Qingfang Wu

Ensemble Kalman Filter based state estimation in 2D shallow water equations using Lagrangian sensing and state augmentation

We present a state estimation method for two-dimensional shallow water equations in rivers using Lagrangian drifter positions as measurements. The aim of this method is to compensate for the lack of knowledge of upstream and downstream boundary conditions in rivers that causes inaccuracy in the velocity field estimation by releasing drifters equipped with GPS receivers. The drifters report their positions and thus provide additional information of the state of the river. This information is incorporated into shallow water equations by using Ensemble Kalman Filtering (EnKF). The proposed method is based on the discretization of the governing nonlinear equations using the finite element method in unstructured meshes. We incorporate the drifter positions into the unknown state, which directly exploits the Langrangian nature of the measurements. The performance of the method is assessed with twin experiments.

Quadratic programming based data assimilation with passive drifting sensors for shallow water flows

We present a method for assimilating Lagrangian sensor measurement data into a shallow water equation model. The underlying estimation problem (in which the dynamics of the system are represented by a system of partial differential equations) relies on the formulation of a minimisation of an error functional, which represents the mismatch between the estimate and the measurements. The corresponding so-called variational data assimilation problem is formulated as a quadratic programming problem with linear constraints. For the hydrodynamics application of interest, data is obtained from drifting sensors that gather position and velocity. The data assimilation method refines the estimate of the initial conditions of the hydrodynamic system. The method is implemented using a new sensor network hardware platform for gathering flow information from a river, which is presented in this article for the first time. Validation of the results is performed by comparing them to an estimate derived from an independent set of static sensors, some of which were deployed as part of our field experiments.

Kalman filter based estimation of flow states in open channels using Lagrangian sensing

In this article, we investigate real-time estimation of flow state in open channels using the measurements obtained from Lagrangian sensors (drifters). One-dimensional Shallow Water Equations (SWE), also known as Saint-Venant equations, are used as the mathematical model for the flow. After linearizing and discretizing the PDEs using an explicit linear scheme, we construct a linear state-space model of the flow. The Kalman filter is then used to estimate the states by incorporating the measurements obtained from passive drifters. Drifters which are equipped with GPS recievers move with the flow and report their position at every time step. The position of the drifters at every time step are used to approximate the average velocity of the flow at the corresponding locations and time step. The method is implemented in simulation on a section of the Sacramento river in California using real data and the results are validated with a two-dimensional simulation of the river. Finally, the performance of the method using Lagrangian sensors is compared to the case of using Eulerian sensors.

Quadratic Programming based data assimilation with passive drifting sensors for shallow water flows

We present a method for assimilating Lagrangian sensor measurement data into a Shallow Water Equation model. Using our method, the variational data assimilation problem is formulated as a Quadratic Programming problem with linear constraints. Drifting sensors that gather position and velocity information in the modeled system can then be used to refine the estimate of the initial conditions of the system. A new sensor network hardware platform for gathering flow information is presented. We summarize the results of a field experiment designed to demonstrate the capabilities of our assimilation method with data gathered from the sensors. Validation of the results is performed by comparing them to an estimate derived from an independent set of static sensors.

Parameter identification for the shallow water equation using modal decomposition

A parameter identification problem for systems governed by first-order, linear hyperbolic partial differential equations subjected to periodic forcing is investigated. The problem is posed as a PDE constrained optimization problem with data of the problem given by the measured input and output variables at the boundary of the domain. By using the governing equations in the frequency domain, a spatially dependent transfer matrix relating the input variables to the output variables is obtained. It is shown that by considering a finite number of dominant oscillatory modes of the input, an accurate representation of the output can be obtained. This converts the original PDE constrained optimization problem to one without any constraints. The optimal parameters can be identified using standard nonlinear programming. The utility of the proposed approach is illustrated by considering a river reach in the Sacramento-San-Joaquin Delta, California, that is subjected to tidal forcing. The dynamics of the reach are modeled by linearized Saint-Venant equations. The available data is the flow variables measured upstream and downstream of the reach. The parameter identification problem is to estimate the average free-surface width, the bed slope, the friction coefficient and the steady-state boundary conditions. It is shown that the estimated model gives an accurate prediction of the flow variables at an intermediate location within the reach.

Inverse modeling for open boundary conditions in channel network

An inverse modeling problem for systems of networked one dimensional shallow water equations subject to periodic forcing is investigated. The problem is described as a PDE-constrained optimization problem with the objective of minimizing the norm of the difference between the observed variables and model outputs. After linearizing and discretizing the governing equations using an implicit discretization scheme, linear constraints are constructed which leads to a quadratic programming formulation of the state estimation problem. The usefulness of the proposed approach is illustrated with a channel network in the Sacramento San-Joaquin Delta in California, subjected to tidal forcing from the San Francisco Bay. The dynamics of the hydraulic system are modeled by the linearized Saint-Venant equations. The method is designed to integrate drifter data as they float in the domain. The inverse modeling problem consists in estimating open boundary conditions from sensor measurements at other locations in the network. It is shown that the proposed method gives an accurate estimation of the flow state variables at the boundaries and intermediate locations.

Heterogeneous Fleets of Active and Passive Floating Sensors for River Studies

Lagrangian sensing for tracing hydrodynamic trajectories is an innovative approach for studying estuarial environments. Actuated Lagrangian sensors are capable of avoiding obstacles and navigating when active and retain a passive hydrodynamic profile that is suited for Lagrangian sensing when passive. A heterogeneous fleet of actuated and passive drifting sensors is presented. Data assimilation using a high-performance computing HPC cluster that runs the ensemble Kalman filter EnKF is an essential component of the estuarial state estimation system. The performance of the mixed capability fleet and the data assimilation backend is evaluated in the context of a landmark 96-unit river study in the Sacramento-San Joaquin Delta region of California.

Variational Lagrangian data assimilation in open channel networks

This article presents a data assimilation method in a tidal system, where data from both Lagrangian drifters and Eulerian flow sensors were fused to estimate water velocity. The system is modeled by first-order, hyperbolic partial differential equations subject to periodic forcing. The estimation problem can then be formulated as the minimization of the difference between the observed variables and model outputs, and eventually provide the velocity and water stage of the hydrodynamic system. The governing equations are linearized and discretized using an implicit discretization scheme, resulting in linear equality constraints in the optimization program. Thus, the flow estimation can be formed as an optimization problem and efficiently solved. The effectiveness of the proposed method was substantiated by a large-scale field experiment in the Sacramento-San Joaquin River Delta in California. A fleet of 100 sensors developed at the University of California, Berkeley, were deployed in Walnut Grove, CA, to collect a set of Lagrangian data, a time series of positions as the sensors moved through the water. Measurements were also taken from Eulerian sensors in the region, provided by the United States Geological Survey. It is shown that the proposed method can effectively integrate Lagrangian and Eulerian measurement data, resulting in a suited estimation of the flow variables within the hydraulic system.

Data reconciliation of an open channel flow network using modal decomposition

This article presents a method to estimate flow variables for an open channel network governed by the linearized Saint-Venant equations and subject to periodic forcing. The discharge at the upstream end of the system and the stage at the downstream end of the system are defined as the model input; the flow properties at selected internal locations, as well as the other external boundary conditions, are defined as the output. Both inputs and outputs are affected by noise and we use the model to re-estimate this data. A spatially-dependent transfer matrix in the frequency domain is constructed to relate the model input and output using modal decomposition. A data reconciliation technique is used to incorporate the error in the measured data and results in a set of reconciliated external boundary conditions; subsequently, the flow properties at any location in the system can be accurately constructed from the input measurements. The applicability and effectiveness of the method is demonstrated with a case study of the river flow subject to tidal forcing in the Sacramento-San Joaquin Delta in California. We used existing USGS sensors placed in the Delta as measurement points, and deploy our own sensors at selected locations to produce data used for the validation. The proposed method gives an accurate estimation of the flow properties at intermediate locations within the channel network.