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Xiaodong Luo and Ibrahim Hoteit

Abstract

This study considers the data assimilation problem in coupled systems, which consists of two components (subsystems) interacting with each other through certain coupling terms. A straightforward way to tackle the assimilation problem in such systems is to concatenate the states of the subsystems into one augmented state vector, so that a standard ensemble Kalman filter (EnKF) can be directly applied. This work presents a divided state-space estimation strategy, in which data assimilation is carried out with respect to each individual subsystem, involving quantities from the subsystem itself and correlated quantities from other coupled subsystems. On top of the divided state-space estimation strategy, the authors also consider the possibility of running the subsystems separately. Combining these two ideas, a few variants of the EnKF are derived. The introduction of these variants is mainly inspired by the current status and challenges in coupled data assimilation problems and thus might be of interest from a practical point of view. Numerical experiments with a multiscale Lorenz 96 model are conducted to evaluate the performance of these variants against that of the conventional EnKF. In addition, specific for coupled data assimilation problems, two prototypes of extensions of the presented methods are also developed in order to achieve a trade-off between efficiency and accuracy.

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Xiaodong Luo and Ibrahim Hoteit

Abstract

The ensemble Kalman filter (EnKF) is an efficient algorithm for many data assimilation problems. In certain circumstances, however, divergence of the EnKF might be spotted. In previous studies, the authors proposed an observation-space-based strategy, called residual nudging, to improve the stability of the EnKF when dealing with linear observation operators. The main idea behind residual nudging is to monitor and, if necessary, adjust the distances (misfits) between the real observations and the simulated ones of the state estimates, in the hope that by doing so one may be able to obtain better estimation accuracy.

In the present study, residual nudging is extended and modified in order to handle nonlinear observation operators. Such extension and modification result in an iterative filtering framework that, under suitable conditions, is able to achieve the objective of residual nudging for data assimilation problems with nonlinear observation operators. The 40-dimensional Lorenz-96 model is used to illustrate the performance of the iterative filter. Numerical results show that, while a normal EnKF may diverge with nonlinear observation operators, the proposed iterative filter remains stable and leads to reasonable estimation accuracy under various experimental settings.

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Xiaodong Luo and Ibrahim Hoteit

Abstract

This article examines the influence of covariance inflation on the distance between the measured observation and the simulated (or predicted) observation with respect to the state estimate. In order for the aforementioned distance to be bounded in a certain interval, some sufficient conditions are derived, indicating that the covariance inflation factor should be bounded in a certain interval, and that the inflation bounds are related to the maximum and minimum eigenvalues of certain matrices. Implications of these analytic results are discussed, and a numerical experiment is presented to verify the validity of the analysis conducted.

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Xiaodong Luo and Ibrahim Hoteit

Abstract

A robust ensemble filtering scheme based on the H filtering theory is proposed. The optimal H filter is derived by minimizing the supremum (or maximum) of a predefined cost function, a criterion different from the minimum variance used in the Kalman filter. By design, the H filter is more robust than the Kalman filter, in the sense that the estimation error in the H filter in general has a finite growth rate with respect to the uncertainties in assimilation, except for a special case that corresponds to the Kalman filter.

The original form of the H filter contains global constraints in time, which may be inconvenient for sequential data assimilation problems. Therefore a variant is introduced that solves some time-local constraints instead, and hence it is called the time-local H filter (TLHF). By analogy to the ensemble Kalman filter (EnKF), the concept of ensemble time-local H filter (EnTLHF) is also proposed. The general form of the EnTLHF is outlined, and some of its special cases are discussed. In particular, it is shown that an EnKF with certain covariance inflation is essentially an EnTLHF. In this sense, the EnTLHF provides a general framework for conducting covariance inflation in the EnKF-based methods. Some numerical examples are used to assess the relative robustness of the TLHF–EnTLHF in comparison with the corresponding KF–EnKF method.

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Ibrahim Hoteit, Xiaodong Luo, and Dinh-Tuan Pham

Abstract

This paper investigates an approximation scheme of the optimal nonlinear Bayesian filter based on the Gaussian mixture representation of the state probability distribution function. The resulting filter is similar to the particle filter, but is different from it in that the standard weight-type correction in the particle filter is complemented by the Kalman-type correction with the associated covariance matrices in the Gaussian mixture. The authors show that this filter is an algorithm in between the Kalman filter and the particle filter, and therefore is referred to as the particle Kalman filter (PKF).

In the PKF, the solution of a nonlinear filtering problem is expressed as the weighted average of an “ensemble of Kalman filters” operating in parallel. Running an ensemble of Kalman filters is, however, computationally prohibitive for realistic atmospheric and oceanic data assimilation problems. For this reason, the authors consider the construction of the PKF through an “ensemble” of ensemble Kalman filters (EnKFs) instead, and call the implementation the particle EnKF (PEnKF). It is shown that different types of the EnKFs can be considered as special cases of the PEnKF. Similar to the situation in the particle filter, the authors also introduce a resampling step to the PEnKF in order to reduce the risk of weights collapse and improve the performance of the filter. Numerical experiments with the strongly nonlinear Lorenz-96 model are presented and discussed.

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Boujemaa Ait-El-Fquih and Ibrahim Hoteit

Abstract

This work addresses the state–parameter filtering problem for dynamical systems with relatively large-dimensional state and low-dimensional parameters’ vector. A Bayesian filtering algorithm combining the strengths of the particle filter (PF) and the ensemble Kalman filter (EnKF) is proposed. At each assimilation cycle of the proposed EnKF–PF, the PF is first used to sample the parameters’ ensemble followed by the EnKF to compute the state ensemble conditional on the resulting parameters’ ensemble. The proposed scheme is expected to be more efficient than the traditional state augmentation techniques, which suffer from the curse of dimensionality and inconsistency that is particularly pronounced when the state is a strongly nonlinear function of the parameters. In the new scheme, the EnKF and PF interact via their ensembles’ members, in contrast with the recently introduced two-stage EnKF–PF (TS–EnKF–PF), which exchanges point estimates between EnKF and PF while requiring almost double the computational load. Numerical experiments are conducted with the Lorenz-96 model to assess the behavior of the proposed filter and to evaluate its performances against the joint PF, joint EnKF, and TS–EnKF–PF. Numerical results suggest that the EnKF–PF performs best in all tested scenarios. It was further found to be more robust, successfully estimating both state and parameters in different sensitivity experiments.

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Bo Liu, Boujemaa Ait-El-Fquih, and Ibrahim Hoteit

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The Bayesian filtering problem for data assimilation is considered following the kernel-based ensemble Gaussian mixture filtering (EnGMF) approach introduced by Anderson and Anderson. In this approach, the posterior distribution of the system state is propagated with the model using the ensemble Monte Carlo method, providing a forecast ensemble that is then used to construct a prior Gaussian mixture (GM) based on the kernel density estimator. This results in two update steps: a Kalman filter (KF)-like update of the ensemble members and a particle filter (PF)-like update of the weights, followed by a resampling step to start a new forecast cycle. After formulating EnGMF for any observational operator, the influence of the bandwidth parameter of the kernel function on the covariance of the posterior distribution is analyzed. Then the focus is on two aspects: (i) the efficient implementation of EnGMF with (relatively) small ensembles, where a new deterministic resampling strategy is proposed preserving the first two moments of the posterior GM to limit the sampling error; and (ii) the analysis of the effect of the bandwidth parameter on contributions of KF and PF updates and on the weights variance. Numerical results using the Lorenz-96 model are presented to assess the behavior of EnGMF with deterministic resampling, study its sensitivity to different parameters and settings, and evaluate its performance against ensemble KFs. The proposed EnGMF approach with deterministic resampling suggests improved estimates in all tested scenarios, and is shown to require less localization and to be less sensitive to the choice of filtering parameters.

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Sabique Langodan, Luigi Cavaleri, Yesubabu Viswanadhapalli, and Ibrahim Hoteit

Abstract

The Red Sea is a narrow, elongated basin that is more than 2000 km long. This deceivingly simple structure offers very interesting challenges for wind and wave modeling, not easily, if ever, found elsewhere. Using standard meteorological products and local wind and wave models, this study explores how well the general and unusual wind and wave patterns of the Red Sea could be reproduced. The authors obtain the best results using two rather opposite approaches: the high-resolution Weather Research Forecasting (WRF) local model and the slightly enhanced surface winds from the global European Centre for Medium-Range Weather Forecasts model. The reasons why these two approaches produce the best results and the implications on wave modeling in the Red Sea are discussed. The unusual wind and wave patterns in the Red Sea suggest that the currently available wave model source functions may not properly represent the evolution of local fields. However, within limits, the WAVEWATCH III wave model, based on Janssen’s and also Ardhuin’s wave model physics, provides very reasonable results in many cases. The authors also discuss these findings and outline related future work.

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Thang M. Luong, Hari P. Dasari, and Ibrahim Hoteit

Abstract

The city of Jeddah, Saudi Arabia, is characterized by a hot and arid desert climate. On occasion, however, extreme precipitation events have led to flooding that caused extensive damage to human life and infrastructure. This study investigates the effect of incorporating an urban canopy model and urban land cover when simulating severe weather events over Jeddah using the Weather Research and Forecasting (WRF) Model at a convective-permitting scale (1.5-km resolution). Two experiments were conducted for 10 heavy rainfall events associated with the dominant large-scale patterns favoring convection over Jeddah: (i) an “urban” experiment that included the urban canopy model and modern-day land cover and (ii) a “desert” experiment that replaced the city area with its presettlement, natural land cover. The results suggest that urbanization plays an important role in modifying rainfall around city area. The urban experiment enhances the amount of rainfall by 26% on average over the Jeddah city area relative to the desert experiment in these extreme events. The changes in model-simulated precipitation are primarily tied to a nocturnal heat-island effect that modifies the planetary boundary layer and atmospheric instability of the convective events.

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Naila F. Raboudi, Boujemaa Ait-El-Fquih, and Ibrahim Hoteit

Abstract

The ensemble Kalman filter (EnKF) is widely used for sequential data assimilation. It operates as a succession of forecast and analysis steps. In realistic large-scale applications, EnKFs are implemented with small ensembles and poorly known model error statistics. This limits their representativeness of the background error covariances and, thus, their performance. This work explores the efficiency of the one-step-ahead (OSA) smoothing formulation of the Bayesian filtering problem to enhance the data assimilation performance of EnKFs. Filtering with OSA smoothing introduces an updated step with future observations, conditioning the ensemble sampling with more information. This should provide an improved background ensemble in the analysis step, which may help to mitigate the suboptimal character of EnKF-based methods. Here, the authors demonstrate the efficiency of a stochastic EnKF with OSA smoothing for state estimation. They then introduce a deterministic-like EnKF-OSA based on the singular evolutive interpolated ensemble Kalman (SEIK) filter. The authors show that the proposed SEIK-OSA outperforms both SEIK, as it efficiently exploits the data twice, and the stochastic EnKF-OSA, as it avoids observational error undersampling. They present extensive assimilation results from numerical experiments conducted with the Lorenz-96 model to demonstrate SEIK-OSA’s capabilities.

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