Sequential Monte Carlo methods for data assimilation in strongly nonlinear dynamics.

A localized ensemble Kalman filter (EnKF) method is developed to assimilate transient flow data to calibrate a heterogeneous conductivity field. To update conductivity value at a point in a study domain, instead of assimilating all the measurements in the study domain, only limited measurement data in an area around the point are used for the conductivity updating in the localized EnKF method. The localized EnKF is proposed to solve the problems of the filter divergence usually existing in a data assimilation method without localization. The developed method is applied, in a synthetical two dimensional case, to calibrate a heterogeneous conductivity field by assimilating transient hydraulic head data. The simulations by the data assimilation with and without localized EnKF are compared. The study results indicate that the hydraulic conductivity field can be updated efficiently by the localized EnKF, while it cannot be by the EnKF. The covariance inflation and localization are found to solve the problem of the filter divergence efficiently. In comparison with the EnKF method without localization, the localized EnKF method needs smaller ensemble size to achieve stabilized results. The simulation results by the localized EnKF method are much more sensitive to conductivity correlation length than to the localization radius. The developed localized EnKF method provides an approach to improve EnKF method in conductivity calibration.

Assimilating real-time sensor data into large-scale spatial-temporal simulations, such as simulations of wildfires, is a promising technique for improving simulation results. This asks for advanced data assimilation methods that can work with the complex structures and nonlinear behaviors associated with the simulation models. This article presents a data assimilation framework using Sequential Monte Carlo (SMC) methods for wildfire spread simulations. The models and algorithms of the framework are described, and experimental results are provided. This work demonstrates the feasibility of applying SMC methods to data assimilation of wildfire spread simulations. The developed framework can potentially be generalized to other application areas where sophisticated simulation models are used.

The particle filter (PF) and the ensemble Kalman filter (EnKF) are two promising and popularly adopted types of ensemble‐based data assimilation methods for paleoclimate reconstruction. However, no systematic comparison between them has been attempted. We compare these two uncertainty based methods in pseudoproxy experiments where synthetic seasonal mean sea surface temperature observations are assimilated. Their skills are evaluated with regards to local, hemispherically averaged and globally averaged analysis error, and their ability to capture large‐scale modes of variability. It is found that the EAKF (Ensemble Adjustment Kalman filter, a variant of EnKF) performs better than the PF with only one third of the ensemble size, despite PF's theoretical superiority in allowing for non‐Gaussian statistics and nonlinear dynamics. The success of the EAKF is attributed to the facts that (1) Gaussian assumption is somewhat appropriate for this application; (2) The EAKF is less sensitive to sampling errors than the PF due to the different methodological natures. Sixteen members are enough to estimate accurate covariance for the EAKF, but 48 (even 96) members still underrepresent the state space of high‐dimensional system for the PF. Our study highlights the importance of a large localization radius in the application of the EnKF to paleoclimate reconstruction due to the sparse proxy network and suggests that additional techniques, such as localization or clustered particle filter, are needed to improve the PF for paleoclimate reconstruction, in addition to the simple importance resampling currently adopted by most research.

The ensemble Kalman filter (EnKF) is a Monte Carlo-based implementation of the Kalman filter (KF) for extremely high-dimensional, possibly nonlinear, and non-Gaussian state estimation problems. Its ability to handle state dimensions in the order of millions has made the EnKF a popular algorithm in different geoscientific disciplines. Despite a similarly vital need for scalable algorithms in signal processing, e.g., to make sense of the ever increasing amount of sensor data, the EnKF is hardly discussed in our field.This self-contained review is aimed at signal processing researchers and provides all the knowledge to get started with the EnKF. The algorithm is derived in a KF framework, without the often encountered geoscientific terminology. Algorithmic challenges and required extensions of the EnKF are provided, as well as relations to sigma point KF and particle filters. The relevant EnKF literature is summarized in an extensive survey and unique simulation examples, including popular benchmark problems, complement the theory with practical insights. The signal processing perspective highlights new directions of research and facilitates the exchange of potentially beneficial ideas, both for the EnKF and high-dimensional nonlinear and non-Gaussian filtering in general.

We propose a sequential Monte Carlo (SMC) method to efficiently and accurately compute cut-Bayesian posterior quantities of interest, variations of standard Bayesian approaches constructed primarily to account for model misspecification. We prove finite sample concentration bounds for estimators derived from the proposed method along with a linear tempering extension and apply these results to a realistic setting where a computer model is misspecified. We then illustrate the SMC method for inference in a modular chemical reactor example that includes submodels for reaction kinetics, turbulence, mass transfer, and diffusion. The samples obtained are commensurate with a direct-sampling approach that consists of running multiple Markov chains, with computational efficiency gains using the SMC method. Overall, the SMC method presented yields a novel, rigorous approach to computing with cut-Bayesian posterior distributions.

The regular and repetitive nature of the sequential Monte Carlo (SMC) method makes it very attractive for implementation using parallel and pipelined architectures. This paper develops a VLSI architecture for the hardware implementation of the SMC algorithm using bootstrap filter. A flat fading wireless channel is considered as our framework on which the channel estimator is designed and implemented using the SMC method. The design and verification activities at the algorithm level, architecture level, and the circuit level are reviewed. The proposed architecture is verified with an FPGA implementation.

This paper develops a parallel implementation of the ensemble Kalman filter (EnKF) based on shrinkage covariancematrix estimation. The EnKF is a sequential Monte Carlo method for parameter and state estimation of highly nonlinearmodels. In the context of ensemble based methods, background error correlations are estimated via anensemble of model realizations. In practice, the model dimension (number of model components) is several timeslarger than the ensemble size owing to the computational effort involved in one single model propagation. Thisconstraint yields to spurious correlations (correlation between distant model components in space) in the backgrounderror estimations. Commonly, localization methods are utilized in order to reduce the impact of spuriouscorrelations. For instance, correlations of model components are reduced based on some function of their physicaldistances. Some EnKF implementations are formulated in such manner that, localization methods are implicitlyinvolved in the assimilation of observations. In this context, one of the best ensemble Kalman filter implementationsis the local ensemble transform Kalman filter (LETKF). In the LETKF, each model component is surrounded by alocal box of size r and then, the information contained within the local domain (i.e., observed components) is used toperform the assimilation. However, for sparse observational networks, r can become sufficiently large with respectto the ensemble size and therefore, the local analysis corrections can be impacted by spurious correlations. Wepropose an ensemble Kalman filter implementation in which the background error covariance matrix is estimated viathe Rao-Blackwell Ledoit and Wolf (RBLW) estimator. This estimator has proven to be better conditioned than thetrue (background) error covariance matrix. In addition, the explicit computation of the RBLW estimator is notrequired in our formulation. This implies enormous savings in terms of memory considering the typical order ofbackground error covariance matrices O(10e8) Furthermore, the proposed EnKF implementation can be efficientlyperformed in parallel: each model component is surrounded by a local box, a local background error covariancematrix is estimated based on the RBLW estimator and then, the ensemble Kalman filter equations are locally used inorder to compute the local analysis corrections. After the local assimilation, each model component is mapped backonto the full model domain from where the global analysis solution is obtained. In operational data assimilation, EnKF implementations are attractive owing to their potential for parallelization. Experiments are performed making use of the Atmospheric General Circulation Model SPEEDY. The proposed method is coded in FORTRAN and MPI is used for the local assimilation of observations. The predicted variables are the zonal and meridional wind components, the temperature and the specific humidity. For each model variable, 8 layers with the T-63 resolution (192 x 96 grid points) are considered which provide a total number of 589,824 model components. The ensemble size is 94. The number of processors utilized are 96, 256, 512, 768, 1024 and 1536. The results reveal that, the use of the RBLW estimator can mitigate the impact of spurious correlations under sparse observational networks and even more, for different number of processors, the computational effort of the proposed EnKF implementation is comparable to that of the LETKF where no covariance estimation is performed. In terms of accuracy, the proposed implementation performs better than the LETKF formulation in the Root Mean Square Error sense.

The requirement for Terrain Referenced Navigation (TRN) also known as Terrain Aided Navigation (TAN) arises when Global Navigation Satellite System (GNSS) signals are unavailable, jammed or blocked. In recent years, research on the application of TRN to aerial and underwater vehicles has been increased rapidly with the developments in the accuracy of digital terrain elevation database (DTED). Since the land and sea floor profiles are inherently nonlinear, TRN becomes a nonlinear estimation problem. Because of the highly nonlinear and non-Gaussian problem, linear or linearized estimation techniques such as Kalman or Extended Kalman Filter (EKF) do not work properly for many terrain profiles. Hence, this paper focuses on the Sequential Monte Carlo (SMC) method namely the particle filter (PF) for dealing with nonlinearities and different types of probability distributions even multi modal. Two different PF implementations are studied, Sequential Importance Sampling with effective resampling (SIS-R) and Sampling Importance Resampling (SIR). The major difference between these implementations is that in SIS-R algorithm, resampling is only performed using effective sample size when actually required. Therefore, the SIS-R algorithm has less computational demands than the SIR algorithm for a given number of particles. Both algorithms are tested for an aircraft scenario over a commercial map. Simulations with different number of particles and inertial measurement units (IMUs) having various error measures are performed and investigated. Monte Carlo (MC) simulations show that both algorithms yield similar results in terms of horizontal position accuracy and convergence. Hereby with this paper, it is presented that besides requiring less computational load, the SIS-R implementation works accurately on TRN problem.

Abstract. Over the years data assimilation methods have been developed to obtain estimations of uncertain model parameters by taking into account a few observations of a model state. The most reliable Markov chain Monte Carlo (MCMC) methods are computationally expensive. Sequential ensemble methods such as ensemble Kalman filters and particle filters provide a favorable alternative. However, ensemble Kalman filter has an assumption of Gaussianity. Ensemble transform particle filter does not have this assumption and has proven to be highly beneficial for an initial condition estimation and a small number of parameter estimations in chaotic dynamical systems with non-Gaussian distributions. In this paper we employ ensemble transform particle filter (ETPF) and ensemble transform Kalman filter (ETKF) for parameter estimation in nonlinear problems with 1, 5, and 2500 uncertain parameters and compare them to importance sampling (IS). The large number of uncertain parameters is of particular interest for subsurface reservoir modeling as it allows us to parameterize permeability on the grid. We prove that the updated parameters obtained by ETPF lie within the range of an initial ensemble, which is not the case for ETKF. We examine the performance of ETPF and ETKF in a twin experiment setup, where observations of pressure are synthetically created based on the known values of parameters. For a small number of uncertain parameters (one and five) ETPF performs comparably to ETKF in terms of the mean estimation. For a large number of uncertain parameters (2500) ETKF is robust with respect to the initial ensemble, while ETPF is sensitive due to sampling error. Moreover, for the high-dimensional test problem ETPF gives an increase in the root mean square error after data assimilation is performed. This is resolved by applying distance-based localization, which however deteriorates a posterior estimation of the leading mode by largely increasing the variance due to a combination of less varying localized weights, not keeping the imposed bounds on the modes via the Karhunen–Loeve expansion, and the main variability explained by the leading mode. A possible remedy is instead of applying localization to use only leading modes that are well estimated by ETPF, which demands knowledge of which mode to truncate.

Ensemble methods such as the Ensemble Kalman Filter (EnKF) are widely used for data assimilation in large-scale geophysical applications, as for example in numerical weather prediction. There is a growing interest for physical models with higher and higher resolution, which brings new challenges for data assimilation techniques because of the presence of non-linear and non-Gaussian features that are not adequately treated by the EnKF. We propose two new localized algorithms based on the Ensemble Kalman Particle Filter, a hybrid method combining the EnKF and the Particle Filter (PF) in a way that maintains scalability and sample diversity. Localization is a key element of the success of EnKF in practice, but it is much more challenging to apply to PFs. The algorithms that we introduce in the present paper provide a compromise between the EnKF and the PF while avoiding some of the problems of localization for pure PFs. Numerical experiments with a simplified model of cumulus convection based on a modified shallow water equation show that the proposed algorithms perform better than the local EnKF. In particular, the PF nature of the method allows to capture non-Gaussian characteristics of the estimated fields such as the location of wet and dry areas.

The unstructured-grid finite-volume community ocean model (FVCOM) offers the potential to build a high-resolution operational prediction system because it can effectively adapt to complex boundaries and flexibly perform local mesh encryption. A matching assimilation system is an indispensable part of an operational forecast system, while the existing assimilation method of the FVCOM [e.g., the ensemble Kalman filter (EnKF)] has difficulty accommodating strong nonlinear model/observation operators and non-Gaussian errors. The localized weighted EnKF (LWEnKF) is a particle filter (PF) that combines the advantages of the EnKF and the PF, which can effectively address nonlinearity and non-Gaussianity and can provide a more accurate initial field for models. An observation preprocessing system and a new data assimilation system based on the LWEnKF for the FVCOM are developed. For the preprocessing system, a K-dimensional tree-based observation thinning method is proposed that can accommodate representative observation efficiently for unstructured grids. Through a 1-month global ocean data assimilation experiment, the effect of our assimilation system is compared with the EnKF and the Mercator analysis forecast product, which comes from a reduced-order Kalman filter system. The statistical verification includes five metrics for various sea surface fields, temperature, and salinity profiles. The results show that the system performs better than the EnKF, although there is still a gap with Mercator in some aspects.

Flood modeling and forecasting are crucial for managing and preparing for extreme flood events, such as those in the Tropical Andes. In this context, assimilating streamflow data is essential. Data Assimilation (DA) seeks to combine errors between forecasting models and discharge measurements through the updating of model states. This study aims to assess the applicability and performance of streamflow DA in a sub-daily forecasting system of the Peruvian Tropical Andes using the Ensemble Kalman Filter (EnKF) and Particle Filter (PF) algorithms. The study was conducted in a data-sparse Andean basin during the period February–March 2022. For this purpose, the lumped GR4H rainfall–runoff model was run forward with 100 ensemble members in four different DA experiments based on IMERG-E and GSMaP-NRT precipitation sources and assimilated real-time hourly discharges at the basin outlet. Ensemble modeling with EnKF and PF displayed that perturbation introduced by GSMaP-NRT’-driven experiments reduced the model uncertainties more than IMERG-E’ ones, and the reduction in high-flow subestimation was more notable for the GSMaP-NRT’+EnKF configuration. The ensemble forecasting framework from 1 to 24 h proposed here showed that the updating of model states using DA techniques improved the accuracy of streamflow prediction at least during the first 6–8 h on average, especially for the GSMaP-NRT’+EnKF scheme. Finally, this study benchmarks the application of streamflow DA in data-sparse basins in the Tropical Andes and will support the development of more accurate climate services in Peru.

Online Estimation of DSGE Models

Model and data reduction for data assimilation: Particle filters employing projected forecasts and data with application to a shallow water model

This paper develops a Rao-Blackwellised particle filtering algorithm for blind system identification. The state space model under consideration uses a time-varying autoregressive (AR) model for the sources, and a time-varying finite impulse response (FIR) model for the channel. The multi-sensor measurements result from the convolution of the sources with the channels in the presence of additive noise. A numerical approximation to the optimal Bayesian solution for the nonlinear sequential state estimation problem is implemented using sequential Monte Carlo (SMC) methods. The Rao-Blackwellisation technique is applied to improve the efficiency of the particle filter by marginalizing out the AR and FIR coefficients from the joint posterior distribution. Simulation results are given to verify the performance of the proposed method.