Error Covariance Matrix Research Articles

Adaptive Kalman filter-based information fusion in electrical impedance tomography for a two-phase flow

Electrical impedance tomography (EIT) is a severely ill-posed nonlinear inverse problem. In order to obtain solutions with physical meaning, the inverse of the model of measurements requires the combination of information from various sources. This paper proposes a new approach through Kalman filtering for adaptive integration of EIT measurements, Tikhonov regularization and evolution models for the characterization of a two-phase air–water fluid flow. The Tikhonov regularization factor is embedded into the observation error covariance matrix, thus allowing for individual adjustment for each of the regularization equations. The filter outputs for different evolution models—random walk, advective and advective–diffusive—are compared in terms of estimate convergence and physical meaning. With the random walk evolution model the analysis of experimental data shows that the proposed information fusion strategy provides fewer artifacts, enabling a more effective identification of the phase interfaces. When the other two evolution models are incorporated into the Kalman filter and compared with the random walk model, faster and more accurate estimates of the flow are obtained even away from the electrodes, as well as sharper phase interfaces are identified. The results suggest that the reason for this improved performance is the fused information from the upstream–downstream dynamics of the advective and advective–diffusive models with the outer-inner structure influence of measurements.

An improved adaptive unscented kalman filtering for state of charge online estimation of lithium-ion battery

Precise state of charge (SOC) estimation is crucial to assure safe and reliable operation of lithium-ion battery in electric vehicles. Adaptive unscented Kalman filter (AUKF) has been intensively applied to estimate SOC due to its features of self-correction and high accuracy. Nevertheless, the estimation by traditional AUKF cannot proceed when error covariance matrix is non-positive definite, greatly influencing the stability of SOC estimation. To address this issue, an improved AUKF is proposed in this paper. Firstly, the forgetting factor recursive least square is employed to online identify parameters of electrical equivalent circuit model. With these identified parameters, an improved AUKF, whose Cholesky decomposition for error covariance matrix of tradition AUKF is replaced by singular value decomposition, is applied here to provide online accurate SOC estimation. The feasibility of the proposed method is verified by experimental data under Federal Urban Driving Schedule test. The validation results of robustness present that the algorithm has satisfactory robustness against inaccurate initial SOC. Moreover, through the comparison with traditional AUKF, it can be easily concluded that the proposed method can achieve precise and stable SOC estimation even though error covariance matrix is non-positive definite.

Gray-Box Parsimonious Subspace Identification of Hammerstein-Type Systems

In this article, a gray-box parsimonious subspace identification method using sampled dynamical and steady-state data is proposed for block-oriented Hammerstein-type systems. Compared with the conventional subspace identification methods based on over-parameterized models, the proposed method assumes parsimonious models, where the number of parameters is as minimal as possible to ensure the model accuracy, especially for highly nonlinear models. Generally, the available dynamical data do not contain adequate information on the low-frequency characteristics of the system. To improve the accuracy of model identification, the steady-state information of the system is taken into account, and a multiregularization method is developed, where the whole model parameters are estimated in a hierarchical, iterative manner from two sets of data: the dynamical input–output data and the steady-state input–output data. The use of parsimonious models and hierarchical estimation can significantly reduce the size of the associated parameter estimation error covariance matrix, thus improving the model accuracy and decreasing the variance of the estimated parameters, compared with the over-parametrization methods only using dynamical data. The effectiveness and merits are demonstrated by a simulation example and a real-world example.

A Fuzzy-Logic-Based Covariance Localization Method in Data Assimilation

In ensemble data assimilation systems, the impracticalities of full sampling and systematic error often lead to spurious correlations between two variables with low actual correlations. To solve these problems, researchers have previously proposed a covariance localization (CL) method, which mainly involves the Schur product between a state error covariance matrix and a distance-based correlation matrix. Although this CL method can reduce spurious correlations to a certain extent, observational data remain difficult to be used effectively, which results in unreasonable assimilation. In this study, we develop a new CL method coupled with a fuzzy logic control algorithm, which we call the covariance fuzzy (CF) method. The proposed CF method is a distance-based localization method with “fuzzy” vanishing correlations in data assimilation (DA) systems. To verify the effectiveness of the new algorithm, we conducted a set of experiments using an ensemble Kalman filter (EnKF) that combines the nonlinear Lorenz-96 model or the quasi-geostrophic (QG) models. First, the performances of the CL and CF methods are discussed with respect to different strength forcings, ensemble sizes, and covariance inflation factors. The experimental results show that the proposed CF method can obtain a more effective observation weight than the CL method and can reduce the errors caused by spurious correlations. Additionally, using power spectral density (PSD) as a performance evaluation index, the robustness of the proposed fuzzy logic localization method is demonstrated. However, the application of the fuzzy logic-based localization methodology to a real atmospheric model remains to be tested.

Assimilation of Satellite Microwave Observations over the Rainbands of Tropical Cyclones

AbstractA novel Bayesian Monte Carlo integration (BMCI) technique was developed to retrieve geophysical variables from satellite microwave radiometer data in the presence of tropical cyclones. The BMCI technique includes three steps: generating a stochastic database, simulating satellite brightness temperatures using a radiative transfer model, and retrieving geophysical variables such as profiles of temperature, relative humidity, and cloud liquid and ice water content from real observations. The technique also provides uncertainty estimates for each retrieval and can output the error covariance matrix of selected parameters. The measurements from the Advanced Technology Microwave Sounder (ATMS) on board Suomi National Polar-Orbiting Partnership (Suomi NPP) and the Global Precipitation Measurement (GPM) Microwave Imager (GMI) were used as input. A new technique was developed to correct the ATMS and GMI observations for the beam-filling effect, which is due to small-scale variability of precipitation and clouds when compared with the instrument footprint and also the nonlinear relation between the brightness temperature and precipitation. In addition, the assimilation of the BMCI retrievals into the NASA GEOS model is discussed for Hurricane Maria. The results show that assimilating the BMCI retrievals can influence the dynamical features of the cyclone, including a stronger warm core, a symmetric eye, and vertically aligned wind columns. Two possible factors that may limit the impact of the BMCI retrievals include 1) the resolution of the model (about 25 km), which was too coarse to show the potential of the BMCI data in improving the representation of tropical storms in the model forecast, and 2) the data assimilation system not being able to consider vertically correlated observation errors.

Forecasting the production side of GDP

AbstractWe evaluate the forecasting performance of time series models for the production side of gross domestic product (GDP)—that is, for the sectoral real value‐added series summing up to aggregate output. We focus on two strategies to model a large number of interdependent time series simultaneously: a Bayesian vector autoregressive model (BVAR) and a factor model structure; and compare them to simple aggregate and disaggregate benchmarks. We evaluate point and density forecasts for aggregate GDP and the cross‐sectional distribution of sectoral real value‐added growth in the euro area and Switzerland. We find that the factor model structure outperforms the benchmarks in most tests, and in many cases also the BVAR. An analysis of the covariance matrix of the sectoral forecast errors suggests that the superiority can be traced back to the ability to capture sectoral comovement more accurately.

Randomized Consensus-Based Distributed Kalman Filtering Over Wireless Sensor Networks

This article is concerned with developing a novel distributed Kalman filtering algorithm over wireless sensor networks based on randomized consensus strategy. Compared with centralized algorithm, distributed filtering techniques require less computation per sensor and lead to more robust estimation since they simply use the information from the neighboring nodes in the network. However, poor local sensor estimation caused by limited observability and network topology changes, which interfere the global consensus, are challenging issues. Motivated by this observation, we propose a novel randomized gossip based distributed Kalman filtering algorithm. Information exchange and computation in the proposed algorithm can be carried out in an arbitrarily connected network of sensors. In addition, the computational burden can be distributed for a sensor, which communicates with a stochastically selected neighbor at each clock step under schemes of gossip algorithm. In this case, the error covariance matrix changes stochastically at every clock step; thus, the convergence is considered in a probabilistic sense. We provide the mean square convergence analysis of the proposed algorithm. Under a sufficient condition, we show that the proposed algorithm is quite appealing as it achieves better mean square error performance theoretically than the noncooperative decentralized Kalman filtering algorithm. Examples and simulations are provided to illustrate the theoretical results.

소방관의 실내 측위를 위한 두 발에 장착된 보행항법장치의 입체구적 포인트 기반 위치 영역 융합

The ZUPT (Zero-velocity UPdaTe)-based PDR (Pedestrian Dead Reckoning) mounted on the foot calculates navigation information through the INS (Inertial Navigation System) algorithm, detects the moment when the foot touches the ground through the ZVD (Zero-Velocity Detection) algorithm, and corrects the INS errors through zero-velocity correction filtering. Through this, it is possible to estimate fairly stable position information of a pedestrian. However, the accumulation of errors over time is inevitable. In this paper, we propose a technique to reduce the accumulation of errors through the fusion of PDRs mounted on both feet. After converting the position information of the two PDRs into cubature points, the body position is estimated using the points. The two PDRs are then updated using the body position information as a measurement of the filter. Here, the measurement error covariance matrix is set using the structural constraint of the body and feet to reduce the phenomenon that the position difference between the two PDRs increases over time. The performance of the proposed technique is verified through experiments.

The ABC-DA system (v1.4): a variational data assimilation system for convective-scale assimilation research with a study of the impact of a balance constraint

Abstract. Following the development of the simplified atmospheric convective-scale “toy” model (the ABC model, named after its three key parameters: the pure gravity wave frequency A, the controller of the acoustic wave speed B, and the constant of proportionality between pressure and density perturbations C), this paper introduces its associated variational data assimilation system, ABC-DA. The purpose of ABC-DA is to permit quick and efficient research into data assimilation methods suitable for convective-scale systems. The system can also be used as an aid to teach and demonstrate data assimilation principles. ABC-DA is flexible and configurable, and is efficient enough to be run on a personal computer. The system can run a number of assimilation methods (currently 3DVar and 3DFGAT have been implemented), with user configurable observation networks. Observation operators for direct observations and wind speeds are part of the current system, and these can, for example, be expanded relatively easily to include operators for Doppler winds. A key feature of any data assimilation system is how it specifies the background error covariance matrix. ABC-DA uses a control variable transform method to allow this to be done efficiently. This version of ABC-DA mirrors many operational configurations by modelling multivariate error covariances with uncorrelated control parameters, each with special uncorrelated spatial patterns. The software developed performs (amongst other things) model runs, calibration tasks associated with the background error covariance matrix, testing and diagnostic tasks, single data assimilation runs, and multi-cycle assimilation/forecast experiments, and it also has associated visualisation software. As a demonstration, the system is used to tackle a scientific question concerning the role of geostrophic balance (GB) to model background error covariances between mass and wind fields. This question arises because although GB is a very useful mechanism that is successfully exploited in larger-scale assimilation systems, its use is questionable at convective scales due to the typically larger Rossby numbers where GB is not so relevant. A series of identical twin experiments is done in cycled assimilation configurations. One experiment exploits GB to represent mass–wind covariances in a mirror of an operational set-up (with use of an additional vertical regression (VR) step, as used operationally). This experiment performs badly where error accumulates over time. Two further experiments are done: one that does not use GB and another that does but without the VR step. Turning off GB impairs the performance, and turning off VR improves the performance in general. It is concluded that there is scope to further improve the way that the background error covariance matrices are represented at convective scale. Ideas for further possible developments of ABC-DA are discussed.

High-dimensional general linear hypothesis tests via non-linear spectral shrinkage

We are interested in testing general linear hypotheses in a high-dimensional multivariate linear regression model. The framework includes many well-studied problems such as two-sample tests for equality of population means, MANOVA and others as special cases. A family of rotation-invariant tests is proposed that involves a flexible spectral shrinkage scheme applied to the sample error covariance matrix. The asymptotic normality of the test statistic under the null hypothesis is derived in the setting where dimensionality is comparable to sample sizes, assuming the existence of certain moments for the observations. The asymptotic power of the proposed test is studied under various local alternatives. The power characteristics are then utilized to propose a data-driven selection of the spectral shrinkage function. As an illustration of the general theory, we construct a family of tests involving ridge-type regularization and suggest possible extensions to more complex regularizers. A simulation study is carried out to examine the numerical performance of the proposed tests.

Background Error in WRF Model

WRF model have been tuned and tested over Georgia’s territory for years. First time in Georgia theprocess of data assimilation in Numerical weather prediction is developing. This work presents how forecasterror statistics appear in the data assimilation problem through the background error covariance matrix – B, wherethe variances and correlations associated with model forecasts are estimated. Results of modeling of backgrounderror covariance matrix for control variables using WRF model over Georgia with desired domain configurationare discussed and presented. The modeling was implemented in two different 3DVAR systems (WRFDA andGSI) and results were checked by pseudo observation benchmark cases using also default global and regional BEmatrixes. The mathematical and physical properties of the covariances are also reviewed.

A conjugate BFGS method for accurate estimation of a posterior error covariance matrix in a linear inverse problem

AbstractOne effective data assimilation/inversion method is the four‐dimensional variational method (4D‐Var). However, it is a non‐trivial task for a conventional 4D‐Var to estimate a posterior error covariance matrix. This study proposes a method to estimate a posterior error covariance matrix applied to the linear inverse problem of an atmospheric constituent. The method was constructed within a 4D‐Var framework using a quasi‐Newton method with the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm. The proposed method was constructed such that conjugacy among the set of increment vector pairs was ensured. It is theoretically demonstrated that, when this conjugate property is coupled with preconditioning, an analytical solution of a posterior error covariance matrix could be obtained from the same number of vector pairs as observations. Furthermore, to accelerate the speed of convergence, the method can be coupled with an ensemble approach. By performing a simple advection test, it was confirmed that the proposed method could obtain an analytical matrix of the posterior error covariance within the same number of iterations as the observations. Furthermore, the method was also evaluated using an atmospheric CO2 inverse problem, which demonstrated its practical utility. The evaluation revealed that the proposed method could provide accurate estimates not only of the diagonal but also of the off‐diagonal elements of the posterior error covariance matrix. Although far more expensive than optimal state estimation, the computational efficiency was found to be reasonable for practical use, especially in conjunction with an ensemble approach. The accurate estimation of a posterior error covariance matrix resulting from the proposed method could provide valuable quantitative information regarding the uncertainties of estimated variables as well as the observational impacts, which would be beneficial for designing observation networks. Furthermore, error correlations derived from the estimated off‐diagonal elements could benefit the interpretation of optimised parameter variations.

The impact of error covariance matrix structure of GRACE’s gravity solution on the mass flux estimates of Greenland ice sheet

The error variance-covariance matrices of the monthly GRACE gravity field models are usually well-structured (e.g., order-leading) to contain the error information of the monthly gravity field models, and they are important information to further improve the accuracy of the estimated mass transportation by a post-processing scheme. Intensive studies have been performed to understand the impact of different approximations of the full error variance-covariance matrix on the mass estimates obtained with the conventional GRACE-post processing methods (e.g., the de-striping method). In this study, based on the variants of the mascon approach which treat monthly GRACE solutions as input, we consider four different structures to the error variance-covariance matrix, (i) full matrix, (ii) block diagonal matrix, (iii) diagonal matrix, and (iv) identity matrix, and examine their impact on the mass transport estimates in Greenland. Using both synthetic and real data, we analyze the results at four temporal scales: (i) the long-term decadal scale, (ii) the inter-annual scale, (iii) the seasonal scale, and (iv) the monthly scale. Based on the synthetic study, we find that for the recovery of the long-term trend, the application of the diagonal structure obtains the best estimates. This is caused by the amplification of the model error in the mascon approach when considering the full and block diagonal structures of error variance-covariance matrix, not due to any imperfection of them. Therefore, we emphasize that one should be aware of mascon model deficiency in the mascon approach. Furthermore, the best inter-annual, seasonal and monthly mass estimates are derived by considering the block diagonal and full structures. This is caused by the behavior of the bias and the unique parameterization error in the case of different structures. A similar finding is also presented in the real data case. Finally, our analysis denotes the necessity of releasing the block diagonal structure together with the official monthly gravity field model for the GRACE Follow-On mission, instead of releasing only the diagonal structure as done for the GRACE mission.

Performance evaluation of different computational methods to estimate Wood’s lactation curve by nonlinear mixed-effects models

Nonlinear mixed-effects models allow modeling repeated measures over time. The fixed effects of these models allow incorporating covariates, whereas the random effects reflect the multiple sources of heterogeneity and correlation between and within the units. To estimate the parameters of these models, it is necessary to use iterative processes, which can be done through different approaches, some of which are applied by the statistical software SAS. In this work, through simulation, we studied the performance of the estimators of a Wood’s incomplete gamma function, obtained through three different methods: linearization method through a Taylor series expansion on the empirical best linear unbiased predictor of random effects, applied by the NLINMIX macro, and expansion around the expected value of random effects, applied by both the NLINMIX macro and the NLMIXED procedure. We also investigated the impact of an incorrect specification of the covariance matrix for random errors. The linearization method through a Taylor series expansion on the empirical best linear unbiased predictor of random effects, applied by the NLINMIX macro, provided estimators with good performance, approximately normally distributed and with biases lower than those obtained with the other methods, even when the covariance matrix for random errors was incorrectly specified.

GENERIC LINEAR ARRAY SCANNER MODELING OF SPECTRAL IMAGING SYSTEMS CONTAINING LIMITED METADATA

Abstract. The National Geospatial-Intelligence Agency (NGA) designed the Generic Linear Array Scanner (GLAS) model for geopositioning images from both airborne and spaceborne linear array scanning systems, including pushbroom, whiskbroom, and panoramic sensors. Providers of hyperspectral imagery (HSI) historically have not populated products with high fidelity metadata to support downstream photogrammetric processing. To demonstrate recommended metadata population and exploitation using the GLAS model, NGA has generated example HSI products using data collected by NASA’s EO-1 Hyperion sensor and provided courtesy of the U.S. Geological Survey. This paper provides novel techniques for: 1) generating reasonably accurate initial approximations for GLAS metadata as a function of per-image metadata consisting of only timing information and the latitude and longitude values of the four corners of the image; and 2) identifying a vector of adjustable parameters and reasonable values for its a priori error covariance matrix that enable corrections to the metadata during a bundle adjustment. The paper describes applying these techniques to fourteen overlapping Hyperion images of the Alps, running a bundle adjustment as a function of tie points and optional ground control points, and demonstrating superior results to the previous polynomial based approach as quantified by the 3D errors at several ground check points.

Control Optimization of Stochastic Systems Based on Adaptive Correction CKF Algorithm

Standard cubature Kalman filter (CKF) algorithm has some disadvantages in stochastic system control, such as low control accuracy and poor robustness. This paper proposes a stochastic system control method based on adaptive correction CKF algorithm. Firstly, a nonlinear time-varying discrete stochastic system model with stochastic disturbances is constructed. The control model is established by using the CKF algorithm, the covariance matrix of standard CKF is optimized by square root filter, the adaptive correction of error covariance matrix is realized by adding memory factor to the filter, and the disturbance factors in nonlinear time-varying discrete stochastic systems are eliminated by multistep feedback predictive control strategy, so as to improve the robustness of the algorithm. Simulation results show that the state estimation accuracy of the proposed adaptive cubature Kalman filter algorithm is better than that of the standard cubature Kalman filter algorithm, and the proposed adaptive correction CKF algorithm has good control accuracy and robustness in the UAV control test.

INS/GPS 통합 항법 시스템의 정지/비행 중 정렬 수행 시 자세 및 가속도계 바이어스의 가관측도에 대한 이론적 분석

This paper presents a theoretical analysis on the degree of observability of attitude and accelerometer bias under static and in-flight alignment by solving the error covariance matrix explicitly. By following the Kalman filter covariance update process, the elements of this matrix can be resolved under simple assumptions, and provide a good conformity to the simulation results. The solution consists of the magnitude of gravity, the given acceleration, and the initial covariance. By using this formula, the covariance of the estimation error, which represents the degree of observability, can be examined accurately before the estimation process. For the initial attitude covariance, set by the coarse alignment, it is shown that the error covariance of the roll, pitch, and accelerometer bias in the x and y axes decrease by half of their respective initial values under static alignment. Moreover, the estimation error variance of the yaw is shown to decrease significantly under in-flight alignment.

Simultaneously Learning Corrections and Error Models for Geometry-Based Visual Odometry Methods

This paper fosters the idea that deep learning methods can be used to complement classical visual odometry pipelines to improve their accuracy and to associate uncertainty models to their estimations. We show that the biases inherent to the visual odometry process can be faithfully learned and compensated for, and that a learning architecture associated with a probabilistic loss function can jointly estimate a full covariance matrix of the residual errors, defining an error model capturing the heteroscedasticity of the process. Experiments on autonomous driving image sequences assess the possibility to concurrently improve visual odometry and estimate an error associated with its outputs.

The Impact of Stochastic Physics-Based Hybrid GSI/EnKF Data Assimilation on Hurricane Forecasts Using EMC Operational Hurricane Modeling System

The National Oceanic and Atmospheric Administration’s (NOAA) cloud-permitting high-resolution operational Hurricane Weather and Research Forecasting (HWRF) model includes the sophisticated hybrid grid-point statistical interpolation (GSI) and Ensemble Kalman Filter (EnKF) data assimilation (DA) system, which allows assimilating high-resolution aircraft observations in tropical cyclone (TC) inner core regions. In the operational HWRF DA system, the flow-dependent background error covariance matrix is calculated from the HWRF self-cycled 40-member ensemble. This DA system has proved to provide improved initial TC structure and therefore improved TC track and intensity forecasts. However, the uncertainties from the model physics are not taken into account in the FY2017 version of the HWRF DA system. In order to further improve the HWRF DA system, the stochastic physics perturbations are introduced in the HWRF DA, including the cumulus convection scheme, the planetary boundary layer (PBL) scheme, and model surface physics (drag coefficient), for HWRF-based ensembles. This study shows that both TC initial conditions and TC track and intensity forecast skills are improved by adding stochastic model physics in the HWRF self-cycled DA system. It was found that the improvements in the TC initial conditions and forecasts are the results of ensemble spread increases which realistically represent the model background error covariance matrix in HWRF DA. For all 2016 Atlantic storms, the TC track and intensity forecast skills are improved by about ~3% and 6%, respectively, compared to the control experiment. The case study shows that the stochastic physics in HWRF DA is especially helpful for those TCs that have inner-core high-resolution aircraft observations, such as tail Doppler radar (TDR) data.

A Rapid and Adaptive Alignment under Mooring Condition Using Adaptive EKF and CNN-Based Learning.

Alignment of the inertial navigation system (INS) in the mooring environment should take into account the movements of the waves or wind. The alignment of the INS is performed through an extended Kalman filter (EKF) using zero velocity as a measurement. However, in the mooring condition, this is not perfect stationary, thus the measurement error covariance matrix should be adjusted. In addition, if the measurement error covariance matrix is fixed to one value, the alignment time may take longer or the performance may be reduced depending on the change in mooring conditions. To solve this problem, we propose an alignment method using adaptive Kalman filter and convolution neural network (CNN)-based learning. The proposed method was verified for the superiority of alignment time and accuracy through Monte Carlo simulation in a mooring environment.