Time-optimal path planning in dynamic flows using level set equations: theory and schemes

We develop an accurate partial differential equation-based methodology that predicts the time-optimal paths of autonomous vehicles navigating in any continuous, strong, and dynamic ocean currents, obviating the need for heuristics. The goal is to predict a sequence of steering directions so that vehicles can best utilize or avoid currents to minimize their travel time. Inspired by the level set method, we derive and demonstrate that a modified level set equation governs the time-optimal path in any continuous flow. We show that our algorithm is computationally efficient and apply it to a number of experiments. First, we validate our approach through a simple benchmark application in a Rankine vortex flow for which an analytical solution is available. Next, we apply our methodology to more complex, simulated flow fields such as unsteady double-gyre flows driven by wind stress and flows behind a circular island. These examples show that time-optimal paths for multiple vehicles can be planned even in the presence of complex flows in domains with obstacles. Finally, we present and support through illustrations several remarks that describe specific features of our methodology.     

Comparison of polynomial chaos and Gaussian process surrogates for uncertainty quantification and correlation estimation of spatially distributed open-channel steady flows

Data assimilation is widely used to improve flood forecasting capability, especially through parameter inference requiring statistical information on the uncertain input parameters (upstream discharge, friction coefficient) as well as on the variability of the water level and its sensitivity with respect to the inputs. For particle filter or ensemble Kalman filter, stochastically estimating probability density function and covariance matrices from a Monte Carlo random sampling requires a large ensemble of model evaluations, limiting their use in real-time application. To tackle this issue, fast surrogate models based on polynomial chaos and Gaussian process can be used to represent the spatially distributed water level in place of solving the shallow water equations. This study investigates the use of these surrogates to estimate probability density functions and covariance matrices at a reduced computational cost and without the loss of accuracy, in the perspective of ensemble-based data assimilation. This study focuses on 1-D steady state flow simulated with MASCARET over the Garonne River (South-West France). Results show that both surrogates feature similar performance to the Monte-Carlo random sampling, but for a much smaller computational budget; a few MASCARET simulations (on the order of 10–100) are sufficient to accurately retrieve covariance matrices and probability density functions all along the river, even where the flow dynamic is more complex due to heterogeneous bathymetry. This paves the way for the design of surrogate strategies suitable for representing unsteady open-channel flows in data assimilation.     

Time-optimal path planning in dynamic flows using level set equations: realistic applications

The level set methodology for time-optimal path planning is employed to predict collision-free and fastest-time trajectories for swarms of underwater vehicles deployed in the Philippine Archipelago region. To simulate the multiscale ocean flows in this complex region, a data-assimilative primitive-equation ocean modeling system is employed with telescoping domains that are interconnected by implicit two-way nesting. These data-driven multiresolution simulations provide a realistic flow environment, including variable large-scale currents, strong jets, eddies, wind-driven currents, and tides. The properties and capabilities of the rigorous level set methodology are illustrated and assessed quantitatively for several vehicle types and mission scenarios. Feasibility studies of all-to-all broadcast missions, leading to minimal time transmission between source and receiver locations, are performed using a large number of vehicles. The results with gliders and faster propelled vehicles are compared. Reachability studies, i.e., determining the boundaries of regions that can be reached by vehicles for exploratory missions, are then exemplified and analyzed. Finally, the methodology is used to determine the optimal strategies for fastest-time pick up of deployed gliders by means of underway surface vessels or stationary platforms. The results highlight the complex effects of multiscale flows on the optimal paths, the need to utilize the ocean environment for more efficient autonomous missions, and the benefits of including ocean forecasts in the planning of time-optimal paths.     

Assessing uncertainty in refraction seismic traveltime inversion using a global inversion strategy

To analyse and invert refraction seismic travel time data, different approaches and techniques have been proposed. One common approach is to invert first‐break travel times employing local optimization approaches. However, these approaches result in a single velocity model, and it is difficult to assess the quality and to quantify uncertainties and non‐uniqueness of the found solution. To address these problems, we propose an inversion strategy relying on a global optimization approach known as particle swarm optimization. With this approach we generate an ensemble of acceptable velocity models, i.e., models explaining our data equally well. We test and evaluate our approach using synthetic seismic travel times and field data collected across a creeping hillslope in the Austrian Alps. Our synthetic study mimics a layered near‐surface environment, including a sharp velocity increase with depth and complex refractor topography. Analysing the generated ensemble of acceptable solutions using different statistical measures demonstrates that our inversion strategy is able to reconstruct the input velocity model, including reasonable, quantitative estimates of uncertainty. Our field data set is inverted, employing the same strategy, and we further compare our results with the velocity model obtained by a standard local optimization approach and the information from a nearby borehole. This comparison shows that both inversion strategies result in geologically reasonable models (in agreement with the borehole information). However, analysing the model variability of the ensemble generated using our global approach indicates that the result of the local optimization approach is part of this model ensemble. Our results show the benefit of employing a global inversion strategy to generate near‐surface velocity models from refraction seismic data sets, especially in cases where no detailed a priori information regarding subsurface structures and velocity variations is available.     

Short-term optimal operation of water systems using ensemble forecasts

Short-term water system operation can be realized using Model Predictive Control (MPC). MPC is a method for operational management of complex dynamic systems. Applied to open water systems, MPC provides integrated, optimal, and proactive management, when forecasts are available. Notwithstanding these properties, if forecast uncertainty is not properly taken into account, the system performance can critically deteriorate.Ensemble forecast is a way to represent short-term forecast uncertainty. An ensemble forecast is a set of possible future trajectories of a meteorological or hydrological system. The growing ensemble forecasts’ availability and accuracy raises the question on how to use them for operational management.The theoretical innovation presented here is the use of ensemble forecasts for optimal operation. Specifically, we introduce a tree based approach. We called the new method Tree-Based Model Predictive Control (TB-MPC). In TB-MPC, a tree is used to set up a Multistage Stochastic Programming, which finds a different optimal strategy for each branch and enhances the adaptivity to forecast uncertainty. Adaptivity reduces the sensitivity to wrong forecasts and improves the operational performance.TB-MPC is applied to the operational management of Salto Grande reservoir, located at the border between Argentina and Uruguay, and compared to other methods.     

Polynomial chaos expansion for global sensitivity analysis applied to a model of radionuclide migration in a randomly heterogeneous aquifer

We perform global sensitivity analysis (GSA) through polynomial chaos expansion (PCE) on a contaminant transport model for the assessment of radionuclide concentration at a given control location in a heterogeneous aquifer, following a release from a near surface repository of radioactive waste. The aquifer hydraulic conductivity is modeled as a stationary stochastic process in space. We examine the uncertainty in the first two (ensemble) moments of the peak concentration, as a consequence of incomplete knowledge of (a) the parameters characterizing the variogram of hydraulic conductivity, (b) the partition coefficient associated with the migrating radionuclide, and (c) dispersivity parameters at the scale of interest. These quantities are treated as random variables and a variance-based GSA is performed in a numerical Monte Carlo framework. This entails solving groundwater flow and transport processes within an ensemble of hydraulic conductivity realizations generated upon sampling the space of the considered random variables. The Sobol indices are adopted as sensitivity measures to provide an estimate of the role of uncertain parameters on the (ensemble) target moments. Calculation of the indices is performed by employing PCE as a surrogate model of the migration process to reduce the computational burden. We show that the proposed methodology (a) allows identifying the influence of uncertain parameters on key statistical moments of the peak concentration (b) enables extending the number of Monte Carlo iterations to attain convergence of the (ensemble) target moments, and (c) leads to considerable saving of computational time while keeping acceptable accuracy.     

LABORATORY EXPERIMENTS ON NAVIGATION-INDUCED BED SHEAR STRESSES AND SEDIMENT RESUSPENSION

IINTRODUCTIONDependingonflowandoperatingconditions,navigationtrafficmaycausesignificantresuspensionofdepositedsediment.Jnanumberofsituationsresuspensionofdepositedsedimentcanhavesevereenvironmentalrepercussions.Forinstance,ifthesedimentcontainscontaminants,thecontaminantsmaybereentrainedwiththesediment,taintingthewaterquality(Erdmannetal.,1994).Inothersituations,..evedincreasesintheamountofcleansuspendedsedimentcanbedetrimentalforaquaticplantsandanimals(Garcfaetal.,1998).Inordertoassesst…     

Enhanced bed load sediment transport by unsteady flows in a degrading channel

Laboratory flume experiments were done to investigate bed load sediment transport by both steady and unsteady flows in a degrading channel. The bed, respectively composed of uniform sand, uniform gravel, or sand-gravel mixtures, always undergoes bulk degradation. It is found that both uniform and non-uniform bed load transport is enhanced greatly by unsteady flows as compared to their volume-equivalent steady flows. This enhancement effect is evaluated by means of an enhancement factor, which is shown to be larger with a coarser bed and lower discharges. Also, the fractional transport rates of gravel and sand in non-uniform sand-gravel mixtures are compared with their uniform counterparts under both steady and unsteady flows. The sand is found to be able to greatly promote the transport of gravel, whilst the gravel considerably hinders the transport of sand. Particularly, the promoting and hindering impacts are more pronounced at lower discharges and tend to be weakened by flow unsteadiness.     

Evolution of scouring process downstream of grade‐control structures under steady and unsteady flows

For many incised channels, one of the most common strategies is to install some hard structures, such as grade‐control structures (GCSs), in the riverbed to resist further incision. In this study, a series of experiments, including both steady and unsteady flow conditions, were conducted to investigate the scouring process downstream of a GCS. Three distinct phases, including the initial, developing and equilibrium phases, during the evolution of scour holes were identified. In addition, a semi‐empirical method was proposed to predict the equilibrium scour‐hole profile for the scour countermeasure design. In general, the comparisons between the experimental and simulated results are reasonably consistent. As the studies on temporal variation of the scour depth at GCSs caused by floods are limited, the effect of flood hydrograph shapes on the scour downstream of GCSs without upstream sediment supply was also investigated experimentally in this study. Based on the dimensional analysis and the concept of superposition, a methodology is proposed to simulate the time evolution of the maximum scour depth downstream of a GCS for steady flows. Moreover, the proposed scheme predicts reasonably well the temporal variations of the maximum scour depth for unsteady flows with both single and multiple peak. Copyright © 2012 John Wiley & Sons, Ltd.     

Development and testing of a coupled ocean–atmosphere mesoscale ensemble prediction system

A coupled ocean–atmosphere mesoscale ensemble prediction system has been developed by the Naval Research Laboratory. This paper describes the components and implementation of the system and presents baseline results from coupled ensemble simulations for two tropical cyclones. The system is designed to take into account major sources of uncertainty in: (1) non-deterministic dynamics, (2) model error, and (3) initial states. The purpose of the system is to provide mesoscale ensemble forecasts for use in probabilistic products, such as reliability and frequency of occurrence, and in risk management applications. The system components include COAMPS® (Coupled Ocean/Atmosphere Mesoscale Prediction System) and NCOM (Navy Coastal Ocean Model) for atmosphere and ocean forecasting and NAVDAS (NRL Atmospheric Variational Data Assimilation System) and NCODA (Navy Coupled Ocean Data Assimilation) for atmosphere and ocean data assimilation. NAVDAS and NCODA are 3D-variational (3DVAR) analysis schemes. The ensembles are generated using separate applications of the Ensemble Transform (ET) technique in both the atmosphere (for moving or non-moving nests) and the ocean. The atmospheric ET is computed using wind, temperature, and moisture variables, while the oceanographic ET is derived from ocean current, temperature, and salinity variables. Estimates of analysis error covariance, which is used as a constraint in the ET, are provided by the ocean and atmosphere 3DVAR assimilation systems. The newly developed system has been successfully tested for a variety of configurations, including differing model resolution, number of members, forecast length, and moving and fixed nest options. Results from relatively coarse resolution (～27-km) ensemble simulations of Hurricanes Hanna and Ike demonstrate that the ensemble can provide valuable uncertainty information about the storm track and intensity, though the ensemble mean provides only a small amount of improved predictive skill compared to the deterministic control member.     

Uncertainty assessment of a model for biological nitrogen and phosphorus removal: Application to a large wastewater treatment plant

In the last few years, the use of mathematical models in WasteWater Treatment Plant (WWTP) processes has become a common way to predict WWTP behaviour. However, mathematical models generally demand advanced input for their implementation that must be evaluated by an extensive data-gathering campaign, which cannot always be carried out. This fact, together with the intrinsic complexity of the model structure, leads to model results that may be very uncertain. Quantification of the uncertainty is imperative. However, despite the importance of uncertainty quantification, only few studies have been carried out in the wastewater treatment field, and those studies only included a few of the sources of model uncertainty. Seeking the development of the area, the paper presents the uncertainty assessment of a mathematical model simulating biological nitrogen and phosphorus removal. The uncertainty assessment was conducted according to the Generalised Likelihood Uncertainty Estimation (GLUE) methodology that has been scarcely applied in wastewater field. The model was based on activated-sludge models 1 (ASM) and 2 (ASM2). Different approaches can be used for uncertainty analysis. The GLUE methodology requires a large number of Monte Carlo simulations in which a random sampling of individual parameters drawn from probability distributions is used to determine a set of parameter values. Using this approach, model reliability was evaluated based on its capacity to globally limit the uncertainty. The method was applied to a large full-scale WWTP for which quantity and quality data was gathered. The analysis enabled to gain useful insights for WWTP modelling identifying the crucial aspects where higher uncertainty rely and where therefore, more efforts should be provided in terms of both data gathering and modelling practises.     

Interface dynamics in randomly heterogeneous porous media

We consider the dynamics of a fluid interface in heterogeneous porous media, whose hydraulic properties are uncertain. Modeling hydraulic conductivity as a random field of given statistics allows us to predict the interface dynamics and to estimate the corresponding predictive uncertainty by means of statistical moments. The novelty of our approach to obtaining the interface statistics consists of dynamically mapping the Cartesian coordinate system onto a coordinate system associated with the moving front. This transforms a difficult problem of deriving closure relationships for highly nonlinear stochastic flows with free surfaces into a relatively simple problem of deriving stochastic closures for linear flows in domains with fixed boundaries. We derive a set of deterministic equations for the statistical moments of the interfacial dynamics, which hold in one and two spatial dimensions, and analyze their solutions for one-dimensional flow.     

Modelling of solute transport in rivers under different flow rates: A case study without transient storage

A methodology to derive solute transport models at any flow rate is presented. The novelty of the proposed approach lies in the assessment of uncertainty of predictions that incorporate parameterisation based on flow rate. A simple treatment of uncertainty takes into account heteroscedastic modelling errors related to tracer experiments performed over a range of flow rates, as well as the uncertainty of the observed flow rates themselves. The proposed approach is illustrated using two models for the transport of a conservative solute: a physically based, deterministic, advection-dispersion model (ADE), and a stochastic, transfer function based, active mixing volume model (AMV). For both models the uncertainty of any parameter increases with increasing flow rate (reflecting the heteroscedastic treatment of modelling errors at different observed flow rates), but in contrast the uncertainty of travel time, computed from the predicted model parameters, was found to decrease with increasing flow rate.     

Data-Worth Assessment for a Three-Dimensional Optimal Design in Nonlinear Groundwater Systems

Groundwater model predictions are often uncertain due to inherent uncertainties in model input data. Monitored field data are commonly used to assess the performance of a model and reduce its prediction uncertainty. Given the high cost of data collection, it is imperative to identify the minimum number of required observation wells and to define the optimal locations of sampling points in space and depth. This study proposes a design methodology to optimize the number and location of additional observation wells that will effectively measure multiple hydrogeological parameters at different depths. For this purpose, we incorporated Bayesian model averaging and genetic algorithms into a linear data-worth analysis in order to conduct a three-dimensional location search for new sampling locations. We evaluated the methodology by applying it along a heterogeneous coastal aquifer with limited hydrogeological data that is experiencing salt water intrusion (SWI). The aim of the model was to identify the best locations for sampling head and salinity data, while reducing uncertainty when predicting multiple variables of SWI. The resulting optimal locations for new observation wells varied with the defined design constraints. The optimal design (OD) depended on the ratio of the start-up cost of the monitoring program and the installation cost of the first observation well. The proposed methodology can contribute toward reducing the uncertainties associated with predicting multiple variables in a groundwater system.     

Comparison of sparse‐grid geometric and random sampling methods in nonlinear inverse solution uncertainty estimation

A new uncertainty estimation method, which we recently introduced in the literature, allows for the comprehensive search of model posterior space while maintaining a high degree of computational efficiency. The method starts with an optimal solution to an inverse problem, performs a parameter reduction step and then searches the resulting feasible model space using prior parameter bounds and sparse‐grid polynomial interpolation methods. After misfit rejection, the resulting model ensemble represents the equivalent model space and can be used to estimate inverse solution uncertainty. While parameter reduction introduces a posterior bias, it also allows for scaling this method to higher dimensional problems. The use of Smolyak sparse‐grid interpolation also dramatically increases sampling efficiency for large stochastic dimensions. Unlike Bayesian inference, which treats the posterior sampling problem as a random process, this geometric sampling method exploits the structure and smoothness in posterior distributions by solving a polynomial interpolation problem and then resampling from the resulting interpolant. The two questions we address in this paper are 1) whether our results are generally compatible with established Bayesian inference methods and 2) how does our method compare in terms of posterior sampling efficiency. We accomplish this by comparing our method for two electromagnetic problems from the literature with two commonly used Bayesian sampling schemes: Gibbs’ and Metropolis‐Hastings. While both the sparse‐grid and Bayesian samplers produce compatible results, in both examples, the sparse‐grid approach has a much higher sampling efficiency, requiring an order of magnitude fewer samples, suggesting that sparse‐grid methods can significantly improve the tractability of inference solutions for problems in high dimensions or with more costly forward physics.     

A probabilistic framework for floodplain mapping using hydrological modeling and unsteady hydraulic modeling

ABSTRACT

Prediction of design hydrographs is key in floodplain mapping using hydraulic models, which are either steady state or unsteady. The former, which require only an input peak, substantially overestimate the volume of water entering the floodplain compared to the more realistic dynamic case simulated by the unsteady models that require the full hydrograph. Past efforts to account for the uncertainty of boundary conditions using unsteady hydraulic modeling have been based largely on a joint flood frequency–shape analysis, with only a very limited number of studies using hydrological modeling to produce the design hydrographs. This study therefore presents a generic probabilistic framework that couples a hydrological model with an unsteady hydraulic model to estimate the uncertainty of flood characteristics. The framework is demonstrated on the Swannanoa River watershed in North Carolina, USA. Given its flexibility, the framework can be applied to study other sources of uncertainty in other hydrological models and watersheds.     

集合资料同化中方差滤波技术研究及试验

本文基于YH4DVAR业务系统构建了集合资料同化试验平台,利用10个集合样本统计得到的流依赖背景误差能显著改进业务应用中背景误差方差的结构和大小.但是受样本数的限制,背景误差方差的集合估计值中引入了大量的随机取样噪声.为了降低噪声对估计值的影响,本文采用谱滤波方法,根据信号和噪声尺度的统计特征构造一个低通滤波器来滤除背景误差方差估计值中的大部分随机取样噪声.在2013年第九号台风"飞燕"的集合方差滤波试验中,10个样本的滤波结果优于30个样本的集合估计值.谱滤波方法的成功应用有效降低了集合资料同化系统对集合样本数的要求,将是集合资料同化系统未来业务化运行的一项不可或缺的关键技术.     

Long waves due to a symmetrical wind stress on the surface of a rotating ocean

Abstract

The problem of unsteady long waves generated by any horizontal and symmetrically distributed, time-periodic surface wind on a rotating ocean is analysed for large times and distances. Uniform asymptotic estimates of the surface displacement in the unsteady state are obtained. The steady-state wave and velocity fields at any distance are also determined. Some characteristics of the unsteady and steady motions are described. Also noted are the features that distinguish the motion from its one-dimensional analogue for which a non-uniform analysis in the unsteady state along with a large-distance form of the surface elevation are already known.     

Forecasting search areas using ensemble ocean circulation modeling

We investigate trajectory forecasting as an application of ocean circulation ensemble modeling. The ensemble simulations are performed weekly, starting with assimilation of data for various variables from multiple sensors on a range of observational platforms. The ensemble is constructed from 100 members, and member no. 1 is designed as a standard (deterministic) simulation, providing us with a benchmark for the study. We demonstrate the value of the ensemble approach by validating simulated trajectories using data from ocean surface drifting buoys. We find that the ensemble average trajectories are generally closer to the observed trajectories than the corresponding results from a deterministic forecast. We also investigate an alternative model in which velocity perturbations are added to the deterministic results and ensemble mean results, by a first-order stochastic process. The parameters of the stochastic model are tuned to match the dispersion of the ensemble approach. Search areas from the stochastic model give a higher hit ratio of the observations than the results based on the ensemble. However, we find that this is a consequence of a positive skew of the area distribution of the convex hulls of the ensemble trajectory end points.