Additive Bias Research Articles

Adaptive sensor-fault tolerant control for a class of multivariable uncertain nonlinear systems

This paper deals with the active fault tolerant control (AFTC) problem for a class of multiple-input multiple-output (MIMO) uncertain nonlinear systems subject to sensor faults and external disturbances. The proposed AFTC method can tolerate three additive (bias, drift and loss of accuracy) and one multiplicative (loss of effectiveness) sensor faults. By employing backstepping technique, a novel adaptive backstepping-based AFTC scheme is developed using the fact that sensor faults and system uncertainties (including external disturbances and unexpected nonlinear functions caused by sensor faults) can be on-line estimated and compensated via robust adaptive schemes. The stability analysis of the closed-loop system is rigorously proven using a Lyapunov approach. The effectiveness of the proposed controller is illustrated by two simulation examples.

Bias corrections for some asymmetric kernel estimators

Several asymmetric kernel (AK) estimators of a density with support [0,∞) have been suggested in the recent fifteen years. In this paper, additive and nonnegative bias correction techniques, originally developed for the standard kernel estimator, are applied to some AK estimators when the underlying density has a fourth order derivative. The major contribution is to study asymptotic properties of new AK estimators corresponding to the limits of improved estimators. Simulation studies are conducted to illustrate the finite sample performance of the proposed estimators.

Weak lensing measurements in simulations of radio images

We present a study of weak lensing shear measurements for simulated galaxy images at radio wavelengths. We construct a simulation pipeline into which we can input galaxy images of known shapelet ellipticity, and with which we then simulate observations with eMERLIN and the international LOFAR array. The simulations include the effects of the clean algorithm, uv sampling, observing angle and visibility noise, and produce realistic restored images of the galaxies. We apply a shapelet-based shear measurement method to these images and test our ability to recover the true source shapelet ellipticities. We model and deconvolve the effective point spread function, and find suitable parameters for clean and shapelet decomposition of galaxies. We demonstrate that ellipticities can be measured faithfully in these radio simulations, with no evidence of an additive bias and a modest (10 per cent) multiplicative bias on the ellipticity measurements. Our simulation pipeline can be used to test shear measurement procedures and systematics for the next generation of radio telescopes.

Intercomparison of daytime stratospheric NO<sub>2</sub> satellite retrievals and model simulations

Abstract. This paper evaluates the agreement between stratospheric NO2 retrievals from infrared limb sounders (Michelson Interferometer for Passive Atmospheric Sounding (MIPAS) and High Resolution Dynamics Limb Sounder (HIRDLS)) and solar UV/VIS backscatter sensors (Ozone Monitoring Instrument (OMI), Scanning Imaging Absorption Spectrometer for Atmospheric Cartography (SCIAMACHY) limb and nadir) over the 2005–2007 period and across the seasons. The observational agreement is contrasted with the representation of NO2 profiles in 3-D chemical transport models such as the Whole Atmosphere Community Climate Model (WACCM) and TM4. A conclusion central to this work is that the definition of a reference for stratospheric NO2 columns formed by consistent agreement among SCIAMACHY, MIPAS and HIRDLS limb records (all of which agree to within 0.25 × 1015 molecules cm−2 or better than 10%) allows us to draw attention to relative errors in other data sets, e.g., (1) WACCM overestimates NO2 densities in the extratropical lower stratosphere, particularly in the springtime and over northern latitudes by up to 35% relative to limb observations, and (2) there are remarkable discrepancies between stratospheric NO2 column estimates from limb and nadir techniques, with a characteristic seasonally and latitudinally dependent pattern. We find that SCIAMACHY nadir and OMI stratospheric columns show overall biases of −0.5 × 1015 molecules cm−2 (−20%) and +0.6 × 1015 molecules cm−2 (+20%) relative to limb observations, respectively. It is argued that additive biases in nadir stratospheric columns are not expected to affect tropospheric retrievals significantly, and that they can be attributed to errors in the total slant column density, related either to algorithmic or instrumental effects. In order to obtain accurate and long-term time series of stratospheric NO2, an effort towards the harmonization of currently used differential optical absorption spectroscopy (DOAS) approaches to nadir retrievals becomes essential, as well as their agreement to limb and ground-based observations, particularly now that limb techniques are giving way to nadir observations as the next generation of climate and air quality monitoring instruments pushes forth.

Effect of Carouseling on Angular Rate Sensor Error Processes

Carouseling is an efficient method to mitigate the measurement errors of inertial sensors, particularly MEMS gyroscopes. In this article, the effect of carouseling on the most significant stochastic error processes of a MEMS gyroscope, i.e., additive bias, white noise, 1/f noise, and rate random walk, is investigated. Variance propagation equations for these processes under averaging and carouseling are defined. Furthermore, a novel approach to generating 1/f noise is presented. The experimental results show that carouseling reduces the contributions of additive bias, 1/f noise, and rate random walk significantly in comparison with plain averaging, which can be utilized to improve the accuracy of dead reckoning systems.

Modeling and forecasting the additive bias corrected extreme value volatility estimator

In this paper, we provide a framework to model and forecast daily volatility based on the newly proposed additive bias corrected extreme value volatility estimator (the Add RS estimator). The theoretical framework of the additive bias corrected extreme value volatility estimator is based on the closed form solution for the joint probability of the running maximum and the terminal value of the random walk. Using the opening, high, low and closing prices of S&P 500, CAC 40, IBOVESPA and S&P CNX Nifty indices, we find that the logarithm of the Add RS estimator is approximately Gaussian and that a simple linear Gaussian long memory model can be applied to forecast the logarithm of the Add RS estimator. The forecast evaluation analysis indicates that the conditional Add RS estimator provides better forecasts of realized volatility than alternative range-based and return-based models.

The power spectrum of systematics in cosmic shear tomography and the bias on cosmological parameters

Cosmic shear tomography has emerged as one of the most promising tools to both investigate the nature of dark energy and discriminate between General Relativity and modified gravity theories. In order to successfully achieve these goals, systematics in shear measurements have to be taken into account; their impact on the weak lensing power spectrum has to be carefully investigated in order to estimate the bias induced on the inferred cosmological parameters. To this end, we develop here an efficient tool to compute the power spectrum of systematics by propagating, in a realistic way, shear measurement, source properties and survey setup uncertainties. Starting from analytical results for unweighted moments and general assumptions on the relation between measured and actual shear, we derive analytical expressions for the multiplicative and additive bias, showing how these terms depend not only on the shape measurement errors, but also on the properties of the source galaxies (namely, size, magnitude and spectral energy distribution). We are then able to compute the amplitude of the systematics power spectrum and its scaling with redshift, while we propose a multigaussian expansion to model in a non-parametric way its angular scale dependence. Our method allows to self-consistently propagate the systematics uncertainties to the finally observed shear power spectrum, thus allowing us to quantify the departures from the actual spectrum. We show that even a modest level of systematics can induce non-negligible deviations, thus leading to a significant bias on the recovered cosmological parameters.

Simulation of Sensor Fault Diagnosis for Wind Turbine Generators DFIG and PMSM Using Kalman Filter

The fault detection and isolation of generators used in wind turbines gathering interest as to maximize the reliability and avail of distributed energy systems with recent unmatched growth in construction of offshore wind farms. In particular it is interested in performing fault detection and isolation (FDI) of incipient faults affecting the measurements of the three-phase signals (currents) in a controlled DFIG and PMSG. Although different authors have dealt with FDI for sensors in induction machines and in DFIGs, most of them rely on the machine model with constant parameters. However, the parameter uncertainties due to changes in the operating conditions will produce degradation in the performance of such FDI systems. The robust techniques to detect faults are exist but there is a need of extra sensor. This paper proposed a systematic methodology for the design of sensor FDI systems with the following characteristics: i) capable of detecting and isolating incipient additive (bias) faults, ii) robust against changes in the references/disturbances affecting the controlled DFIG and PMSG as well as modeling/parametric uncertainties, iii) residual generation system based on a multi-observer strategy to enhance the isolation process, The designed sensor FDI systems have been validated using measured voltages, as well as simulated data from a controlled DFIG. First the state space models of DFIG and PMSM explained followed by kalman filter introduction and current sensor fault detection using a bank of kalman filter named dedicated Observer Scheme and generalized Observer scheme to detect simultaneous and multiple faults was theorized and simulated using MATLAB simulation tool .The simulation results were summarized with and without Sensor fault.

Model Calibration via Deformation

Dynamical computer models often exhibit space-time features that are partially misaligned or misshapen when compared to observational data. Whether due to approximate numerical schemes, incomplete physics, or estimated boundary conditions, the goal of calibrating these models to field data involves optimally aligning model output with observed features. The traditional approach to correcting model discrepancy is to introduce an additive and/or multiplicative bias. Especially for dynamical models, systematic bias may alternatively be viewed as deformation bias. We introduce an expanded approach to model calibration in the presence of space-time feature discrepancy. Borrowing ideas from the image warping literature, we propose a nonlinear deformation of the computer model that optimally aligns with observed images; probabilistically this manifests as a transformation of model coordinate space with a variational penalty on the likelihood function. We apply the approach to a dynamical magnetosphere-ionosphere...

On improving convergence rate of Bernstein polynomial density estimator

This paper is concerned with the Bernstein estimator [Vitale, R.A. (1975), ‘A Bernstein Polynomial Approach to Density Function Estimation’, in Statistical Inference and Related Topics, ed. M.L. Puri, 2, New York: Academic Press, pp. 87–99] to estimate a density with support [0, 1]. One of the major contributions of this paper is an application of a multiplicative bias correction [Terrell, G.R., and Scott, D.W. (1980), ‘On Improving Convergence Rates for Nonnegative Kernel Density Estimators’, The Annals of Statistics, 8, 1160–1163], which was originally developed for the standard kernel estimator. Moreover, the renormalised multiplicative bias corrected Bernstein estimator is studied rigorously. The mean squared error (MSE) in the interior and mean integrated squared error of the resulting bias corrected Bernstein estimators as well as the additive bias corrected Bernstein estimator [Leblanc, A. (2010), ‘A Bias-reduced Approach to Density Estimation Using Bernstein Polynomials’, Journal of Nonparametric Statistics, 22, 459–475] are shown to be O(n−8/9) when the underlying density has a fourth-order derivative, where n is the sample size. The condition under which the MSE near the boundary is O(n−8/9) is also discussed. Finally, numerical studies based on both simulated and real data sets are presented.

Im3shape: a maximum likelihood galaxy shear measurement code for cosmic gravitational lensing

We present and describe im3shape, a new publicly available galaxy shape measurement code for weak gravitational lensing shear. im3shape performs a maximum likelihood fit of a bulge-plus-disc galaxy model to noisy images, incorporating an applied point spread function. We detail challenges faced and choices made in its design and implementation, and then discuss various limitations that affect this and other maximum likelihood methods. We assess the bias arising from fitting an incorrect galaxy model using simple noise-free images and find that it should not be a concern for current cosmic shear surveys. We test im3shape on the GREAT08 Challenge image simulations, and meet the requirements for upcoming cosmic shear surveys in the case that the simulations are encompassed by the fitted model, using a simple correction for image noise bias. For the fiducial branch of GREAT08 we obtain a negligible additive shear bias and sub-two percent level multiplicative bias, which is suitable for analysis of current surveys. We fall short of the sub-percent level requirement for upcoming surveys, which we attribute to a combination of noise bias and the mis-match between our galaxy model and the model used in the GREAT08 simulations. We meet the requirements for current surveys across all branches of GREAT08, except those with small or high noise galaxies, which we would cut from our analysis. Using the GREAT08 metric we we obtain a score of Q=717 for the usable branches, relative to the goal of Q=1000 for future experiments. The code is freely available from https://bitbucket.org/joezuntz/im3shape

Rethinking human visual attention: Spatial cueing effects and optimality of decisions by honeybees, monkeys and humans

Visual attention is commonly studied by using visuo-spatial cues indicating probable locations of a target and assessing the effect of the validity of the cue on perceptual performance and its neural correlates. Here, we adapt a cueing task to measure spatial cueing effects on the decisions of honeybees and compare their behavior to that of humans and monkeys in a similarly structured two-alternative forced-choice perceptual task. Unlike the typical cueing paradigm in which the stimulus strength remains unchanged within a block of trials, for the monkey and human studies we randomized the contrast of the signal to simulate more real world conditions in which the organism is uncertain about the strength of the signal. A Bayesian ideal observer that weights sensory evidence from cued and uncued locations based on the cue validity to maximize overall performance is used as a benchmark of comparison against the three animals and other suboptimal models: probability matching, ignore the cue, always follow the cue, and an additive bias/single decision threshold model. We find that the cueing effect is pervasive across all three species but is smaller in size than that shown by the Bayesian ideal observer. Humans show a larger cueing effect than monkeys and bees show the smallest effect. The cueing effect and overall performance of the honeybees allows rejection of the models in which the bees are ignoring the cue, following the cue and disregarding stimuli to be discriminated, or adopting a probability matching strategy. Stimulus strength uncertainty also reduces the theoretically predicted variation in cueing effect with stimulus strength of an optimal Bayesian observer and diminishes the size of the cueing effect when stimulus strength is low. A more biologically plausible model that includes an additive bias to the sensory response from the cued location, although not mathematically equivalent to the optimal observer for the case stimulus strength uncertainty, can approximate the benefits of the more computationally complex optimal Bayesian model. We discuss the implications of our findings on the field's common conceptualization of covert visual attention in the cueing task and what aspects, if any, might be unique to humans.

Origins of weak lensing systematics, and requirements on future instrumentation (or knowledge of instrumentation)

The first half of this paper explores the origin of systematic biases in the measurement of weak gravitational lensing. Compared to previous work, we expand the investigation of PSF instability and fold in for the first time the effects of non-idealities in electronic imaging detectors and imperfect galaxy shape measurement algorithms. Together, these now explain the additive A(l) and multiplicative M(l) systematics typically reported in current lensing measurements. We find that overall performance is driven by a product of a telescope/camera's *absolute performance*, and our *knowledge about its performance*. The second half of this paper propagates any residual shear measurement biases through to their effect on cosmological parameter constraints. Fully exploiting the statistical power of Stage IV weak lensing surveys will require additive biases A<1.8e-12 and multiplicative biases M<4.0e-3. These can be allocated between individual budgets in hardware, calibration data and software, using results from the first half of the paper. If instrumentation is stable and well-calibrated, we find extant shear measurement software from GREAT10 already meet requirements on galaxies detected at S/N=40. Averaging over a population of galaxies with a realistic distribution of sizes, it also meets requirements for a 2D cosmic shear analysis from space. If used on fainter galaxies or for 3D cosmic shear tomography, existing algorithms would need calibration on simulations to avoid introducing bias at a level similar to the statistical error. Requirements on hardware and calibration data are discussed in more detail in a companion paper. Our analysis is intentionally general, but is specifically being used to drive the hardware and ground segment performance budget for the design of the European Space Agency's recently-selected Euclid mission.

A New Outlier Detection Method Considering Outliers As Model Errors

Outlier detection is an important task for fitting a model to a set of data. Two different outlier detection approaches are given as tests for outliers and robust methods. For these approaches, usually outliers are considered as additive bias terms neglected in the original adjustment model. However, there is another approach that outlier is considered as a model error in the Gauss–Markov model. This model error is represented as an unknown parameter. As it cannot be known before which observation includes outlier; this method is applied on the data for each observation separately and tested with t-test or F-test. It is successful if the sample includes only one outlier. To detect multiple outliers more successfully, in this article, a new outlier detection method is introduced. In this method, all the possible combinations of multiple outliers are considered as model errors and it is accepted that the smallest variance of them gives the solution for a certain number of outliers, then the estimated model errors are detected by comparing with a critical value. The critical value is chosen as 3σo. To compare the results of the new method, with those of the Least Median of Squares (LMS) and Huber M-estimators, Monte Carlo simulation technique is used for linear regression. The Mean Success Rate is proposed to measure the reliabilities of the methods. We showed that the new method is robust and includes the property of high breakdown point as LMS; and more efficient than LMS.

Evaporation-precipitation variability over Indian Ocean and its assessment in NCEP Climate Forecast System (CFSv2)

An attempt has been made to explore all the facets of Evaporation-Precipitation (E-P) distribution and variability over the Indian Ocean (IO) basin using Objectively Analyzed air-sea Fluxes (OAFlux) data and subsequently a thorough assessment of the latest version of National Centers for Environment Prediction (NCEP) Climate Forecast System (CFS) version-2 is done. This study primarily focuses on two fundamental issues, first, the core issue of pervasive cold SST bias in the CFS simulation in the context of moisture flux exchange between the atmosphere and the ocean and second, the fidelity of the model in simulating mean and variability of E-P and its elemental components associated with the climatic anomalies occurring over the Indian and the Pacific ocean basin. Valuation of evaporation and precipitation, the two integral component of E-P, along with the similar details of wind speed, air-sea humidity difference ( $$\Updelta Q$$ ) and Sea Surface Temperature (SST) are performed. CFS simulation is vitiated by the presence of basin wide systematic positive bias in evaporation, $$\Updelta Q$$ and similar negative bias in wind speed and SST. Bifurcation of the evaporation bias into its components reveals that bias in air humidity ( $$\hbox{Q}_{a}$$ ) is basically responsible for the presence of pervasive positive evaporation bias. The regions where CFS does not adhere to the observed wind-evaporation and $$\hbox{Q}_{a}$$ -evaporation relation was found to lie over the northern Arabian Sea (AS), the western Bay of Bengal (BoB) and the western Equatorial IO. Evaporation bias is found to control a significant quantum of cold SST bias over most of the basin owing to its intimate association with SST in a coupled feedback system. This area is stretched over the almost entire north IO, north of $$15^{\circ}\hbox{S}$$ excluding a small equatorial strip, where the evaporation bias may essentially explain 20–100 % of cold SST bias. This percentage is maximum over the western IO, central AS and BoB. The CFS simulation comply the distinct feature of the observed mean annual cycle of evaporation and precipitation, but with the additive systematic bias over most of the region. El Nino and negative Indian Ocean Dipole (NIOD) seems to have much better control over the interannual variability of evaporation in the CFS simulation, contrary to the observation where El Nino and positive Indian Ocean Dipole (PIOD) has the larger say. Both El Nino and PIOD (La Nina and NIOD) have the negative (positive) influence on the basin wide evaporation with the exception over a limited region and this relation holds for the twain. The seasonal (JJA and SON) locking of El Nino and PIOD to evaporation and precipitation is displayed by the north-south and east-west asymmetric correlation pattern respectively and this is much perspicuous in the observation as compared to the CFS. The conjoined influence of El Nino and PIOD on evaporation (precipitation) reveals the dominance of PIOD (PIOD + El Nino) response in case of both the observation as well as the CFS. This study will lead a way forward to rectify the ubiquitous cold SST bias in the CFS simulation and help in establishing the credibility of the CFS in terms of seasonal predictability.

Geolocation Performance With Biased Range Measurements

We study geolocation based on biased range estimates. Positive bias arises using time delay ranging methods in a multipath fading environment, when the line of sight direct path is severely attenuated. We model the range measurement as contaminated with Gaussian noise and an additive nonnegative bias term, and consider deterministic and random bias cases. We develop weighted least squares (WLS) and maximum likelihood (ML) geolocation estimators, and show that in general they yield biased geolocation estimates. A perturbation analysis technique is applied to find bias and mean square error (MSE) expressions for the WLS and MLE algorithms. MLE generally outperforms WLS, because MLE exploits knowledge of the range measurement bias distribution. The location error expressions are functions of the measurement bias and variance, as well as the network geometry. These results are useful to study achievable geolocation performance, and are applied to optimize sensor placement for improving the overall geolocation accuracy. We also develop the Cramér-Rao bound (CRB) on geolocation for our model. The CRB is a bound on unbiased estimation, whereas the geolocation algorithms may be biased, and we show how the estimators approach the CRB in certain cases. Numerical examples are presented to verify the analysis and study some cases of interest.

Gender-Related Invariance of the Beck Depression Inventory II for Taiwanese Adolescent Samples

The Chinese version of Beck Depression Inventory II (BDI-II-C) is one of the most used instruments to measure the severity of depression in Taiwan. The scarce literature regarding its psychometric properties (e.g., measurement invariance) highlighted the need and significance for such an investigation. The purpose of this study was to examine the gender-related measurement invariance of the BDI-II-C in an adolescent sample facing an entrance examination in the following two ways: (a) examining configural, metric, and scalar invariance using multigroup confirmatory factor analyses and (b) estimating the effects of any detected noninvariance on mean differences. The participants included 827 (416 boys and 411 girls) Taiwanese adolescents. Results indicate that measurement invariance was established at the level of configural, metric, and partial scalar invariance. Seven noninvariant intercepts (Items 2, 3, 7, 9, 10, 12, and 19) were identified, showing that there was differential additive response style bias for the BDI-II-C across gender groups. Additionally, the results demonstrated that the noninvariance had significant effects on interpretation based on gender latent mean difference as well as observed mean difference.

Measuring longitudinal change in the hippocampal formation from in vivo high-resolution T2-weighted MRI

The hippocampal formation (HF) is a brain structure of great interest because of its central role in learning and memory, and its associated vulnerability to several neurological disorders. In vivo oblique coronal T2-weighted MRI with high in-plane resolution (~0.5 mm × 0.5 mm), thick slices (~2.0 mm), and a field of view tailored to imaging the hippocampal formation (denoted HF-MRI in this paper) has been advanced as a useful imaging modality for detailed hippocampal morphometry. Cross-sectional analysis of volume measurements derived from HF-MRI has shown the modality's promise to yield sensitive imaging-based biomarker for neurological disorders such as Alzheimer's disease. However, the utility of this modality for making measurements of longitudinal change has not yet been demonstrated. In this paper, using an unbiased deformation-based morphometry (DBM) pipeline, we examine the suitability of HF-MRI for estimating longitudinal change by comparing atrophy rates measured in the whole hippocampus from this modality with those measured from more common isotropic (~1 mm³) T1-weighted MRI in the same set of individuals, in a cohort of healthy controls and patients with cognitive impairment. While measurements obtained from HF-MRI were largely consistent with those obtained from T1-MRI, HF-MRI yielded slightly larger group effect of greater atrophy rates in patients than in controls. The estimated minimum sample size required for detecting a 25% change in patients' atrophy rate in the hippocampus compared to the control group with a statistical power β=0.8 was N=269. For T1-MRI, the equivalent sample size was N=325. Using a dataset of test-retest scans, we show that the measurements were free of additive bias. We also demonstrate that these results were not a confound of certain methodological choices made in the DBM pipeline to address the challenges of making longitudinal measurements from HF-MRI, using a region of interest (ROI) around the HF to globally align serial images, followed by slice-by-slice deformable registration to measure local volume change. Additionally, we present a preliminary study of atrophy rate measurements within hippocampal subfields using HF-MRI. Cross-sectional differences in atrophy rates were detected in several subfields.

Using a MEMS gyroscope to measure the Earth’s rotation for gyrocompassing applications

In this paper, a method and system for gyrocompassing based on a low-cost micro-electro-mechanical (MEMS) gyroscope are described. The proposed setup is based on the choice of a gyroscope with specified bias instability better than 2 deg h-1 and on careful error compensation. The gyroscope is aligned parallel to the local level, which helps to eliminate the g-sensitivity effect but also sacrifices a fraction of the Earth’s rotation rate that can be observed. The additive bias is compensated for by rotating the sensor mechanically and by extended Kalman filtering. In this paper, it is demonstrated that the proposed system is capable of observing the Earth’s rotation, and the north finding results show that a two-sigma accuracy of 4.03° was attained at latitude 61°N. With current MEMS gyroscopes, the system requires hours of time to achieve this accuracy, but the results demonstrate the theoretical accuracy potential of these small self-contained, low-cost sensors.

Filtering skill for turbulent signals for a suite of nonlinear and linear extended Kalman filters

The filtering skill for turbulent signals from nature is often limited by errors due to utilizing an imperfect forecast model. In particular, real-time filtering and prediction when very limited or no a posteriori analysis is possible (e.g. spread of pollutants, storm surges, tsunami detection, etc.) introduces a number of additional challenges to the problem. Here, a suite of filters implementing stochastic parameter estimation for mitigating model error through additive and multiplicative bias correction is examined on a nonlinear, exactly solvable, stochastic test model mimicking turbulent signals in regimes ranging from configurations with strongly intermittent, transient instabilities associated with positive finite-time Lyapunov exponents to laminar behavior. Stochastic Parameterization Extended Kalman Filter (SPEKF), used as a benchmark here, involves exact formulas for propagating the mean and covariance of the augmented forecast model including the unresolved parameters. The remaining filters use the same nonlinear forecast model but they introduce model error through different moment closure approximations and/or linear tangent approximation used for computing the second-order statistics of the augmented stochastic forecast model. A comprehensive study of filter performance is carried out in the presence of various moment closure errors which are enhanced by additional model errors due to incorrect parameters inducing additive and multiplicative stochastic biases. The estimation skill of the unresolved stochastic parameters is also discussed and it is shown that the linear tangent filter, despite its popularity, is completely unreliable in many turbulent regimes for both parameter estimation and filtering; moreover, regimes of filter divergence for the linear tangent filter are identified. The results presented here provide useful guidelines for filtering turbulent, high-dimensional, spatially extended systems with more general model errors, as well as for designing more skillful methods for superparameterization of unresolved intermittent processes in complex multi-scale models. They also provide unambiguous benchmarks for the capabilities of linear and nonlinear extended Kalman filters using incorrect statistics on an exactly solvable test bed with rich and realistic dynamics.