an outlier and a cluster of M-1 members. Previous works may suggest that this issue represents a failure of EnSRFs; this work dispels that notion. It is shown that ensemble clustering can be reverted also due to nonlinear processes, in particular the alternation between nonlinear expansion and compression of the ensemble for different regions of the attractor. Some EnSRFs that use random rotations have been developed to overcome this issue; these formulations are analyzed and their advantages and disadvantages with respect to common EnSRFs are discussed. The third and last part contains the implementation of the Robert-Asselin-Williams (RAW) filter in an atmospheric model. The RAW filter is an improvement to the widely popular Robert-Asselin filter that successfully suppresses spurious computational waves while avoiding any distortion in the mean value of the function. Using statistical significance tests both at the local and field level, it is shown that the climatology of the SPEEDY model is not modified by the changed time stepping scheme; hence, no retuning of the parameterizations is required. It is found the accuracy of the medium-term forecasts is increased by using the RAW filter.

Ensemble clustering in deterministic ensemble Kalman filters

Ensemble clustering (EC) can arise in data assimilation with ensemble square root filters (EnSRFs) using non-linear models: an M-member ensemble splits into a single outlier and a cluster of M−1 members. The stochastic Ensemble Kalman Filter does not present this problem. Modifications to the EnSRFs by a periodic resampling of the ensemble through random rotations have been proposed to address it. We introduce a metric to quantify the presence of EC and present evidence to dispel the notion that EC leads to filter failure. Starting from a univariate model, we show that EC is not a permanent but transient phenomenon; it occurs intermittently in non-linear models. We perform a series of data assimilation experiments using a standard EnSRF and a modified EnSRF by a resampling though random rotations. The modified EnSRF thus alleviates issues associated with EC at the cost of traceability of individual ensemble trajectories and cannot use some of algorithms that enhance performance of standard EnSRF. In the non-linear regimes of low-dimensional models, the analysis root mean square error of the standard EnSRF slowly grows with ensemble size if the size is larger than the dimension of the model state. However, we do not observe this problem in a more complex model that uses an ensemble size much smaller than the dimension of the model state, along with inflation and localisation. Overall, we find that transient EC does not handicap the performance of the standard EnSRF.

A low-order model investigation of the analysis of gravity waves in the ensemble Kalman filter.

The behavior of the ensemble Kalman filter (EnKF) is examined in the context of a model that exhibits a nonlinear chaotic (slow) vortical mode coupled to a linear (fast) gravity wave of a given amplitude and frequency. It is shown that accurate recovery of both modes is enhanced when covariances between fast and slow normal-mode variables (which reflect the slaving relations inherent in balanced dynamics) are modeled correctly. More ensemble members are needed to recover the fast, linear gravity wave than the slow, vortical motion. Although the EnKF tends to diverge in the analysis of the gravity wave, the filter divergence is stable and does not lead to a great loss of accuracy. Consequently, provided the ensemble is large enough and observations are made that reflect both time scales, the EnKF is able to recover both time scales more accurately than optimal interpolation (OI), which uses a static error covariance matrix. For OI it is also found to be problematic to observe the state at a frequency that is a subharmonic of the gravity wave frequency, a problem that is in part overcome by the EnKF.However, error in themodeled gravity wave parameters can be detrimental to the performance of the EnKF and remove its implied advantages, suggesting that a modified algorithm or a method for accounting for model error is needed.

Ensemble transform Kalman-Bucy filters

Two recent works have adapted the Kalman–Bucy filter into an ensemble setting. In the first formulation, the ensemble of perturbations is updated by the solution of an ordinary differential equation (ODE) in pseudo-time, while the mean is updated as in the standard Kalman filter. In the second formulation, the full ensemble is updated in the analysis step as the solution of single set of ODEs in pseudo-time. Neither requires matrix inversions except for the frequently diagonal observation error covariance. We analyse the behaviour of the ODEs involved in these formulations. We demonstrate that they stiffen for large magnitudes of the ratio of background error to observational error variance, and that using the integration scheme proposed in both formulations can lead to failure. A numerical integration scheme that is both stable and is not computationally expensive is proposed. We develop transform-based alternatives for these Bucy-type approaches so that the integrations are computed in ensemble space where the variables are weights (of dimension equal to the ensemble size) rather than model variables. Finally, the performance of our ensemble transform Kalman–Bucy implementations is evaluated using three models: the 3-variable Lorenz 1963 model, the 40-variable Lorenz 1996 model, and a medium complexity atmospheric general circulation model known as SPEEDY. The results from all three models are encouraging and warrant further exploration of these assimilation techniques.

Estimating correlated observation error statistics using an ensemble transform Kalman filter

For certain observing types, such as those that are remotely sensed, the observation errors are correlated and these correlations are state- and time-dependent. In this work, we develop a method for diagnosing and incorporating spatially correlated and time-dependent observation error in an ensemble data assimilation system. The method combines an ensemble transform Kalman filter with a method that uses statistical averages of background and analysis innovations to provide an estimate of the observation error covariance matrix. To evaluate the performance of the method, we perform identical twin experiments using the Lorenz ’96 and Kuramoto-Sivashinsky models. Using our approach, a good approximation to the true observation error covariance can be recovered in cases where the initial estimate of the error covariance is incorrect. Spatial observation error covariances where the length scale of the true covariance changes slowly in time can also be captured. We find that using the estimated correlated observation error in the assimilation improves the analysis.

An Ensemble Kalman Smoother for Nonlinear Dynamics

It is for mally proved that the general smoother for nonlinear dynamics can be for mulated as a sequential method, that is, obser vations can be assimilated sequentially during a for ward integration. The general filter can be derived from the smoother and it is shown that the general smoother and filter solutions at the final time become identical, as is expected from linear theor y. Then, a new smoother algorithm based on ensemble statistics is presented and examined in an example with the Lorenz equations. The new smoother can be computed as a sequential algorithm using only for ward-in-time model integrations. It bears a strong resemblance with the ensemble Kalman filter . The difference is that ever y time a new dataset is available during the for ward integration, an analysis is computed for all previous times up to this time. Thus, the first guess for the smoother is the ensemble Kalman filter solution, and the smoother estimate provides an improvement of this, as one would expect a smoother to do. The method is demonstrated in this paper in an intercomparison with the ensemble Kalman filter and the ensemble smoother introduced by van Leeuwen and Evensen, and it is shown to be superior in an application with the Lorenz equations. Finally , a discussion is given regarding the properties of the analysis schemes when strongly non-Gaussian distributions are used. It is shown that in these cases more sophisticated analysis schemes based on Bayesian statistics must be used.

Analysis Scheme in the Ensemble Kalman Filter

This paper discusses an important issue related to the implementation and interpretation of the analysis scheme in the ensemble Kalman filter . I t i s shown that the obser vations must be treated as random variables at the analysis steps. That is, one should add random perturbations with the correct statistics to the obser vations and generate an ensemble of obser vations that then is used in updating the ensemble of model states. T raditionally , this has not been done in previous applications of the ensemble Kalman filter and, as will be shown, this has resulted in an updated ensemble with a variance that is too low . This simple modification of the analysis scheme results in a completely consistent approach if the covariance of the ensemble of model states is interpreted as the prediction error covariance, and there are no further requirements on the ensemble Kalman filter method, except for the use of an ensemble of sufficient size. Thus, there is a unique correspondence between the error statistics from the ensemble Kalman filter and the standard Kalman filter approach

Assimilation of Geosat Altimeter Data for the Agulhas Current Using the Ensemble Kalman Filter with a Quasigeostrophic Model

The ring-shedding process in the Agulhas Current is studied using the ensemble Kalman filter to assimilate geosat altimeter data into a two-layer quasigeostrophic ocean model. The properties of the ensemble Kalman filter are further explored with focus on the analysis scheme and the use of gridded data. The Geosat data consist of 10 fields of gridded sea-surface height anomalies separated 10 days apart that are added to a climatic mean field. This corresponds to a huge number of data values, and a data reduction scheme must be applied to increase the efficiency of the analysis procedure. Further, it is illustrated how one can resolve the rank problem occurring when a too large dataset or a small ensemble is used.

A Floating-point Extended Kalman Filter Implementation for Autonomous Mobile Robots

Localization and Mapping are two of the most important capabilities for autonomous mobile robots and have been receiving considerable attention from the scientific computing community over the last 10 years. One of the most efficient methods to address these problems is based on the use of the Extended Kalman Filter (EKF). The EKF simultaneously estimates a model of the environment (map) and the position of the robot based on odometric and exteroceptive sensor information. As this algorithm demands a considerable amount of computation, it is usually executed on high end PCs coupled to the robot. In this work we present an FPGA-based architecture for the EKF algorithm that is capable of processing two-dimensional maps containing up to 1.8 k features at real time (14 Hz), a three-fold improvement over a Pentium M 1.6 GHz, and a 13-fold improvement over an ARM920T 200 MHz. The proposed architecture also consumes only 1.3% of the Pentium and 12.3% of the ARM energy per feature.

Estudo, proposta e previsão de desempenho de um filtro de energia para experimentos com alta resolução de comprimento de onda em fontes pulsantes de nêutrons

Em fontes de nêutrons por spallation para pesquisa de materiais, o comprimento de onda dos nêutrons é geralmente determinado pelos tempos de percurso (TOF) dos nêutrons desde a fonte até o detector. A precisão atingível é limitada pelo fato de o tempo de emissão do pulso característico do sistema alvo/moderador ser diferente de zero (a situação ideal fictícia seria a emissão se ocorresse na forma de um impulso). “Moderadores acoplados” (elementos usados para produzir feixes de alta intensidade com nêutrons de baixa energia) apresentam um decaimento de intensidade em função do tempo muito longo, ao longo de todo o espectro usado nos experimentos. Por este motivo, “moderadores desacoplados”, os quais produzem feixes com intensidade mais reduzida, são freqüentemente usados para instrumentos que requerem alta resolução. Neste trabalho, propusemos e analisamos uma nova técnica de filtragem dinâmica de feixes de nêutrons polarizados de baixa energia para experimentos que utilizam TOF na determinação do comprimento de onda. O dispositivo consiste de um sistema ótico polarizador/analisador e um inversor de spin seletivo em energia, o qual funciona por ressonância espacial do spin. Variando a condição de ressonância em sincronia com a estrutura temporal do pulso de nêutrons (através do controle de campos magnéticos), o filtro pode ajustar a resolução de energia (ou de comprimento de onda) de pulsos de banda larga em tais experimentos, separando os nêutrons com a correta relação “TOF/comprimento de onda” dos demais Um método para o cálculo de desempenho do sistema foi apresentado em um Trabalho Individual (TI) (PARIZZI et al., 2002 - i), do qual se fará aqui uma breve revisão além de amplo uso na otimização dos parâmetros a serem ajustados para o projeto do filtro. Os resultados finais mostram que ganhos consideráveis em resolução podem ser obtidos com a aplicação desta técnica em experimentos de reflectometria por tempo de percurso, sem que para tal seja necessário comprometer a intensidade do feixe usado pelo mesmo instrumento quando operado em um modo de baixa resolução, dando ao usuário do instrumento a opção de escolher a relação ótima entre intensidade e resolução para seu experimento. Como parte da conclusão desta dissertação, é apresentada uma proposta de parâmetros para a construção deste tipo de filtro e previsão de desempenho da configuração proposta, baseada no software de modelamento desenvolvido.

Bloqueio adenosinérgico e filtro sensorial

Esta dissertação é estruturada em três partes. Na primeira, revisa-se a noção de filtro sensorial, com enfoque particular no paradigma do P50, uma técnica eletroneurofisiológica de extremo valor para a investigação da neurobiologia básica subjacente aos defeitos de processamento sensorial que caracterizam algumas doenças mentais, e mais particularmente a esquizofrenia, sobre a qual dedica-se especial interesse. Na segunda, revisa-se a hipótese, proposta recentemente por Lara e Souza (2000), de hipofunção adenosinérgica como disfunção bioquímica básica na esquizofrenia, à luz das evidências mais recentes. Na terceira, desenvolve-se um trabalho experimental original com o intuito de investigar a hipótese hipoadenosinérgica da esquizofrenia. Trata-se de um desafio farmacológico, de um ensaio clínico cruzado onde 13 voluntários hígidos foram submetidos a tratamento com teofilina (um antagonista não-seletivo dos receptores de adenosina do tipo A1 e A2A) e a placebo, em dois momentos diferentes, tendo se avaliado os seus potenciais evocados de acordo com o paradigma do P50 antes (em seu valor basal) e após o tratamento, levantando-se uma curva de efeito com base no tempo. Paralelamente, avaliaram-se 17 pacientes com diagnóstico estabelecido de esquizofrenia, clinicamente estáveis, em acompanhamento ambulatorial e em uso de medicação neuroléptica típica, com a intenção de fornecer um grupo adicional de comparação e de replicar os achados prévios de falha de supressão do componente P50 na esquizofrenia, um aspecto fundamental para demonstrar o domínio da metodologia experimental, que foi aqui empregada pela primeira vez em nosso meio. Este estudo foi capaz de mostrar que a indução de um estado transitório de hipofunção adenosinérgica em indivíduos normais, mostra perda da supressão. Em outras palavras, que déficits no processamento da informação auditiva, que não existiam nos indivíduos normais, foram provocados pela utilização de teofilina, que, bloqueando os receptores de adenosina A1 e A2A, provocou um estado hipoadenosinérgico transitório. A disfunção provocada pela teofilina foi da mesma ordem de grandeza da verificada nos pacientes com esquizofrenia. Estes resultados fornecem evidência que corroboram o modelo de hipofunção adenosinérgica para a esquizofrenia.

Racionalidade imperfeita e inércia inflacionária : uma nova estimação da Curva de Phillips para o Brasil

O presente trabalho estima uma nova relação entre emprego e inflação para o Brasil, tendo como pano de fundo hipóteses propostas pelo arcabouço neo-keynesiano. Quatro hipóteses são testadas e sustentadas ao longo de todo o trabalho: i) os agentes não possuem racionalidade perfeita; ii) a imperfeição no processo de formação das expectativas dos agentes pode ser um fator determinante no componente inercial da inflação brasileira; iii) a inflação possui um componente inercial autônomo, que não depende de choques verificados em mercados isolados; e, iv) relações não-lineares entre inflação e desemprego são capazes de fornecer melhores explicações para o comportamento da economia nos últimos 12 anos. Enquanto as duas primeiras hipóteses são verificadas através de um modelo com mudanças markovianas, as duas últimas são testadas a partir da estimação de uma Curva de Phillips convexa, estimadas pelo Filtro de Kalman. Entretanto, mesmo fazendo uso destas estimativas, as funções de resposta da política monetária apresentam as mesmas propriedades de estimativas tradicionais para o Brasil.