This article is motivated by the practical problem of highway traffic estimation using velocity measurements from GPS enabled mobile devices such as cell phones. In order to simplify the estimation procedure, a velocity model for highway traffic is constructed, which results in a dynamical system in which the observation operator is linear. This article presents a new scalar hyperbolic partial differential equation (PDE) model for traffic velocity evolution on highways, based on the seminal Lighthill-Whitham-Richards (LWR) PDE for density. Equivalence of the solution of the new velocity PDE and the solution of the LWR PDE is shown for quadratic flux functions. Because this equivalence does not hold for general flux functions, a discretized model of velocity evolution based on the Godunov scheme applied to the LWR PDE is proposed. Using an explicit instantiation of the weak boundary conditions of the PDE, the discrete velocity evolution model is generalized to a network, thus making the model applicable to arbitrary highway networks. The resulting velocity model is a nonlinear and nondifferentiable discrete time dynamical system with a linear observation operator, for which a Monte Carlo based ensemble Kalman filtering data

We formulate a four dimensional Ensemble Kalman Filter (4D-LETKF) that minimizes a cost function similar to that in a 4D-VAR method. Using per-fect model experiments with the Lorenz-96 model, we compare assimilation of simulated asynchronous observations with 4D-VAR and 4D-LETKF. We find that both schemes have comparable error when 4D-LETKF is performed sufficiently frequently and when 4D-VAR is performed over a sufficiently long analysis time window. We explore how the error depends on the time between analyses for 4D-LETKF and the analysis time window for 4D-VAR. 1

ABSTRACT A variant of a least squares ensemble (Kalman) filter that is suitable for implementation on parallel architectures is presented. This parallel ensemble filter produces results that are identical to those from sequential algorithms already described in the literature when forward observation operators that relate the model state vector to the expected value of observations are linear (although actual results may differ due to floating point arithmetic round-off error). For nonlinear forward observation operators, the sequential and parallel algorithms solve different linear approximations to the full problem but produce qualitatively similar results. The parallel algorithm can be implemented to produce identical answers with the state variable prior ensembles arbitrarily partitioned onto a set of processors for the assimilation step (no caveat on round-off is needed for this result). Example implementations of the parallel algorithm are described for environments with low (high) communication latency and cost. Hybrids of these implementations and the traditional sequential ensemble filter can be designed to optimize performance for a variety of parallel computing environments. For large models on machines with good communications, it is possible to implement the parallel algorithm to scale efficiently to thousands of processors while bit-wise reproducing the results from a single processor implementation. Timing results on several Linux clusters are presented from an implementation appropriate for machines with low-latency communication. Most ensemble Kalman filter variants that have appeared in the literature differ only in the details of how a prior ensemble estimate of a scalar observation is updated given an observed value and the observational error distribution. These details do not impact other parts of either the sequential or parallel filter algorithms here, so a variety of ensemble filters including ensemble square root and perturbed observations filters can be used with all the implementations described.

A wildfire model is formulated based on balance equations for energy and fuel, where the fuel loss due to combustion corresponds to the fuel reaction rate. The resulting coupled partial differential equations have coefficients which can be approximated from prior measurements of wildfires. An Ensemble Kalman Filter technique is then used to assimilate temperatures measured at selected points into running wildfire simulations. The assimilation technique is able to modify the simulations to track the measurements correctly even if the simulations were started with an erroneous ignition location that is quite far away from the correct one.

Abstract. We report on an ongoing effort to build a Dynamic Data Driven Application System (DDDAS) for short-range forecast of weather and wildfire behavior from real-time weather data, images, and sensor streams. The system changes the forecast as new data is received. We encapsulate the model code and apply an ensemble Kalman filter in time-space with a highly parallel implementation. In this paper, we discuss how we will demonstrate that our system works using a DDDAS testbed approach and data collected from an earlier fire. 1

A new type of ensemble filter is proposed, which combines an ensemble Kalman filter (EnKF) with the ideas of morphing and registration from image processing. This results in filters suitable for nonlinear problems whose solutions exhibit moving coherent features, such as thin interfaces in wildfire modeling. The ensemble members are represented as the composition of one common state with a spatial transformation, called registration mapping, plus a residual. A fully automatic registration method is used that requires only gridded data, so the features in the model state do not need to be identified by the user. The morphing EnKF operates on a transformed state consisting of the registration mapping and the residual. Essentially, the morphing EnKF uses intermediate states obtained by morphing instead of linear combinations of the states. 1

assimilation

We present an efficient variation of the Local Ensemble Kalman Filter (Ott et al. 2002, 2004) and the results of perfect model tests with the Lorenz-96 model. This scheme is locally analogous to performing the Ensemble Transform Kalman Filter (Bishop et al. 2001). We also include a four-dimensional extension of the scheme to allow for asynchronous observations. 1.

Convergence of the ensemble Kalman filter in the limit for large ensembles to the Kalman filter is proved. In each step of the filter, convergence of the ensemble sample covariance follows from a weak law of large numbers for exchangeable random variables, Slutsky’s theorem gives weak convergence of ensemble members, and L p bounds on the ensemble then give L p convergence.

Formation properties are one of the key factors in numerical modeling of flow and transport in geologic formations in spite of the fact that they may not be completely characterized. The incomplete knowledge or uncertainty in the description of the formation properties leads to uncertainty in simulation results. In this study, the ensemble Kalman filter (EnKF) approach is used for continuously updating model parameters such as hydraulic conductivity and model variables such as pressure head while simultaneously providing an estimate of the uncertainty through assimilating dynamic and static measurements, without resorting to the explicit computation of the covariance or the Jacobian of the state variables. A two-dimensional example is built to demonstrate the capability of EnKF and to analyze its sensitivity with respect to different factors such as the number of realizations, measurement timings, and initial guesses. An additional example is given to illustrate the applicability of EnKF to three-dimensional problems and to examine the model predictability after dynamic data assimilation. It is found from these examples that EnKF provides an efficient approach for obtaining satisfactory estimation of the hydraulic conductivity field with dynamic measurements. After data assimilation the conductivity field matches the reference field very well, and different kinds of incorrect prior knowledge of the formation properties may also be rectified to a certain extent.