Uncertain Initial Conditions Research Articles

DESIGN OF A CONFIDENCE INTERVAL OBSERVER -APPLICATION TO VEHICLE POSITIONING

State estimation methods allow the vehicle position and velocity to be reconstructed by combining information from sensors and vehicle model. From a security point of view, position and velocity have to be known with a high level of confidence in order, for example, to avoid vehicle collision. In this paper, a confidence interval observer is developed to enclose positioning variables with some confidence degree (or integrity level). Based on a vehicle model and intervals bounding, with some associated probability, the uncertain initial conditions, inputs, model parameters and measurements, a predictor provides intervals for the state estimate trajectory between two measurement times. At each measurement time, confidence intervals from the sensors and the predictor are combined with union and intersection operations so as to satisfy the specified integrity level. Finally, the shortest non-empty intervals are chosen among the safe intervals. This method is illustrated with simulation tests based on an autonomous underwater vehicle described by a nonlinear model.

Tomographic filtering of geodynamic models: Implications for model interpretation and large‐scale mantle structure

The resolution operator is a critical accompaniment to tomographic models of the mantle. facilitates the comparison between conceptual three‐dimensional velocity models and tomographic models because it can filter these theoretical models to the spatial resolution of the tomographic model. We compute for the tomographic model S20RTS (Ritsema et al., 1999, 2004) and two companion models that are based on the same data but derived with different norm damping values. The three models explain (within measurement uncertainty) S‐SKS and S‐SKKS travel times equally well. To demonstrate how artifacts distort tomographic images and complicate model interpretation, we apply to (1) a thermochemical and (2) an isochemical model of convection in the mantle that feature different patterns of shear velocity heterogeneity in the deep mantle if we assume that shear velocity heterogeneity is caused by temperature variations only. suppresses short‐wavelength structures, removes strong velocity gradients, and introduces artificial stretching and tilting of velocity anomalies. Temperature anomalies in the thermochemical model resemble the spatial extent of low seismic velocity anomalies and the shear velocity spectrum in the D” region better than the isochemical model. However, the thermochemical model overpredicts the amplitude of shear velocity variation and places the African and Pacific anomalies imperfectly. We suspect that inaccurate velocity scaling laws and uncertain initial conditions control these mismatches. Extensive hypothesis testing is required to identify successful models.

DISCRETE-TIME INTERVAL OBSERVERS APPLIED TO CULTURES OF PHYTOPLANKTON

Based on intervals bounding the uncertain initial conditions, inputs, model parameters and measurements, interval observers provide guaranteed intervals for the state trajectory. However, most of the published studies focus on methods relying on continuous-time on-line measurements, or at least, with relatively fast sampling. In this study, interval state estimation methods are proposed in the situation, quite common in biological systems, where measurements are only available at discrete, and possible rare, times. The attention is focused on defining predictors preserving the boundedness of the state variables between two measurement times assuming bounded uncertainties. The methods are tested with experimental data from continuous cultures of the green algae Dunaliella tertiolecta.

Finite Horizon <formula formulatype="inline"><tex>${H}^{\infty}$</tex> </formula> Filtering With Initial Condition

We consider the problem of finite horizon H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sup> filtering with uncertain initial conditions. An H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sup> norm-like performance measure that explicitly accounts for the effect of initial condition is proposed. Necessary and sufficient conditions are derived for the existence of an estimator that achieves a pre-specified value of this performance measure

A Component-Resampling Approach for Estimating Probability Distributions from Small Forecast Ensembles

In many meteorological and climatological modeling applications, the availability of en- sembles of predictions containing very large numbers of members would substantially ease statistical analyses and validations. This study describes and demonstrates an objective approach for generating large ensembles of "additional" realizations from smaller ensembles, where the additional ensem- ble members share important first-and second-order statistical characteristics and some dynamic relations within the original ensemble. By decomposing the original ensemble members into as- suredly independent time-series components (using a form of principal component decomposition) that can then be resampled randomly and recombined, the component-resampling procedure gener- ates additional time series that follow the large and small scale structures in the original ensemble members, without requiring any tuning by the user. The method is demonstrated by applications to operational medium-range weather forecast ensembles from a single NCEP weather model and application to a multi-model, multi-emission-scenarios ensemble of 21st Century climate-change projections. 1. Problem Ensembles of weather and climate forecasts from a single model often are used to describe prediction uncertainties associated with uncertain initial conditions. Super-ensembles of forecasts from several models or with several external climate forcings can be used to describe prediction uncertainties associated with model differences and indeterminate future forcings. In weather- and climate-prediction applications, several studies have argued that ensemble means are typically better predictors than are any individual members of the contributing ensembles (e.g., Krishnamurti et al., 2000; Richardson, 2001; Zhu et al., 2002). This finding even- tually may be found to extend also to climate-change projections. However, en- semble forecasts also presumably contain information about higher order statistics of prediction uncertainties. Typically, members of an ensemble of weather or cli- mate predictions diverge as time from the initial conditions passes. Under some dynamical conditions, the members can also converge temporarily. Both of these behaviors are much as one might expect from the actual (unknown) forecast uncer- tainties, with general increases attributable to sensitive dependencies of dynamical

Reduction of overestimation in interval arithmetic simulation of biological wastewater treatment processes

A novel interval arithmetic simulation approach is introduced in order to evaluate the performance of biological wastewater treatment processes. Such processes are typically modeled as dynamical systems where the reaction kinetics appears as additive nonlinearity in state. In the calculation of guaranteed bounds of state variables uncertain parameters and uncertain initial conditions are considered. The recursive evaluation of such systems of nonlinear state equations yields overestimation of the state variables that is accumulating over the simulation time. To cope with this wrapping effect, innovative splitting and merging criteria based on a recursive uncertain linear transformation of the state variables are discussed. Additionally, re-approximation strategies for regions in the state space calculated by interval arithmetic techniques using disjoint subintervals improve the simulation quality significantly if these regions are described by several overlapping subintervals. This simulation approach is used to find a practical compromise between computational effort and simulation quality. It is pointed out how these splitting and merging algorithms can be combined with other methods that aim at the reduction of overestimation by applying consistency techniques. Simulation results are presented for a simplified reduced-order model of the reduction of organic matter in the activated sludge process of biological wastewater treatment.

Parameterizing Spatial Crop Models with Inverse Modeling: Sources of Error and Unexpected Results

The literature to date on inverse modeling-based parameterization of spatial (distributed) crop models using yield data is dominated by one-dimensional uncoupled models, in which the landscape is divided into cells that do not interchange water. We studied three possible sources of error for uncoupled and spatially coupled distributed crop models: (1) error from biases (towards unrepresentatively wet or dry weather) in the sampling of years of yield data used for the parameterization process, (2) errors due to lack of knowledge of initial soil water conditions, and (3) error from lack of spatial coupling and water transport among different landscape locations. We show analytical evidence that the spatiotemporal infiltration behavior of a simple spatially coupled water balance model cannot be reproduced through a modification of the parameters of an uncoupled model. The corresponding yield prediction limitations of the uncoupled model are confirmed, using an example, both at the parameter estimation and the validation stages. In our example, however, parameter error due to weather biases and the error from lack of knowledge of initial conditions greatly impacted the predictive capability of the spatially coupled model, and had less effect on its uncoupled counterpart. This effect of biased weather has not been reported previously in the crop modeling IM literature. We conclude that the use of spatially coupled distributed crop models requires high-quality data. Practical precision agriculture applications are characterized by uncertain initial conditions and the possibility of biased weather. Under these circumstances, the use of a spatially coupled model may not be justified, especially for low landscape positions.

Discrete-time H∞ filtering with initial condition

We consider the problem of finite horizon discrete time H∞ filtering with uncertain initial condition. An H∞ norm-like performance measure that explicitly accounts for the effect of the initial condition is proposed. Conditions are then derived for the existence and construction of a strictly causal sub-optimal estimator that achieves a pre-specified value of this performance measure. It is also shown that the achievable infimum value of this measure would in general depend on the structure and magnitude of uncertainty present in the initial condition of the system.

Uncertainty Quantification for Systems with Random Initial Conditions Using Wiener–Hermite Expansions

A number of engineering problems, including laminar‐turbulent transition in convectively unstable flows, require predicting the evolution of a nonlinear dynamical system under uncertain initial conditions. The method of Wiener–Hermite expansion is an attractive alternative to modeling methods, which solve for the joint probability density function of the stochastic amplitudes. These problems include the “curse of dimensionality” and closure problems. In this paper, we apply truncated Wiener–Hermite expansions with both fixed and time‐varying bases to a model stochastic system with three degrees of freedom. The model problem represents the combined effects of quadratic nonlinearity and stochastic initial conditions in a generic setting and occurs in related forms in both classical dynamics, turbulence theory, and the nonlinear theory of hydrodynamic stability. In this problem, the truncated Wiener–Hermite expansions give a good account of short‐time behavior, but not of the long‐time relaxation characteristic of this system. It is concluded that successful application of truncated Wiener–Hermite expansions may require special adaptations for each physical problem.

UNCERTAINTY MODELING USING FUZZY ARITHMETIC BASED ON SPARSE GRIDS: APPLICATIONS TO DYNAMIC SYSTEMS

Fuzzy arithmetic provides a powerful tool to introduce uncertainty into mathematical models. With Zadeh's extension principle, one can obtain a fuzzy-valued extension of any real-valued objective function. An efficient and accurate approach to compute expensive multivariate functions of fuzzy numbers is given by fuzzy arithmetic based on sparse grids. In this paper, we illustrate the general applicability of this new method by computing two dynamic systems subjected to uncertain parameters as well as uncertain initial conditions. The first model consists of a system of delay differential equations simulating the periodic outbreak of a disease. In the second model, we consider a multibody mechanism described by an algebraic differential equation system.

Conformant planning via symbolic model checking and heuristic search

In this paper we tackle the problem of Conformant Planning: find a sequence of actions that guarantees goal achievement regardless of an uncertain initial condition and of nondeterministic action effects. Our approach, set in the framework of search in the belief space, is based on two main intuitions. First, symbolic model checking techniques, Binary Decision Diagrams in particular, can be used to represent and expand the search space, and to provide an efficient computational platform. Second, driving the search solely with reachability information is not effective, and the notion of knowledge has to be taken explicitly into account. We make the following contributions. First, we thoroughly analyze reachability heuristics: we formally prove their admissibility, we show how they can be implemented by means of symbolic techniques, and experimentally evaluate them. Second, we analyze the limitations of reachability heuristics, and propose more informed, yet admissible, heuristics, based on a formal notion of knowledge. Third, we present a practical conformant planning algorithm, where the search is explicitly based on the notion of target knowledge, i.e., the amount of information that has to be available in order to reach the goal. The search alternates between the “Acquire Knowledge” mode, where actions have the precise purpose of gathering necessary information, and the “Reach Goal” mode, where actions are directed towards the goal. Finally, we provide a thorough experimental comparison between several conformant planners. Our approach is sometimes able to outperform the competitor systems by orders of magnitude.

Multi-scale assimilation of surface soil moisture data for robust root zone moisture predictions

In the presence of uncertain initial conditions and soil hydraulic properties land surface model performance can be significantly improved by the assimilation of periodic observations of certain state variables, such as the near surface soil moisture as observed from a remote platform. Recently, Montaldo et al. [Water Resour Res 37 (2001) 2889] derived a framework that uses biases between observed and modeled time rates of change of surface soil moisture to quantify biases between modeled and actual root-zone-average soil moisture contents. For very large errors in the saturated conductivity the soil moisture assimilation procedure is continuously working against the drainage errors, resulting in a persistent bias in its predictions. In this paper, we adopt this persistent (directional) bias in soil moisture as evidence of an error in the saturated conductivity. From manipulations of soil water balance equations we derived an expression that quantitatively relates the persistent bias in soil moisture to the estimated error in the saturated hydraulic conductivity. We combined this result with the approach of Montaldo et al. [Water Resour Res 37 (2001) 2889] to form a multi-scale assimilation approach. The multi-scale assimilation system is shown to provide marked improvements in the prediction of root zone soil moisture for a case study using data taken from an experimental catchment near Cork, Ireland. In effect, the root zone moisture is updated to provide a temporal trajectory of the near surface moisture that follows the trajectory of the observed surface moisture, and the hydraulic conductivity is adjusted on the basis of the time averaged corrections applied to the root zone water content. It is anticipated that this approach would be useful in operational forecasting models over large domains, where system parameters would be uncertain and occasional distributed observations would be limited to the near surface zone.

Robust simulation of root zone soil moisture with assimilation of surface soil moisture data

An operational framework is presented for assimilating surface soil moisture remote sensing measurements into a soil‐vegetation‐atmosphere‐transfer (SVAT) model for the robust prediction of root zone moisture time series. The proposed approach is based on analytical treatment of the dynamical equations coupling surface and deeper soil reservoirs. The resulting framework uses biases between observed and modeled time rates of change of surface soil moisture to quantify biases between modeled and actual root zone average soil moisture contents. The approach is based on the popular interactions between soil‐biosphere‐atmosphere (ISBA) force‐restore SVAT model. An experimental data set, collected near Cork, Ireland, is analyzed both for a long data series of 183 days and four short periods that were selected to focus on different hydrometeorological conditions. The results demonstrated that the proposed framework performs uniformly robust over 3 orders of magnitude of misspecification of saturated hydraulic conductivity. In the presence of uncertain initial conditions, the results demonstrated a marked increase in model skill (over the original ISBA model) for periods when average precipitation was less than average potential evaporation.

The probability of containment failure by direct containment heating in Zion

This paper is the first step in the resolution of the direct containment heating (DCH) issue for the Zion nuclear power plant using the risk oriented accident analysis methodology (ROAAM). This paper includes the definition of a probabilistic framework that decomposes the DCH problem into three probability density functions that reflect the most uncertain initial conditions (UO 2 mass, zirconium oxidation fraction, and steel mass). Uncertainties in the initial conditions are significant, but our quantification approach is based on establishing reasonable bounds that are not unnecessarily conservative. To this end, we also make use of the ROAAM ideas of enveloping scenarios and ‘splintering’. Two causal relations (CRs) are used in this framework: CR1 is a model that calculates the peak pressure in the containment as a function of the initial conditions, and CR2 is a model that returns the frequency of containment failure as a function of pressure within the containment. Uncertainty in CR1 is accounted for by the use of two independently developed phenomenological models, the convection-limited containment heating model and the two-cell equilibrium model, and by probabilistically distributing the key parameter in both, which is the ratio of the melt entrainment time to the system blowdown time constant. The two phenomenological models have been compared with an extensive database including recent integral simulations at two different physical scales (1:10-scale in the Surtsey facility at Sandia National Laboratories and 1:40-scale in the COREXIT facility at Argonne National Laboratory). The loads predicted by these models were significantly lower than those from previous parametric calculations. The containment load distributions do not intersect the containment strength (fragility) curve in any significant way, resulting in containment failure probabilities less than 10 −3 for all scenarios considered. Sensitivity analyses did not show any areas of large sensitivity. The feasibility of extrapolating containment loads distributions to most other pressurized water reactors is explored.

Dynamically Stratified Monte Carlo Forecasting

A new method for performing Monte Carlo forecasts is introduced. The method, called dynamic stratification, selects initial perturbations based on a stratification of the error distribution. A simple implementation is presented in which the error distribution used for the stratification is estimated from a linear model derived from a large ensemble of 12-h forecasts with the full dynamical model. The stratification thus obtained is used to choose a small subsample of initial states with which to perform the dynamical Monte Carlo forecasts. Several test cases are studied using a simple two-level general circulation model (GCM) with uncertain initial conditions. It is found that the method provides substantial reductions in the sampling error of the forecast mean and variance when compared to the more traditional approach of choosing the initial perturbations at random. The degree of improvement, however, is sensitive to the nature of the initial error distribution and to the base state. In practice the method may be viable only if the computational burden involved in obtaining an adequate estimate of the error distribution is shared with the data-assimilation procedure.

Stability of learning control with disturbances and uncertain initial conditions

The authors investigate the effects of state disturbances, output noise, and errors in initial conditions on a class of learning control algorithms. They present a simple learning algorithm and exhibit, via a concise proof, bounds on the asymptotic trajectory errors for the learned input and the corresponding state and output trajectories. Furthermore, these bounds are continuous functions of the bounds on the initial condition errors, state disturbances, and output noise, and the bounds are zero in the absence of these disturbances. >

Identification of nth-power velocity damping

This paper demonstrates how the two damping parameters associated with the non-linear nth-power velocity model can be identified from the time series records of the displacement and velocity responses to sinusoidal excitation. No restriction is imposed on the level of damping present and estimates are acquired by minimizing the square of the error between observed responses and those predicted by a linearized model. The problems of noise contamination and uncertain initial conditions are treated by using simulated data and converged estimates are presented with computation times measured in CPU seconds on a VAX 11 780 computer.

Variance estimates for a dynamic eutrophication model of Saginaw Bay, Lake Huron

First‐order variance propagation is used to estimate variance of model output originating from variances of uncertain initial conditions, parameter values, and external load estimates for a nutrient‐phytoplankton‐zooplankton dynamic eutrophication model of one segment of Saginaw Bay, Lake Huron. Coefficients of variation of model output during summer are not unlike those estimated from measurements. The major source of variance is parameter value variances. The most important parameters, in terms of model sensitivity and variance propagation, are identified by analysis of the predicted correlation matrix. Methods for reducing model output variance are suggested.

Growth of large, late-stage planetesimals

The late stage of terrestrial planets' growth determined many of their fundamental properties, including their thermal properties and petrology, their impact records, and possibly the existence of the Moon. A critical result of late-stage models, which bears on observable properties, is the size of the largest planetesimals that grew near, and later impacted,those that became full-size planets. There has been considerable misinterpretation of previous models regarding the relation between the size of planetesimals and their relative velocities. Furthermore, some models neglect the possible decrease in relative velocity as control is transferred from the largest to the second-largest body in an accreation zone. Evidence that Venus helped stir Earth-zone planetismals is not copelling. When models are evaluated, the results are found to depend strongly on uncertain initial conditions. The size of the second-largest planetesimal in the Earth's zone might range from ∼300 to ∼2500 km, with corresponding accretion times of ∼7 × 10 6 and ∼10 8 years, respectively. Both extremes are generated from plausible initial conditions and both seem consistent with observed planetary properties.