scholarly journals Chaotic dynamics and the role of covariance inflation for reduced rank Kalman filters with model error

2018 ◽  
Author(s):  
Colin Grudzien ◽  
Alberto Carrassi ◽  
Marc Bocquet
Keyword(s):  
Ensemble Kalman Filter ◽  
Kalman Filters ◽  
Chaotic Dynamics ◽  
Forecast Error ◽  
Model Error ◽  
High Dimensional ◽  
Reduced Rank ◽  
Nonlinear Approximations ◽  
Complementary Subspace

Abstract. The ensemble Kalman filter and its variants have shown to be robust for data assimilation in high dimensional geophysical models, with localization, using ensembles of extremely small size relative to the model dimension. A reduced rank representation of the estimated covariance, however, leaves a large dimensional complementary subspace unfiltered. Utilizing the dynamical properties of the filtration for the backward Lyapunov vectors, this paper explores a previously unexplained mechanism, describing the intrinsic role of covariance inflation in reduced rank, ensemble based Kalman filters. Our derivation of the forecast error evolution describes the dynamic upwelling of the unfiltered error from outside of the span of the anomalies into the filtered subspace. Analytical results for linear systems explicitly describe the mechanism for the upwelling, and the associated recursive Riccati equation for the forecast error, while nonlinear approximations are explored numerically.

scholarly journals Chaotic dynamics and the role of covariance inflation for reduced rank Kalman filters with model error

2018 ◽  
Vol 25 (3) ◽  
pp. 633-648 ◽  
Author(s):  
Colin Grudzien ◽  
Alberto Carrassi ◽  
Marc Bocquet
Keyword(s):  
Ensemble Kalman Filter ◽  
Kalman Filters ◽  
Chaotic Dynamics ◽  
Forecast Error ◽  
Theoretical Interpretation ◽  
High Dimensional ◽  
Reduced Rank ◽  
Nonlinear Approximations ◽  
Complementary Subspace

Abstract. The ensemble Kalman filter and its variants have shown to be robust for data assimilation in high dimensional geophysical models, with localization, using ensembles of extremely small size relative to the model dimension. However, a reduced rank representation of the estimated covariance leaves a large dimensional complementary subspace unfiltered. Utilizing the dynamical properties of the filtration for the backward Lyapunov vectors, this paper explores a previously unexplained mechanism, providing a novel theoretical interpretation for the role of covariance inflation in ensemble-based Kalman filters. Our derivation of the forecast error evolution describes the dynamic upwelling of the unfiltered error from outside of the span of the anomalies into the filtered subspace. Analytical results for linear systems explicitly describe the mechanism for the upwelling, and the associated recursive Riccati equation for the forecast error, while nonlinear approximations are explored numerically.

scholarly journals A comparison of assimilation results from the ensemble Kalman Filter and a reduced-rank extended Kalman Filter

2003 ◽  
Vol 10 (6) ◽  
pp. 477-491 ◽  
Author(s):  
X. Zang ◽  
P. Malanotte-Rizzoli
Keyword(s):  
Kalman Filter ◽  
State Space ◽  
Extended Kalman Filter ◽  
Ensemble Kalman Filter ◽  
Forecast Error ◽  
High Viscosity ◽  
Empirical Orthogonal Functions ◽  
Reduced Rank ◽  
Low Viscosity ◽  
Reduced State

Abstract. The goal of this study is to compare the performances of the ensemble Kalman filter and a reduced-rank extended Kalman filter when applied to different dynamic regimes. Data assimilation experiments are performed using an eddy-resolving quasi-geostrophic model of the wind-driven ocean circulation. By changing eddy viscosity, this model exhibits two qualitatively distinct behaviors: strongly chaotic for the low viscosity case and quasi-periodic for the high viscosity case. In the reduced-rank extended Kalman filter algorithm, the model is linearized with respect to the time-mean from a long model run without assimilation, a reduced state space is obtained from a small number (100 for the low viscosity case and 20 for the high viscosity case) of leading empirical orthogonal functions (EOFs) derived from the long model run without assimilation. Corrections to the forecasts are only made in the reduced state space at the analysis time, and it is assumed that a steady state filter exists so that a faster filter algorithm is obtained. The ensemble Kalman filter is based on estimating the state-dependent forecast error statistics using Monte Carlo methods. The ensemble Kalman filter is computationally more expensive than the reduced-rank extended Kalman filter.The results show that for strongly nonlinear case, chaotic regime, about 32 ensemble members are sufficient to accurately describe the non-stationary, inhomogeneous, and anisotropic structure of the forecast error covariance and the performance of the reduced-rank extended Kalman filter is very similar to simple optimal interpolation and the ensemble Kalman filter greatly outperforms the reduced-rank extended Kalman filter. For the high viscosity case, both the reduced-rank extended Kalman filter and the ensemble Kalman filter are able to significantly reduce the analysis error and their performances are similar. For the high viscosity case, the model has three preferred regimes, each with distinct energy levels. Therefore, the probability density of the system has a multi-modal distribution and the error of the ensemble mean for the ensemble Kalman filter using larger ensembles can be larger than with smaller ensembles.

scholarly journals Brief communication: An innovation-based estimation method for model error covariance in Kalman filters

2021 ◽  
Author(s):  
Eviatar Bach ◽  
Michael Ghil
Keyword(s):  
Kalman Filter ◽  
Covariance Matrix ◽  
Ensemble Kalman Filter ◽  
Kalman Filters ◽  
Estimation Method ◽  
Model Error ◽  
High Accuracy ◽  
Covariance Estimation ◽  
Observation Error ◽  
Error Covariance

Abstract. We present a simple innovation-based model error covariance estimation method for Kalman filters. The method is based on Berry and Sauer (2013) and the simplification results from assuming known observation error covariance. We carry out experiments with a prescribed model error covariance using a Lorenz (1996) model and ensemble Kalman filter. The prescribed error covariance matrix is recovered with high accuracy.

Atmospheric Data Assimilation with an Ensemble Kalman Filter: Results with Real Observations

2005 ◽  
Vol 133 (3) ◽  
pp. 604-620 ◽  
Author(s):  
P. L. Houtekamer ◽  
Herschel L. Mitchell ◽  
Gérard Pellerin ◽  
Mark Buehner ◽  
Martin Charron ◽  
...  
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
Initial Conditions ◽  
Forecast Error ◽  
Lower Troposphere ◽  
Model Error ◽  
Atmospheric Data ◽  
Ensemble Mean ◽  
Parameterized Model

Abstract An ensemble Kalman filter (EnKF) has been implemented for atmospheric data assimilation. It assimilates observations from a fairly complete observational network with a forecast model that includes a standard operational set of physical parameterizations. To obtain reasonable results with a limited number of ensemble members, severe horizontal and vertical covariance localizations have been used. It is observed that the error growth in the data assimilation cycle is mainly due to model error. An isotropic parameterization, similar to the forecast-error parameterization in variational algorithms, is used to represent model error. After some adjustment, it is possible to obtain innovation statistics that agree with the ensemble-based estimate of the innovation amplitudes for winds and temperature. Currently, no model error is added for the humidity variable, and, consequently, the ensemble spread for humidity is too small. After about 5 days of cycling, fairly stable global filter statistics are obtained with no sign of filter divergence. The quality of the ensemble mean background field, as verified using radiosonde observations, is similar to that obtained using a 3D variational procedure. In part, this is likely due to the form chosen for the parameterized model error. Nevertheless, the degree of similarity is surprising given that the background-error statistics used by the two procedures are rather different, with generally larger background errors being used by the variational scheme. A set of 5-day integrations has been started from the ensemble of initial conditions provided by the EnKF. For the middle and lower troposphere, the growth rates of the perturbations are somewhat smaller than the growth rate of the actual ensemble mean error. For the upper levels, the perturbation patterns decay for about 3 days as a consequence of diffusive model dynamics. These decaying perturbations tend to severely underestimate the actual error that grows rapidly near the model top.

scholarly journals The Role of Network Science in Glioblastoma

Cancers ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 1045
Author(s):  
Marta B. Lopes ◽  
Eduarda P. Martins ◽  
Susana Vinga ◽  
Bruno M. Costa
Keyword(s):  
Personalized Medicine ◽  
Drug Development ◽  
Information Flow ◽  
Clinical Studies ◽  
Network Science ◽  
Cancer Genomics ◽  
High Dimensional ◽  
Network Discovery ◽  
Software Implementations

Network science has long been recognized as a well-established discipline across many biological domains. In the particular case of cancer genomics, network discovery is challenged by the multitude of available high-dimensional heterogeneous views of data. Glioblastoma (GBM) is an example of such a complex and heterogeneous disease that can be tackled by network science. Identifying the architecture of molecular GBM networks is essential to understanding the information flow and better informing drug development and pre-clinical studies. Here, we review network-based strategies that have been used in the study of GBM, along with the available software implementations for reproducibility and further testing on newly coming datasets. Promising results have been obtained from both bulk and single-cell GBM data, placing network discovery at the forefront of developing a molecularly-informed-based personalized medicine.

The influence of economic policy uncertainty and geopolitical risk on US citizens overseas air passenger travel by regional destination

2020 ◽  
pp. 135481662098119
Author(s):  
James E Payne ◽  
Nicholas Apergis
Keyword(s):  
Economic Policy ◽  
Forecast Error ◽  
Error Variance ◽  
Variance Decomposition ◽  
Air Travel ◽  
Policy Uncertainty ◽  
Economic Policy Uncertainty ◽  
Passenger Travel ◽  
Forecast Error Variance

This research note extends the literature on the role of economic policy uncertainty and geopolitical risk on US citizens overseas air travel through the examination of the forecast error variance decomposition of total overseas air travel and by regional destination. Our empirical findings indicate that across regional destinations, US economic policy uncertainty explains more of the forecast error variance of US overseas air travel, followed by geopolitical risk with global economic policy uncertainty explaining a much smaller percentage of the forecast error variance.

scholarly journals Model error in weather forecasting

2001 ◽  
Vol 8 (6) ◽  
pp. 357-371 ◽  
Author(s):  
D. Orrell ◽  
L. Smith ◽  
J. Barkmeijer ◽  
T. N. Palmer
Keyword(s):  
Prediction Models ◽  
Weather Forecasting ◽  
Initial Conditions ◽  
Forecast Error ◽  
Weather Prediction ◽  
Model Error ◽  
Local Model ◽  
State Dependent ◽  
Rapid Divergence ◽  
Dependent Model

Abstract. Operational forecasting is hampered both by the rapid divergence of nearby initial conditions and by error in the underlying model. Interest in chaos has fuelled much work on the first of these two issues; this paper focuses on the second. A new approach to quantifying state-dependent model error, the local model drift, is derived and deployed both in examples and in operational numerical weather prediction models. A simple law is derived to relate model error to likely shadowing performance (how long the model can stay close to the observations). Imperfect model experiments are used to contrast the performance of truncated models relative to a high resolution run, and the operational model relative to the analysis. In both cases the component of forecast error due to state-dependent model error tends to grow as the square-root of forecast time, and provides a major source of error out to three days. These initial results suggest that model error plays a major role and calls for further research in quantifying both the local model drift and expected shadowing times.

scholarly journals Greedy measurement selection for state estimation

2018 ◽  
Author(s):  
J. Nichols ◽  
Albert Cohen ◽  
Peter Binev ◽  
Olga Mula
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
High Dimensional ◽  
Dimensional Vector ◽  
Physical Processes ◽  
Linear Measurements ◽  
Measurement Selection ◽  
Selection For ◽  
Unknown Solution

Parametric PDEs of the general form $$ \mathcal{P}(u,a)=0 $$ are commonly used to describe many physical processes, where $\mathcal{P}$ is a differential operator, a is a high-dimensional vector of parameters and u is the unknown solution belonging to some Hilbert space V. Typically one observes m linear measurements of u(a) of the form $\ell_i(u)=\langle w_i,u \rangle$, $i=1,\dots,m$, where $\ell_i\in V'$ and $w_i$ are the Riesz representers, and we write $W_m = \text{span}\{w_1,\ldots,w_m\}$. The goal is to recover an approximation $u^*$ of u from the measurements. The solutions u(a) lie in a manifold within V which we can approximate by a linear space $V_n$, where n is of moderate dimension. The structure of the PDE ensure that for any a the solution is never too far away from $V_n$, that is, $\text{dist}(u(a),V_n)\le \varepsilon$. In this setting, the observed measurements and $V_n$ can be combined to produce an approximation $u^*$ of u up to accuracy $$ \Vert u -u^*\Vert \leq \beta^{-1}(V_n,W_m) \, \varepsilon $$ where $$ \beta(V_n,W_m) := \inf_{v\in V_n} \frac{\Vert P_{W_m}v\Vert}{\Vert v \Vert} $$ plays the role of a stability constant. For a given $V_n$, one relevant objective is to guarantee that $\beta(V_n,W_m)\geq \gamma >0$ with a number of measurements $m\geq n$ as small as possible. We present results in this direction when the measurement functionals $\ell_i$ belong to a complete dictionary.

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