Uncertainties In Initial Conditions Research Articles

Uncertainty Quantification Using Generalized Polynomial Chaos Expansion for Nonlinear Dynamical Systems With Mixed State and Parameter Uncertainties

This paper develops a framework for propagation of uncertainties, governed by different probability distribution functions in a stochastic dynamical system. More specifically, it deals with nonlinear dynamical systems, wherein both the initial state and parametric uncertainty have been taken into consideration and their effects studied in the model response. A sampling-based nonintrusive approach using pseudospectral stochastic collocation is employed to obtain the coefficients required for the generalized polynomial chaos (gPC) expansion in this framework. The samples are generated based on the distribution of the uncertainties, which are basically the cubature nodes to solve expectation integrals. A mixture of one-dimensional Gaussian quadrature techniques in a sparse grid framework is used to produce the required samples to obtain the integrals. The familiar problem of degeneracy with high-order gPC expansions is illustrated and insights into mitigation of such behavior are presented. To illustrate the efficacy of the proposed approach, numerical examples of dynamic systems with state and parametric uncertainties are considered which include the simple linear harmonic oscillator system and a two-degree-of-freedom nonlinear aeroelastic system.

Design and practical implementation of kinematic constraints in Inertial Navigation System-Doppler Velocity Log (INS-DVL)-based navigation

Kinematic constrained navigation is a subset of model-aided navigation systems that is attractive because of its independence from extra hardware equipment. In this study, the error-based Kalman Filter is used for the data fusion process through feedback strategy, and kinematic constraints are utilized to improve Inertial Navigation System-Doppler Velocity Log (INS-DVL) navigation performance. Here, the pseudo-measurement method is used for applying kinematic constraints. Real offline data obtained from field tests of an instrumented vessel in shallow water demonstrate that applying kinematic constraints improves navigation performance and robustness against initial condition uncertainty. Repeatability of results is also investigated for three different maneuvers.

Evaluation of the Wyoming Weather Modification Pilot Project (WWMPP) Using Two Approaches: Traditional Statistics and Ensemble Modeling

AbstractThe Wyoming Weather Modification Pilot Project randomized cloud seeding experiment was a crossover statistical experiment conducted over two mountain ranges in eastern Wyoming and lasted for 6 years (2008–13). The goal of the experiment was to determine if cloud seeding of orographic barriers could increase snowfall and snowpack. The experimental design included triply redundant snow gauges deployed in a target–control configuration, covariate snow gauges to account for precipitation variability, and ground-based seeding with silver iodide (AgI). The outcomes of this experiment are evaluated with the statistical–physical experiment design and with ensemble modeling. The root regression ratio (RRR) applied to 118 experimental units provided insufficient statistical evidence (p value of 0.28) to reject the null hypothesis that there was no effect from ground-based cloud seeding. Ensemble modeling estimates of the impact of ground-based seeding provide an alternate evaluation of the 6-yr experiment. The results of the model ensemble approach with and without seeding estimated a mean enhancement of precipitation of 5%, with an inner-quartile range of 3%–7%. Estimating the impact on annual precipitation over these mountain ranges requires results from another study that indicated that approximately 30% of the annual precipitation results from clouds identified as seedable within the seeding experiment. Thus the seeding impact is on the order of 1.5% of the annual precipitation, compared to 1% for the statistical–physical experiment, which was not sufficient to reject the null hypothesis. These results provide an estimate of the impact of ground-based cloud seeding in the Sierra Madre and Medicine Bow Mountains in Wyoming that accounts for uncertainties in both initial conditions and model physics.

Probabilistic forecasting of wind power production losses in cold climates: a case study

Abstract. The problem of icing on wind turbines in cold climates is addressed using probabilistic forecasting to improve next-day forecasts of icing and related production losses. A case study of probabilistic forecasts was generated for a 2-week period. Uncertainties in initial and boundary conditions are represented with an ensemble forecasting system, while uncertainties in the spatial representation are included with a neighbourhood method. Using probabilistic forecasting instead of one single forecast was shown to improve the forecast skill of the ice-related production loss forecasts and hence the icing forecasts. The spread of the multiple forecasts can be used as an estimate of the forecast uncertainty and of the likelihood for icing and severe production losses. Best results, both in terms of forecast skill and forecasted uncertainty, were achieved using both the ensemble forecast and the neighbourhood method combined. This demonstrates that the application of probabilistic forecasting for wind power in cold climates can be valuable when planning next-day energy production, in the usage of de-icing systems and for site safety.

Principles of measurement modeling in optical nonlinear dynamical systems

The task of the paper consists in a creating the principles for measurement modeling in optical nonlinear dynamical systems (lasers, solitons, optical cryptography systems). The measurement model of such systems should contain: the input values, their time and noise dependencies, measurement uncertainties of input quantities and initial conditions, evolution function and forecasting time. It is suggested to use the measurement portrait for the case when the mathematical description of the systems is impossible. Analysis of measurement portrait allows us to calculate the fractal dimension, dynamics of the dynamic variables and the relationship of their values without knowledge of the analytical solutions of the system equations. Bifurcation points and Lyapunov exponents, the forecast time of the dynamics and other quantities can be calculated too. The principles of modeling the measurement process, proposed by the authors, provide theoretical, model and experimental studies of physical phenomena in the optical nonlinear dynamical systems, facilitate the solution of a wide range of problems with creation and management of optical systems, and study the processes of self-organization and dynamics in such systems.

Uncertainty in malaria simulations in the highlands of Kenya: Relative contributions of model parameter setting, driving climate and initial condition errors.

In this study, experiments are conducted to gauge the relative importance of model, initial condition, and driving climate uncertainty for simulations of malaria transmission at a highland plantation in Kericho, Kenya. A genetic algorithm calibrates each of these three factors within their assessed prior uncertainty in turn to see which allows the best fit to a timeseries of confirmed cases. It is shown that for high altitude locations close to the threshold for transmission, the spatial representativeness uncertainty for climate, in particular temperature, dominates the uncertainty due to model parameter settings. Initial condition uncertainty plays little role after the first two years, and is thus important in the early warning system context, but negligible for decadal and climate change investigations. Thus, while reducing uncertainty in the model parameters would improve the quality of the simulations, the uncertainty in the temperature driving data is critical. It is emphasized that this result is a function of the mean climate of the location itself, and it is shown that model uncertainty would be relatively more important at warmer, lower altitude locations.

Control strategy of optimal deployment for spacecraft solar array system with initial state uncertainty

A control strategy combining feedforward control and feedback control is presented for the optimal deployment of a spacecraft solar array system with the initial state uncertainty. A dynamic equation of the spacecraft solar array system is established under the assumption that the initial linear momentum and angular momentum of the system are zero. In the design of feedforward control, the dissipation energy of each revolute joint is selected as the performance index of the system. A Legendre pseudospectral method (LPM) is used to transform the optimal control problem into a nonlinear programming problem. Then, a sequential quadratic programming algorithm is used to solve the nonlinear programming problem and offline generate the optimal reference trajectory of the system. In the design of feedback control, the dynamic equation is linearized along the reference trajectory in the presence of initial state errors. A trajectory tracking problem is converted to a two-point boundary value problem based on Pontryagin’s minimum principle. The LPM is used to discretize the two-point boundary value problem and transform it into a set of linear algebraic equations which can be easily calculated. Then, the closed-loop state feedback control law is designed based on the resulting optimal feedback control and achieves good performance in real time. Numerical simulations demonstrate the feasibility and effectiveness of the proposed control strategy.

Constraints from zoom-in simulations on the protostellar accretion process

AbstractStars are embedded in different environments of Giant Molecular Clouds during their formation phase. Despite this fact, it is common practice to assume an isolated spherical core as the initial condition for models of individual star formation. To avoid the uncertainties of initial and boundary conditions, we use an alternative approach of zoom-in simulations to account for the environment in which protostars form. Our models show that injections of 26Al from a close-by supernova into the young solar system were highly unlikely. Moreover, we find that the accretion process of protostars is heterogeneous and environment-dependent.

Practical Predictability of Supercells: Exploring Ensemble Forecast Sensitivity to Initial Condition Spread

Abstract As convection-allowing ensembles are routinely used to forecast the evolution of severe thunderstorms, developing an understanding of storm-scale predictability is critical. Using a full-physics numerical weather prediction (NWP) framework, the sensitivity of ensemble forecasts of supercells to initial condition (IC) uncertainty is investigated using a perfect model assumption. Three cases are used from the real-time NSSL Experimental Warn-on-Forecast System for Ensembles (NEWS-e) from the 2016 NOAA Hazardous Weather Testbed Spring Forecasting Experiment. The forecast sensitivity to IC uncertainty is assessed by repeating the simulations with the initial ensemble perturbations reduced to 50% and 25% of their original magnitudes. The object-oriented analysis focuses on significant supercell features, including the mid- and low-level mesocyclone, and rainfall. For a comprehensive analysis, supercell location and amplitude predictability of the aforementioned features are evaluated separately. For all examined features and cases, forecast spread is greatly reduced by halving the IC spread. By reducing the IC spread from 50% to 25% of the original magnitude, forecast spread is still substantially reduced in two of the three cases. The practical predictability limit (PPL), or the lead time beyond which the forecast spread exceeds some prechosen threshold, is case and feature dependent. Comparing to past studies reveals that practical predictability of supercells is substantially improved by initializing once storms are well established in the ensemble analysis.

Initial-Condition Dependence and Initial-Condition Uncertainty in Climate Science

Abstract This article examines initial-condition dependence and initial-condition uncertainty for climate projections and predictions. The first contribution is to provide a clear conceptual characterization of predictions and projections. Concerning initial-condition dependence, projections are often described as experiments that do not depend on initial conditions. Although prominent, this claim has not been scrutinized much and can be interpreted differently. If interpreted as the claim that projections are not based on estimates of the actual initial conditions of the world or that what makes projections true are conditions in the world, this claim is true. However, it can also be interpreted as the claim that simulations used to obtain projections are independent of initial-condition ensembles. This article argues that evidence does not support this claim. Concerning initial-condition uncertainty, three kinds of initial-condition uncertainty are identified (two have received little attention from philosophers so far). The first (the one usually discussed) is the uncertainty associated with the spread of the ensemble simulations. The second arises because the theoretical initial ensemble cannot be used in calculations and has to be approximated by finitely many initial states. The third uncertainty arises because it is unclear how long the model should be run to obtain potential initial conditions at pre-industrial times. Overall, the discussion shows that initial-condition dependence and uncertainty in climate science are more complex and important issues than usually acknowledged. 1Introduction2Projections and Predictions 2.1Predictions2.2Projections3Initial-Condition Dependence 3.1Projections 3.1.1Dynamical conditions justifying independence of initial conditions?3.1.2What projections can and cannot provide3.2Predictions4Initial-Condition Uncertainty 4.1Projections4.2Predictions5ConclusionAppendix

Assessing the restoration time of surface water and groundwater systems under groundwater pumping

Since surface water and groundwater systems are fully coupled and integrated, increased groundwater withdrawal during drought may reduce groundwater discharges into the stream, thereby prolonging both systems’ recovery from drought. To analyze watershed response to basin-level groundwater pumping, we propose a modelling framework to understand the resiliency of surface water and groundwater systems using an integrated hydrologic model under transient pumping. The proposed framework incorporates uncertainties in initial conditions to develop robust estimates of restoration times of both surface water and groundwater and quantifies how pumping impacts state variables such as soil moisture. Groundwater pumping impacts over a watershed were also analyzed under different pumping volumes and different potential climate scenarios. Our analyses show that groundwater restoration time is more sensitive to variability in climate forcings as opposed to changes in pumping volumes. After the cessation of pumping, streamflow recovers quickly in comparison to groundwater, which has higher persistence. Pumping impacts on various hydrologic variables were also discussed. Potential for developing optimal conjunctive management plans using seasonal-to-interannual climate forecasts is also discussed.

Reachability analysis of linear dynamic systems with constant, arbitrary, and Lipschitz continuous inputs

This paper sets forth a method for reachability analysis of linear dynamic systems in continuous time that can be used to compute time-domain bounds of states and outputs with floating-point precision. The focus is on the particular initial conditions and inputs that cause state or output trajectories to attain their extreme values in time. Inputs can be constant, arbitrary, or Lipschitz continuous waveforms. Uncertainties in initial conditions and inputs are modeled using zonotopes. The calculations exploit the definition of zonotopes and the modal information of the linear system dynamics.

Effects of parameter and data uncertainty on long-term projections in a model of the global forest sector

This study explored the consequences for long-term projections and impact analysis of the uncertainty in model parameters and initial conditions. Using the Global Forest Products Model, multiple replications of projections were carried out with parameters or initial condition data sampled randomly from their assumed distribution. The results showed that parameter uncertainty led to uncertainty of the projections increasing steadily with the time horizon, and more rapidly than the uncertainty stemming from initial conditions. Among the parameter uncertainties, those in the supply and demand elasticities tended to dominate the uncertainty in the other parameters describing forest growth, manufacturing activities, and trade inertia. In an application to impact analysis it was found that, due only to the uncertainty of the model parameters, and conditional on other assumptions, an assumed rise in global temperature of 2.8 °C over a century caused the forest stock in 2065 to be 2.4% to 4.0% higher in developed countries, and 2.5% to 3.9% lower in developing countries, with 68% probability, a conservative estimate of the true uncertainty given all the other factors involved in such a prediction.

Predictability of Arctic sea ice on weather time scales

The field of Arctic sea ice prediction on “weather time scales” is still in its infancy with little existing understanding of the limits of predictability. This is especially true for sea ice deformation along so-called Linear Kinematic Features (LKFs) including leads that are relevant for marine operations. Here the potential predictability of the sea ice pack in the wintertime Arctic up to ten days ahead is determined, exploiting the fact that sea ice-ocean models start to show skill at representing sea ice deformation at high spatial resolutions. Results are based on ensemble simulations with a high-resolution sea ice-ocean model driven by atmospheric ensemble forecasts. The predictability of LKFs as measured by different metrics drops quickly, with predictability being almost completely lost after 4–8 days. In contrast, quantities such as sea ice concentration or the location of the ice edge retain high levels of predictability throughout the full 10-day forecast period. It is argued that the rapid error growth for LKFs is mainly due to the chaotic behaviour of the atmosphere associated with the low predictability of near surface wind divergence and vorticity; initial condition uncertainty for ice thickness is found to be of minor importance as long as LKFs are initialized at the right locations.

Data assimilation of soil water flow by considering multiple uncertainty sources and spatial–temporal features: a field-scale real case study

Accurate estimates of soil moisture and soil hydraulic parameters via data assimilation largely depend on the quality of the model structure, input data and observations. Often, however, all of this information is subject to uncertainty under real circumstances. This real-case study seeks to understand the effects of different uncertainty sources and observation scales on data assimilation performance. Ensemble Kalman filter method based on the soil water-flow model, a sub-module of soil–water–atmosphere–plant model, is established to simultaneously estimate the model states and parameters. The soil hydraulic parameters are extensively measured or calibrated to examine the parameter estimation accuracy. Furthermore, considering the high spatial and temporal variability of soil moisture observation in the field-scale problem, an analysis of spatiotemporal characteristics is combined with data assimilation. Results indicated that simultaneously considering parameter and initial conditions uncertainty leads to a better soil moisture and parameter estimation than that ignoring initial uncertainty in realistic practice. Unlike the other error sources, an inadequate description to the meteorological forcing has a negative influence on surface soil moisture estimation, which might be attributed to the persistent disturbances of evaporation uncertainty and the lack of observations at shallow soil depth. Moreover, a prior knowledge of spatiotemporal features of soil moisture observation is beneficial to efficiently improve data assimilation performance. It is possible to implement field-scale data assimilation with a few representative points, instead of using the spatial average of all observations at a high cost. The assimilation results highlight the possibly positive outcomes of accounting for the multi-source of uncertainties and emphasize the significant importance of characterizing the spatial–temporal feature of soil moisture for a field-scale application.

Optimal reorientation of a free-floating space robot subject to initial state uncertainties

This paper focuses on the dynamics and optimal reorientation of a free-floating space robot system in the presence of initial state uncertainties. A control strategy combining optimal motion planning and feedback control is presented based on the dynamic model of the system. In the design of the optimal motion planning, Legendre pseudospectral method (LPM) is used to transform the optimal reorientation problem into a nonlinear programming problem. Then, sequential quadratic programming algorithm is employed to solve the nonlinear programming problem and off-line generate the optimal reference trajectory of the system. In the design of feedback control, the state equation is linearized around the reference trajectory obtained by LPM. The tracking control problem is converted into a two-point boundary value problem based on Pontryagin’s maximum principle. Then LPM is used to discretize the two-point boundary value problem and transform it into a set of linear algebraic equations. This process does not require any integration calculations and has good performance in real time. Numerical simulations indicate that the control strategy is effective with good robustness.

Idealized Experiments for Optimizing Model Parameters Using a 4D-Variational Method in an Intermediate Coupled Model of ENSO

Large biases exist in real-time ENSO prediction, which can be attributed to uncertainties in initial conditions and model parameters. Previously, a 4D variational (4D-Var) data assimilation system was developed for an intermediate coupled model (ICM) and used to improve ENSO modeling through optimized initial conditions. In this paper, this system is further applied to optimize model parameters. In the ICM used, one important process for ENSO is related to the anomalous temperature of subsurface water entrained into the mixed layer (Te), which is empirically and explicitly related to sea level (SL) variation. The strength of the thermocline effect on SST (referred to simply as “the thermocline effect”) is represented by an introduced parameter, αTe. A numerical procedure is developed to optimize this model parameter through the 4D-Var assimilation of SST data in a twin experiment context with an idealized setting. Experiments having their initial condition optimized only, and having their initial condition plus this additional model parameter optimized, are compared. It is shown that ENSO evolution can be more effectively recovered by including the additional optimization of this parameter in ENSO modeling. The demonstrated feasibility of optimizing model parameters and initial conditions together through the 4D-Var method provides a modeling platform for ENSO studies. Further applications of the 4D-Var data assimilation system implemented in the ICM are also discussed.

Computational guidance for planetary powered descent using collaborative optimization

An innovative computational guidance framework is proposed for planetary powered descent using collaborative optimization approach. First, the dynamical model and constraints for planetary powered descent are presented. Then, using collaborative optimization strategy, the computational guidance framework for powered descent is formulated as a multi-discipline optimization problem including trajectory optimization, optimal guidance, and system-level optimization. Finally, the computational guidance approach employs three algorithms for respectively solving the three optimization modules to implement numerical simulations. The optimality and robustness of the computational guidance approach are verified with all constraints satisfied even in the presence of initial state uncertainty.

Three-dimensional relative reachable domain with initial state uncertainty in Gaussian distribution

A probability threshold is a confidence level defining a bound outside which the occurrence of a random variable is considered as a rare event. Upon providing such a probability threshold on the initial state uncertainty, relative reachable domain is defined as the minimum positional volume enclosing all the possible relative trajectories resulting from the initial state uncertainty. In this study, the conventional coplanar relative reachable domain with initial state uncertainty in isotropic Gaussian distribution is extended to the three-dimensional case with uncertainty in arbitrary Gaussian distribution. Positional errors in Gaussian distribution are thought to be confined within an error ellipsoid depending on the given probability threshold. Such an error ellipsoid will evolve with time and consequently sweep out a volume, which is the relative reachable domain to be determined. This paper proposed an algorithm of solving the envelope surface of the relative reachable domain for close range relative motion based on the linearized dynamical model and the mathematical definition of an envelope. Moreover, the algorithm is modified to improve the accuracy for long range relative motion. Comparisons between the solved relative reachable domain and the result of 10,000 Monte Carlo runs, which can be regarded as the true result, for different scenarios on circular reference orbits demonstrated the feasibility of the proposed method.

Estimation Techniques for Bilinear Control Systems

Abstract The problem of estimating reachable sets of impulsive control systems with uncertainties in initial states and in the matrix of the system is studied. We assume that the initial states to be unknown but belong to a given star-shaped symmetric nondegenerate polytope. The matrix included in the differential equations of the system dynamics is uncertain and only bounds on admissible values of this matrix coefficients are known. We present here the approach that allow to find ellipsoidal estimates of reachable sets which uses the special structure of the bilinear control system. The algorithm of constructing such ellipsoidal set-valued estimates and numerical simulation results are given.