Received 11 January 2009, accepted 8 April 2009.
Multisite modelling and streamflow prediction are mostly required in water resources design and management. This paper aims at investigating the extent of applicability of Kalmanfilter (KF) for modelling and predicting streamflow records in northern Algeria, which has never been done before. KF method is based upon the recursive least squares concept and has the important property of sequential optimization, which means that the model is adaptively updated as soon as the output of the system is obtained. One of the advantages of the KF technique is that the property of stationarity is not a prerequisite. This allows for changes in model parameters and variances, which is a manner that accommodates the non-linear response of hydrologic systems. Another advantage of KF is that its application is elaborated in the time-domain. This characteristic plays an important role in real time forecasting of hydrologic time series. Besides, the KF algorithm may be initiated with minimum objective information and adjusts itself subsequently whenever more data become available. The main purpose of this paper is to apply the KF approach to the modelling and prediction of multisite streamflow in northern part of Algeria. The data used are the annual streamflow records at 10 hydrometric stations located in the above region. The obtained result is an online prediction operation where the streamflow predictor is not bound to time or space, but rather adapts itself recursively to evolving conditions related to meteorology or other physical systems in the study area. It is observed that the obtained results are satisfactory and the associated errors are quite acceptable.
The main goal of this article is to provide a tutorial-style discus- sion on why traditional synchronization loop architectures, inherited from the analog era, may be abandoned in modern digital receivers and to move forward toward the design and actual use of more flex- ible, robust and powerful Kalmanfilter (KF)-based synchronization schemes. Carrier synchronization is a key process in most electronic devices involved in aerospace systems, and it is typically carried out following a two-stage approach: acquisition and tracking. The first stage detects the presence of the desired signal and provides a coarse estimate of its synchronization parameters, and the second one re- fines those estimates, filtering out noise and tracking any possible time variation . In the present work, we are concerned with the analysis of the carrier phase (CP) tracking problem. Hence, acquisi- tion and time delay synchronization are not discussed.
organizations in water resources management usually have observation wells in aquifers which their water levels are measured regularly. So, using both types of data –observed and model results- to reduce the uncertainty would be a wise approach.
Data assimilation methods are made to combine the observed data of any kind– here, water level data from observation wells- with predictions of the implemented model- here groundwater flow model. In fact, these methods try to find the best combination of observed and model predictions based on the uncertainty involved in each method as the solution . KalmanFilter (KF) is the most common sequential data assimilation method that assimilates a state-space expression of a prediction model with noisy observations to give an estimation of the system state with the least square error . KF is developed based on the state and observation equations that are respectively stochastic representation of model and observations. System state is a set of variables within the system that contain all past information affecting the future behavior of the system .
Keywords: data assimilation, Kalmanfilter, covariance dynamics, parameterisation of analysis
One of the foundations of data assimilation is based on the theory of Kalman filtering. Because of its computational complexity and the extent of required information for its implementation, the Kalmanfilter (KF) has long been recognised as not viable for large dimension problems in geosciences. Alternative formulations, based for example on ensemble methods, have been developed. The ensemble Kalmanfilter (EnKF) was developed by Evensen (1994). The numerous formulations of the sequential algorithm or its smoother version have also had an impact on the variational data assimilation where new algorithms now take advantage of adjoint-free formulation. Considering other ensemble strategies, like particle filter methods in their present formulation, the EnKF is very robust and is used for atmospheric data assimilation with a limited ensemble of few dozen members.
Index Terms—Phase estimation, Extended KalmanFilter, over- sampling, carrier synchronization, GALILEO, BOC.
I. I NTRODUCTION
Synchronization is a fundamental part in modern digital receivers. A synchronizer has to estimate some parameters, such as carrier frequency, carrier phase and timing epoch. This knowledge is required to recover the signal of interest correctly. In this paper we focus our attention on the phase estimation problem. Many methods for estimating the phase introduced by an unknown channel have been proposed over the past decades, from Phase Locked Loops (PLL) to the most sophisticated signal processing techniques. The KalmanFilter (KF) , , presented in early 1960s is one of the mostly used techniques for parameter estimation in linear gaussian problems. We can find extensive discussion on the KF in  and . When dealing with nonlinear filtering problems, the Extended KalmanFilter (EKF) approximates the problem to apply the KF solution. Some contributions show the use of EKF for carrier phase recovery and frequency tracking – . To our knowledge, the EKF has never been applied to oversampled phase estimation for Binary Offset Carrier (BOC) shaped signals.
Standard state estimation techniques, ranging from the linear Kalmanfilter (KF) to nonlinear extended KF (EKF), sigma-point or particle filters, assume a per- fectly known system model, that is, process and measurement functions and system noise statistics (both the distribution and its parameters). This is a strong assumption which may not hold in practice, reason why several approaches have been proposed for robust filtering, mainly because the filter performance is particularly sensitive to different model mismatches. In the context of linear filtering, a solution to cope with possible system matrices mismatch is to use linear constraints. In this contribution we further explore the extension and use of recent results on linearly constrained KF for robust nonlinear filtering under both process and measurement model mismatch. We first investigate how lin- ear equality constraints can be incorporated within the EKF and derive a new linearly constrained extended KF (LCEKF). Then we detail its use to mitigate parametric modeling errors in the nonlinear process and measurement func- tions. Numerical results are provided to show the performance improvement of the new LCEKF for robust vehicle navigation.
Chˆ ateaufort, 78772 Magny Les Hameaux CEDEX.
∗∗ MINES ParisTech, PSL Reasearch University, Centre for Robotics,
60 bd Saint-Michel, 75006 Paris, France.
Abstract: In the aerospace industry the (multiplicative) extended Kalmanfilter (EKF) is the most common method for sensor fusion for guidance and navigation. However, from a theoretical point of view, the EKF has been shown to possess local convergence properties only under restrictive assumptions. In a recent paper, we proved a slight variant of the EKF, namely the invariant extended Kalmanfilter (IEKF), when used as a nonlinear observer, possesses local convergence properties under the same assumptions as those of the linear case, for a class of systems defined on Lie groups. This is especially interesting as the IEKF also retains all the desirable features of the standard EKF, especially its relevant tuning in the presence of noises. In the present paper we provide three examples of engineering interest where the theory is shown to apply, yielding three EKF-like algorithms with guaranteed local convergence properties. Beyond those contributions, the present article is sufficiently accessible to help the practitioner understand through concrete examples the general IEKF theory, and to provide him with guidelines for the design of IEKFs.
In this study we perform ensemble Kalmanfilter (EnKF) data assimilation experiments during some severe dust storm episodes in China using surface observations of dust con- centrations and a realistic model in order to explore the impacts on forecast skills. The EnKF is an advanced and flexible technique for data assimilation which can calculate flow-dependent statistics from the ensemble forecasts and have been widely used in atmospheric and oceanic applica- tions (Evensen, 1994; Houtekamer and Mitchell, 1998, 2001; Mitchell and Houtekamer, 2000, 2002; Houtekamer et al., 2005; Whitaker et al., 2002; Lorenc, 2003; Evensen, 2003, 2006; Hanea et al., 2007). However the EnKF has not been applied in severe dust storm forecasts. In this study, we made an initial effort to explore the potential problems of this issue with EnKF.
I. I NTRODUCTION
Simultaneous Localization and Mapping or SLAM is an active area of research in robotics due to its use in emerging applications such as autonomous driving and piloting, search- and-rescue missions and mobile cartography . SLAM is fundamentally a sensor fusion problem, and as such it is typically handled via an Extended KalmanFilter (EKF), although a number of direct nonlinear designs have also been proposed e.g. , , , , .
Secondly, the topic of “more intelligent engines” [RTO, 2009] is gaining momentum in the community. One of the objectives is to bring additional capabilities in the engine control system. On one hand, the control laws would adapt to the actual condition of the engine (e.g, the acceleration schedule would be modified in the case where the high pressure compressor is degraded). On the other hand, the regulation would be based on so-called virtual sensors. For instance, the thrust predicted by an on-board model would replace current thrust setting parameters such as the fan speed or engine pressure ratio. Both of these strategies require accurate estimates of the health condition and of non-measurable quantities which could be handled by tools similar to the one designed in this thesis. The third area for future research that we propose addresses a number of questions related to modelling. Firstly, we remember that the Kalmanfilter tracks gradual deterioration on the basis of a very coarse model for the temporal evolution of the health parameters. It mainly serves to dampen the measurement noise, but does not take into account the operational history of the engine. However, the wear profile depends to a large extent on engine usage. The development of ageing models based on operational factors, see e.g., [Spieler et al., 2007, Wensky et al., 2010], is an emerging field that will increase the level of information available to track performance changes.
* Correspondence: firstname.lastname@example.org; Tel.: +86-532-6678-1265 Academic Editor: Changzhi Li
Received: 21 May 2016; Accepted: 11 November 2016; Published: 25 November 2016
Abstract: The swell propagation model built on geometric optics is known to work well when simulating radiated swells from a far located storm. Based on this simple approximation, satellites have acquired plenty of large samples on basin-traversing swells induced by fierce storms situated in mid-latitudes. How to routinely reconstruct swell fields with these irregularly sampled observations from space via known swell propagation principle requires more examination. In this study, we apply 3-h interval pseudo SAR observations in the ensemble Kalmanfilter (EnKF) to reconstruct a swell field in ocean basin, and compare it with buoy swell partitions and polynomial regression results. As validated against in situ measurements, EnKF works well in terms of spatial–temporal consistency in far-field swell propagation scenarios. Using this framework, we further address the influence of EnKF parameters, and perform a sensitivity analysis to evaluate estimations made under different sets of parameters. Such analysis is of key interest with respect to future multiple-source routinely recorded swell field data. Satellite-derived swell data can serve as a valuable complementary dataset to in situ or wave re-analysis datasets.
This paper presents a new approach combining the Bayesian Dynamic Linear Models framework with the Switching KalmanFilter theory for detecting anomalies of the behaviour structures. The key aspects are that (1) it enables early anomaly detection, (2) it is robust towards false alarms in real operation condition, and (3) it does not require labeled training data with normal and abnormal conditions. The approach is applied to the horizontal displacement data collected on a dam in Canada. In the case study considered, the method has shown that it was capable to detect the changes in the dam behaviour caused by the refection work. It also provided the specific information about the dam behaviour over time. This new approach offers a promising path toward the large-scale deployment of SHM system for monitoring behaviour of a population of structures.
This paper proposes a unified state-space formulation for parameter estimation of exponential-affine term structure models. This class of models, charaterized by Duffie and Kan (1993), contains models such as Vasicek (1977), Cox, Ingersoll and Ross (1985) and Chen and Scott (1992), among others. The proposed method uses an approximate linear Kalmanfilter which only requires specifying the conditional mean and variance of the system in an approximate sense. The method allows for measurement errors in the observed yields to maturitiy, and can simultaneously deal with many yields on bonds with different maturities. A Monte Carlo study indicates thet the proposed method is a reliable procedure for moderate sample sizes. An empirical analysis of three existing exponential-affine term structure models is carried out using monthly U.S. Treasury yield data with four different maturities. Our test results indicate a strong rejection of all three models.
Similarly, conservation of mass during the ensemble filter update depends on the particular numerical method used to generate the replicates of the updated ensemble and to generate the mean of this ensemble (analysis itself). Mass may not be conserved during the update even when the numerical forecast technique is mass conservative. A number of methods have been proposed to deal with this issue. For example, Jacobs and Ngodock (2003) noted that mass in a simplified 1D ocean model can be conserved when the model error in a representer algorithm is expressed in terms of the mass flux due to uncertainty in ocean depth rather than as additive error in the continuity equation. In land surface hydrology, Pan and Wood (2006) showed how to ensure conserva- tion of total water mass by imposing it as a ‘‘perfect observation’’ in a two-step Kalmanfilter approach. In an ocean data assimilation system, Brankart et al. (2003) imposed conservation of total mass, including positive layer thicknesses, through an a posteriori adjustment to the analyzed state. Positivity can also be ensured by introducing a change of state variables, using techniques such as Gaussian anamorphosis (e.g., Simon and Bertino 2009). In atmospheric data assimilation, nonnegativity of the specific humidity has been imposed as a weak constraint in a three-dimensional variational data as- similation (3D-Var) implementation (Liu and Xue 2006; Liu et al. 2007).
Autonomous Excavation of Rocks Using a Gaussian Process Model and Unscented KalmanFilter
Filippos E. Sotiropoulos and H. Harry Asada, Member, IEEE
Abstract—In large-scale open-pit mining and construction works, excavators must deal with large rocks mixed with gravel and granular soil. Capturing and moving large rocks with the bucket of an excavator requires a high level of skill that only experienced human operators possess. In an attempt to develop autonomous rock excavators, this paper presents a control method that predicts the rock movement in response to bucket operation and computes an optimal bucket movement to capture the rock. The process is highly nonlinear and stochastic. A Gaussian process model, which is nonlinear, non- parametric and stochastic, is used for describing rock behaviors interacting with the bucket and surrounding soil. Experimental data is used directly for identifying the model. An Unscented KalmanFilter (UKF) is then integrated with the Gaussian process model for predicting the rock movements and estimating properties of the rock. A feedback controller that optimizes a cost function is designed based on the rock motion prediction and implemented on a robotic excavator prototype. Experiments demonstrate encouraging results towards autonomous mining and rock excavation.
I. I NTRODUCTION
Tracking is an important technique for a variety of ap- plications, ranging from military such as anti-aircraft and missiles shelter defense systems to very commercial cases like unmanned autopilot navigation, monitoring and security. The main concern in all the applications is to find the object of interest, which is generally called the target, and then to follow it using vision systems. The goal of object tracking in a video stream is to continuously and reliably determine the position of an object against dynamic scenes with presence of noise . To this sake, numerous sophisticated algorithms have been proposed and implemented. For example, con- sidering Gaussian and linear problems, Welch and Bishop  proposed a Kalmanfilter for tracking a user’s pose for interactive interface with virtual environments. The proposed single-constraint-at-a-time (SCAAT) tracking method fused the measurements of different optical sensors in order to im- prove the tracking accuracy and stability. Proposed schemes in  recruited the posterior probability distribution over some scene properties of interest, based on image observa- tions to improve the functionality of object tracking under real working conditions. Other tracking strategies can also be found as Multiple Hypothesis Tracking , kernel-based tracking , and tracking based on optical flow .
The Unscented Kalmanfilter
To overcome the underlying assumptions and flaws inherent to the Extended Kalmanfilter (EKF), an al- ternative algorithm, referred to as the Unscented Kalmanfilter (UKF), was first proposed by Julier et al. [ 50 , 51 ]. The basic difference between the EKF and the UKF lies in the manner in which Gaussian ran- dom variables are represented for the purpose of propagating them through nonlinear system equations. In the EKF, state distribution is approximated by a Gaussian random variable which is then propagated analytically through the first-order linearization of the nonlinear system equations. The UKF addresses this problem by using a deterministic sampling approach based on the mathematical Unscented transfor- mation. The state distribution is then, as with the EKF, approximated by a Gaussian random variable, but is now sampled at a minimal set of carefully chosen sample points. These sample points completely capture the true mean and covariance of the Gaussian random variable, and when propagated though the nonlinear functions, yield exact evaluations of the posterior mean and covariance (while the EKF achieves first-order accuracy only). The reader may refer to [ 50 , 52 ] for more information on UKF performance in predicting mean and covariance of a random variable that undergoes a nonlinear trans- formation. An illustration of the Unscented transformation is shown in Figure C.1.
The matrices H and Φ and the random vectors E t and E t ∗ have explicit (but complex)
A rich literature Guo(1999), Shepard(1994) exists to estimate parameters of the state space models. Such techniques are closely related to statistical data assimilation schemes. In Gaussian state space models, the Kalmanfilter provides an optimal recursive estimate of x(t) from observations Y t = (y 1 (t), . . . , y J (t)). Unfortunately, the nature of the pulse-
Keywords - Needle steering; needle detection; ultrasound; Kalmanfilter
I. I NTRODUCTION
To reduce patient injuries and guarantee procedure success, needle-involved clinical procedures require very fine precision during insertion. This problem has motivated many research for robotizing needle insertion. Among them, needle steering - i.e. fine control of needle curvature for the execution of curved trajectories toward a target - has received increasing interest over the last decade.
Keywords Nonlinear asymptotic observer, Extended KalmanFilter, Contraction theory.
Since the seminal work of Kalman and Bucy  and Luenberger , the problem of building observers for deterministic linear systems has been laid on firm theoretical ground. Yet, when the system is nonlinear, there is no general methods to tackle observer design. Over the last decades, nonlinear observer design has been an active field of research, and several methods have emerged for attacking some specific nonlinearities. In the engineering world, the most popular method is the so-called Extended KalmanFilter (EKF), a natural extension of the Kalmanfilter. The principle is to linearize the system around the trusted (i.e. estimated) trajectory of the system, build a Kalmanfilter for this time-varying linear model, and imple- ment it on the nonlinear model. The EKF is known to yield good results in practice when the guess on the initial state is close enough to the actual state, but possesses no guarantee of convergence in the general case, and indeed can diverge.