Four-dimensional variational data assimilation for high resolution nested models

Four-dimensional variational data assimilation (4D-Var) is used in environmental prediction to estimate the state of a system from measurements. When 4D-Var is applied in the context of high resolution nested models, problems may arise in the representation of spatial scales longer than the domain of the model. In this paper we study how well 4D-Var is able to estimate the whole range of spatial scales present in one-way nested models. Using a model of the one-dimensional advection–diffusion equation we show that small spatial scales that are observed can be captured by a 4D-Var assimilation, but that information in the larger scales may be degraded. We propose a modification to 4D-Var which allows a better representation of these larger scales.

Applying the extended Kalman filter to systems described by nonlinear differential-algebraic equations

This paper describes a method for the state estimation of nonlinear systems described by a class of differential-algebraic equation models using the extended Kalman filter. The method involves the use of a time-varying linearisation of a semi-explicit index one differential-algebraic equation. The estimation technique consists of a simplified extended Kalman filter that is integrated with the differential-algebraic equation model. The paper describes a simulation study using a model of a batch chemical reactor. It also reports a study based on experimental data obtained from a mixing process, where the model of the system is solved using the sequential modular method and the estimation involves a bank of extended Kalman filters.

Optimal control of nonlinear systems represented by differential algebraic equations

An iterative procedure is described for solving nonlinear optimal control problems subject to differential algebraic equations. The procedure iterates on an integrated modified simplified model based problem with parameter updating in such a manner that the correct solution of the original nonlinear problem is achieved.

Ensemble prediction for nowcasting with a convection-permitting model - II: forecast error statistics

A 24-member ensemble of 1-h high-resolution forecasts over the Southern United Kingdom is used to study short-range forecast error statistics. The initial conditions are found from perturbations from an ensemble transform Kalman filter. Forecasts from this system are assumed to lie within the bounds of forecast error of an operational forecast system. Although noisy, this system is capable of producing physically reasonable statistics which are analysed and compared to statistics implied from a variational assimilation system. The variances for temperature errors for instance show structures that reflect convective activity. Some variables, notably potential temperature and specific humidity perturbations, have autocorrelation functions that deviate from 3-D isotropy at the convective-scale (horizontal scales less than 10 km). Other variables, notably the velocity potential for horizontal divergence perturbations, maintain 3-D isotropy at all scales. Geostrophic and hydrostatic balances are studied by examining correlations between terms in the divergence and vertical momentum equations respectively. Both balances are found to decay as the horizontal scale decreases. It is estimated that geostrophic balance becomes less important at scales smaller than 75 km, and hydrostatic balance becomes less important at scales smaller than 35 km, although more work is required to validate these findings. The implications of these results for high-resolution data assimilation are discussed.

A method for merging flow-dependent forecast error statistics from an ensemble with static statistics for use in high resolution variational data assimilation

The background error covariance matrix, B, is often used in variational data assimilation for numerical weather prediction as a static and hence poor approximation to the fully dynamic forecast error covariance matrix, Pf. In this paper the concept of an Ensemble Reduced Rank Kalman Filter (EnRRKF) is outlined. In the EnRRKF the forecast error statistics in a subspace defined by an ensemble of states forecast by the dynamic model are found. These statistics are merged in a formal way with the static statistics, which apply in the remainder of the space. The combined statistics may then be used in a variational data assimilation setting. It is hoped that the nonlinear error growth of small-scale weather systems will be accurately captured by the EnRRKF, to produce accurate analyses and ultimately improved forecasts of extreme events.

Correlations of control variables in variational data assimilation

Variational data assimilation systems for numerical weather prediction rely on a transformation of model variables to a set of control variables that are assumed to be uncorrelated. Most implementations of this transformation are based on the assumption that the balanced part of the flow can be represented by the vorticity. However, this assumption is likely to break down in dynamical regimes characterized by low Burger number. It has recently been proposed that a variable transformation based on potential vorticity should lead to control variables that are uncorrelated over a wider range of regimes. In this paper we test the assumption that a transform based on vorticity and one based on potential vorticity produce an uncorrelated set of control variables. Using a shallow-water model we calculate the correlations between the transformed variables in the different methods. We show that the control variables resulting from a vorticity-based transformation may retain large correlations in some dynamical regimes, whereas a potential vorticity based transformation successfully produces a set of uncorrelated control variables. Calculations of spatial correlations show that the benefit of the potential vorticity transformation is linked to its ability to capture more accurately the balanced component of the flow.

Velocity-based moving mesh methods for nonlinear partial differential equations

This article describes a number of velocity-based moving mesh numerical methods formultidimensional nonlinear time-dependent partial differential equations (PDEs). It consists of a short historical review followed by a detailed description of a recently developed multidimensional moving mesh finite element method based on conservation. Finite element algorithms are derived for both mass-conserving and non mass-conserving problems, and results shown for a number of multidimensional nonlinear test problems, including the second order porous medium equation and the fourth order thin film equation as well as a two-phase problem. Further applications and extensions are referenced.