Statistical-analysis methods are generally derived under the assumption that forecast errors are strictly random and zero in the mean. If the short-term forecast, used as the background field in the statistical-analysis equation, is in fact biased, so will the resulting analysis be biased. The only way to account properly for bias in a statistical analysis is to do so explicitly, by estimating the forecast bias and then correcting the forecast prior to analysis.

We present a rigorous method for estimating forecast bias by means of data assimilation, based on an unbiased subset of the observing system. The result is a sequential bias estimation and correction algorithm, whose implementation involves existing components of operational statistical-analysis systems. The algorithm is designed to perform on-line, in the context of suboptimal data-assimilation methods which are based on approximate information about forecast- and observation-error covariances. The added computational cost of incorporating online bias estimation and correction into an operational system roughly amounts to one additional solution of the statistical-analysis equation, for a limited number of observations. Off-line forecast-bias estimates based on previously produced assimilated-data sets can be produced as well, using an existing analysis system.

We show that our sequential bias estimation algorithm fits into a broader theoretical framework provided by the separate-bias estimation approach of estimation theory. In this framework the bias parameters are defined rather generally and can be used to describe systematic model errors and observational bias as well. We illustrate the performance of the algorithm in a simulated data-assimilation experiment with a one-dimensional forced dissipative shallow-water model. A climate error is introduced into the forecast model via topographic forcing. while random errors are generated by stochastic forcing. In this simple setting our algorithm is well able to estimate and correct the forecast bias caused by this systematic error, and the climate error in the assimilated-data set is virtually eliminated as a result.

Data assimilation in the presence of forecast bias

The Kalman filter (KF) has received a huge interest from the industrial electronics community and has played a key role in many engineering fields since the 1970s, ranging, without being exhaustive, trajectory estimation, state and parameter estimation for control or diagnosis, data merging, signal processing, and so on. This paper provides a brief overview of the industrial applications and implementation issues of the KF in six topics of the industrial electronics community, highlighting some relevant reference papers and giving future research trends.

/pdf/industrial-applications-of-the-kalman-filter-a-review-tqdmwc4feq.pdf

Industrial Applications of the Kalman Filter: A Review

https://hal.archives-ouvertes.fr/hal-00098136/document

Unbiased minimum variance estimation for systems with unknown exogenous inputs

The two-stage Kalman estimator was originally proposed to reduce the computational complexity of the augmented state Kalman filter. It was also applied to the tracking of maneuvering targets by treating the target acceleration as a bias term. Except in certain restrictive conditions, the conventional two-stage estimators are suboptimal in the sense that they are not equivalent to the augmented state filter. In this paper, the authors propose a new two-stage Kalman estimator, i.e., new structure, which is an extension of Friedland's estimator and is optimal in general conditions. In addition, we provide some analytic results to demonstrate the computational advantages of two-stage estimators over augmented ones.

https://ir.nctu.edu.tw:443/bitstream/11536/31598/1/000077975300035.pdf

Optimal solution of the two-stage Kalman estimator

The well-known conventional Kalman filter requires an accurate system model and exact stochastic information. But in a number of situations, the system model has an unknown bias, which may degrade the performance of the Kalman filter or may cause the filter to diverge. The effect of the unknown bias may be more pronounced on the extended Kalman filter (EKF), which is a nonlinear filter. The two-stage extended Kalman filter (TEKF) with respect to this problem has been receiving considerable attention for a long time. Recently, the optimal two-stage Kalman filter (TKF) for linear stochastic systems with a constant bias or a random bias has been proposed by several researchers. A TEKF can also be similarly derived as the optimal TKF. In the case of a random bias, the TEKF assumes that the information of a random bias is known. But the information of a random bias is unknown or partially known in general. To solve this problem, this paper proposes an adaptive two-stage extended Kalman filter (ATEKF) using an adaptive fading EKF. To verify the performance of the proposed ATEKF, the ATEKF is applied to the INS-GPS (inertial navigation system-Global Positioning System) loosely coupled system with an unknown fault bias. The proposed ATEKF tracked/estimated the unknown bias effectively although the information about the random bias was unknown.

Adaptive Two-Stage Extended Kalman Filter for a Fault-Tolerant INS-GPS Loosely Coupled System

A modified decoupled Kalman estimator suitable for use when the bias vector varies as a random-walk process is defined and demonstrated in a practical application consisting of the calibration of a strapdown inertial navigation system. The estimation system accuracy associated with the modified estimator is shown to be essentially the same as that of the generalized partitioned Kalman estimator. Considering that the sensor error random rates assumed in the example are on the order of 5 to 10 times greater than normally associated with contemporary strapdown systems, it may be inferred that inertial navigation systems possessing more typical sensor error random growth characteristics should be amenable to a decoupled estimator approach in a broad spectrum of aided-navigation system applications. This should also be true in a variety of other applications in which the bias vector experiences only limited random variation. >

Separate bias Kalman estimator with bias state noise

An alternate simplified derivation of Friedland's two-stage Kalman estimator is given for a somewhat more general class of problems than considered by Friedland. Friedland's result is also extended to encompass two variations on the basic idea which are of practical interest.

An alternate derivation and extension of Friendland's two-stage Kalman estimator

Addresses two issues concerning the separate-bias Kalman estimator. The first of these issues deals with the derivation of the optimal estimator for the general case in which the bias vector is stochastic in nature, and the second issue deals with defining a suitable suboptimal realization of the generalized estimator.

M. Ignagni

Papers

Separate bias Kalman estimator with bias state noise

An alternate derivation and extension of Friendland's two-stage Kalman estimator

Optimal and suboptimal separate-bias Kalman estimators for a stochastic bias