Sequential data assimilation with a nonlinear quasi‐geostrophic model using Monte Carlo methods to forecast error statistics

In the past few years sequential data assimilation (SDA) methods have emerged as the best possible method at hand to properly treat all sources of error in hydrological modeling. However, very few studies have actually implemented SDA methods using realistic input error models for precipitation. In this study we use particle filtering as a SDA method to propagate input errors through a conceptual hydrologic model and quantify the state, parameter and streamflow uncertainties. Recent progress in satellite‐based precipitation observation techniques offers an attractive option for considering spatiotemporal variation of precipitation. Therefore, we use the PERSIANN‐CCS precipitation product to propagate input errors through our hydrologic model. Some uncertainty scenarios are set up to incorporate and investigate the impact of the individual uncertainty sources from precipitation, parameters and also combined error sources on the hydrologic response. Also probabilistic measures are used to quantify the quality of ensemble prediction.

Estimation of the tropical Atlantic circulation from altimetry data using a reduced-rank stationary Kalman filter

The Kuramoto–Sivashinsky equation plays an important role as a low‐dimensional prototype for complicated fluid dynamics systems having been studied due to its chaotic pattern forming behavior. Up to now, efforts to carry out data assimilation with this 1‐D model were restricted to variational adjoint methods domain and only Chorin and Krause (Proc. Natl. Acad. Sci. 2004; 101(42):15013–15017) tested it using a sequential Bayesian filter approach. In this work we compare three sequential data assimilation methods namely the Kalman filter approach, the sequential Monte Carlo particle filter approach and the maximum likelihood ensemble filter methods. This comparison is to the best of our knowledge novel. We compare in detail their relative performance for both linear and nonlinear observation operators. The results of these sequential data assimilation tests are discussed and conclusions are drawn as to the suitability of these data assimilation methods in the presence of linear and nonlinear observation operators. Copyright © 2009 John Wiley & Sons, Ltd.

In this study, we present a probabilistic urban inundation modeling framework that combines high-resolution process-based modeling with observed information via ensemble data assimilation (DA). We investigate the impact of multivariate flood observations on improving urban inundation prediction accuracy through synthetic experiments. The framework leverages diverse flood observations from both the urban surface and sewer system, integrating them into the modeling process using non-Gaussian sequential DA methods, such as particle filtering. The modeling framework employs the H12 model for integrated 1D sewer network and 2D surface inundation analyses, with synthetic experiments conducted in an urban catchment in Osaka, Japan. Prior to implementing DA, a sensitivity analysis is performed to assess the effects of uncertainties in inundation modeling. Major uncertainty components, such as input forcings and storm drain box efficiency, are perturbed to evaluate their influence. The synthetic DA experiments analyze the influence of various types of flood observations, including urban surface inundation depths and sewer water levels, on the posterior distributions of the model ensemble. Additionally, the impact of observation location and density on DA performance is evaluated. This study demonstrates the potential of utilizing diverse flood observations in data assimilation to enhance the accuracy and reliability of urban flood predictions.

In numerical weather prediction, parameterisations are used to simulate missing physics in the model. These can be due to a lack of scientific understanding or a lack of computing power available to address all the known physical processes. Parameterisations are sources of large uncertainty in a model as parameter values used in these parameterisations cannot be measured directly and hence are often not well known; and the parameterisations themselves are also approximations of the processes present in the true atmosphere. Whilst there are many efficient and effective methods for combined state/parameter estimation in data assimilation (DA), such as state augmentation, these are not effective at estimating the structure of parameterisations. A new method of parameterisation estimation is proposed that uses sequential DA methods to estimate errors in the numerical models at each space-time point for each model equation. These errors are then fitted to pre-determined functional forms of missing physics or parameterisations that are based upon prior information. We applied the method to a one-dimensional advection model with additive model error, and it is shown that the method can accurately estimate parameterisations, with consistent error estimates. Furthermore, it is shown how the method depends on the quality of the DA results. The results indicate that this new method is a powerful tool in systematic model improvement.

The process and measurement noise covariance matrices significantly impact the Extended Kalman Filter (EKF) performance and are often hand-tuned in practice, which usually entails a tedious task. Q-learning, a well-known method in reinforcement learning, has been applied recently to better adapt the noise covariance matrices for the EKF, thanks to its simplicity and capability in handling uncertain environments. Typically, some heuristics are involved in designing the Q-learning-based EKF (QLEKF), such as tuning grid size and covariance matrices values of each state, which inevitably degrades the estimation performance when the heuristics are not suitable. We propose a dynamic grid-based Q-learning EKF (DG-QLEKF) to overcome that drawback, which brings two novelties, an updated ϵ-greedy algorithm and a dynamic grid strategy. The proposed algorithm and strategy can thoroughly exploit arbitrary search scope and find appropriate values of noise covariance matrices. The effectiveness of DGQLEKF, applied in navigation for attitude and bias estimation, is validated through the Monte Carlo method and real flight data from an unmanned aerial vehicle. The DG-QLEKF leads to much more improved state estimation than the QLEKF and traditional EKF.

This paper compares two different recursive tracking techniques for precisely localizing a mobile vehicle in an indoor harsh industrial environment. An extended Kalman filter (EKF) and unscented Kalman filter (UKF), the corresponding algorithms and mathematical models are presented and analysed. Experimental range measurements generated from local positioning radar system are used to test the performance of these algorithms with respect to position and velocity root mean square errors. True and estimated trajectories of the mobile vehicle with associated means and error covariances are illustrated with the number of samples required in each case. Results obtained show that UKF outer performs EKF with respect to positioning accuracy and root mean square error. Both filters show comparable computational complexity with more robustness obtained by applying UKF for non linear estimation since there are no linearization errors as in the case of EKF.

An Extended Kalman Filter (EKF) for the assimilation of remotely sensed near‐surface soil moisture into the Interactions between Surface, Biosphere, and Atmosphere (ISBA) model is described. ISBA is the land surface scheme in Météo‐France's Aire Limitée Adaptation Dynamique développement InterNational (ALADIN) Numerical Weather Prediction (NWP) model, and this work is directed toward providing initial conditions for NWP. The EKF is used to assimilate near‐surface soil moisture observations retrieved from C‐band Advanced Microwave Scanning Radiometer (AMSR‐E) brightness temperatures into ISBA. The EKF can translate near‐surface soil moisture observations into useful increments to the root‐zone soil moisture. If the observation and model soil moisture errors are equal, the Kalman gain for the root‐zone soil moisture is typically 20–30%, resulting in a mean net monthly increment for July 2006 of 0.025 m3 m−3 over ALADIN's European domain. To test the benefit of evolving the background error, the EKF is compared to a Simplified EKF (SEKF), in which the background errors at the time of the analysis are constant. While the Kalman gains for the EKF and SEKF are derived from different model processes, they produce similar soil moisture analyses. Despite this similarity, the EKF is recommended for future work where the extra computational expense can be afforded. The method used to rescale the near‐surface soil moisture data to the model climatology has a greater influence on the analysis than the error covariance evolution approach, highlighting the importance of developing appropriate methods for rescaling remotely sensed near‐surface soil moisture data.

This article reviews the characteristics of forecast error statistics in meteorological data assimilation from the substantial literature on this subject. It is shown how forecast error statistics appear in the data assimilation problem through the background error covariance matrix, B. The mathematical and physical properties of the covariances are surveyed in relation to a number of leading systems that are in use for operational weather forecasting. Different studies emphasize different aspects of B, and the known ways that B can impact the assimilation are brought together. Treating B practically in data assimilation is problematic. One such problem is in the numerical measurement of B, and five calibration methods are reviewed, including analysis of innovations, analysis of forecast differences and ensemble methods. Another problem is the prohibitive size of B. This needs special treatment in data assimilation, and is covered in a companion article (Part II). Examples are drawn from the literature that show the univariate and multivariate structure of the B‐matrix, in terms of variances and correlations, which are interpreted in terms of the properties of the atmosphere. The need for an accurate quantification of forecast error statistics is emphasized. Copyright © 2008 Royal Meteorological Society

A new sequential data assimilation method named “Monte Carlo H∞ filter” is introduced based on H∞ filter technique and Monte Carlo method in this paper. This method applies to nonlinear systems in condition of lacking the statistical properties of observational errors. In order to compare the assimilation capability of Monte Carlo H∞ filter with that of the ensemble Kalman filter (EnKF) in solving practical problems caused by temporal correlation or spatial correlation of observational errors, two numerical experiments are performed by using Lorenz (1963) system and shallow-water equations respectively. The result is that the assimilation capability of the new method is better than that of EnKF method. It is also shown that Monte Carlo H∞ filter assimilation method is effective and suitable to nonlinear systems in that it does not depend on the statistical properties of observational errors and has better robustness than EnKF method when the statistical properties of observational errors are varying. In addition, for the new method, the smallest level factor founded by search method is flow-dependent.

In this work we investigate the use of Extended Kalman Filter (EKF) methods to identify parameters in a nonlinear model. The model, derived in earlier work, describes how a person behaves when he or she is offered a sequence of rewards to carry out a task for which his or her initial attitude is negative. A main conclusion of the paper is that EKF methods can be used to effectively estimate a single parameter, but due to observability problems, estimation of multiple parameters is ineffective. Monte Carlo simulations are used to thoroughly study the performance of the EKF-based estimator to estimate a parameter (related to cognitive dissonance that is experienced by the person making the decision) in cases where only 5, 10, or 20 data samples are available. In addition, preliminary experimental results, based on experiments with 10 data samples, support the validity of the underlying model and demonstrate feasibility of the EKF approach for estimation of the cognitive-dissonance parameter, despite the small size of the data set.

Global positioning technologies such as the Global Positioning System (GPS) are ubiquitously available for different positioning applications. Within indoor environments, coverage of the explicit sensors based on GPS is limited. Developing an indoor location tracking system based on the Received Signal Strength Indicator (RSSI) of the Wireless Local Area Network (WLAN) is considered cost effective method. The widely used technique for estimating the position out of the RSSI measurements is the Extended Kalman Filter (EKF). However, EKF has high computational complexity due to the calculation of Jacobian matrices and suffers from filer instability. In this paper, we propose the Scaled Unscented Kalman Filter (SUKF), which is one of the Sigma Point Kalman Filters (SPKF) family, to overcome the limitations of the EKF. SUKF shall work over the WLAN IEEE 802.11n networks to exploit the RSSI range measurements for localizing and tracking of a mobile node. For performance evaluation, SUKF is compared with the EKF. Results are illustrated using Monte Carlo simulation in MATLAB.

One of the most important challenges involved in the formation flight of satellites is that of state estimation in terms of relative and inertial estimates. In near earth spacecraft GPS measurements are commonly used to solve this problem and for deep space missions the DSN is used. The objective of this paper is to determine sensitivities of relative state estimation errors to varying sensor quality in a GPS-denied environment. This paper considers a Chief-Deputy architecture that will travel in an elliptical orbit with a perigee in Low-Earth Orbit and an apogee in Mid-Earth orbit (MEO). While in the GPS-available zone, the Chief satellite will process GPS measurements to determine its state relative to the Earth-Centered Inertial (ECI) frame. Measurements are taken of range and range-rate of the Deputy relative to the Chief throughout the orbit and processed using an Indirect, Extended Kalman Filter (EKF). The states being estimated are the position and velocity of each satellite, along with sensor biases. A Monte-Carlo simulation is developed to validate the consistency of the EKF covariance estimates. Only two satellites are considered for simplicity, though the formation can be expanded to include an arbitrary number of satellites. Following validation of the EKF, the covariance estimates are used to identify trends in relative state estimation for the two-satellite formation as a function of sensor parameters. Specifically, the sensitivity of state estimation to range and range-rate sensor quality is assessed by analyzing covariance as a function of range and range-rate bias. The resulting analysis provides a preliminary estimate of system performance, enabling the selection of system components to meet future mission requirements.

Inertial Navigation Systems (INS) have an unbounded type of error characteristics. In order to overcome this problem, aiding systems are used to fuse information obtained via additional sensors with the INS solutions. Terrain Aided Navigation (TAN) is an integrated solution method which exploits Digital Terrain Elevation Maps (DTEM) and on board altimeter data to aid INS in case of GNSS outage. In TAN applications, a position measurement is formed by comparing the terrain profiles, which are obtained by using on board altimeters, with a preloaded DTEM. In general, this position measurement is formed by finding the DTEM points which create minimum-mean-square error with the terrain profile that is measured with on board altimeters. Error on the formed position measurement varies with the terrain profile over which the airborne platform flies. This position measurement is used to keep INS errors bounded with various sensor fusion techniques, two of which are Extended Kalman Filtering (EKF) and Gaussian Sum Filtering (GSF). However, error on the position measurement may cause these techniques to produce very erroneous navigation data or to behave in a divergent manner. Probabilistic Measurement Method (PMM) offered in this paper results with more probable position measurements which have less errors. PMM method is used with an EKF structure and takes advantage of position error covariances that are readily available within the EKF. Joint Probability Density Function (JPDF) of horizontal error components are formed and used while the terrain profile is compared with DTEM in order to form a position measurement. In this paper, performances of EKF with conventional positioning, GSF, and EKF with PMM are investigated and evaluated for 10 different flight paths. Flight paths are chosen to have different terrain profiles and the performance evaluation is realized by comparing the mean horizontal and vertical position errors.

Attitude and deformation coupled estimation of flexible satellite using low-cost sensors