A demonstration study of three advanced, sequential data assimilation methods, applied with the nonlinear Miami Isopycnic Coordinate Ocean Model (MICOM), has been performed within the European Commission-funded DIADEM project. The data assimilation techniques considered are the Ensemble Kalman Filter (EnKF), the Ensemble Kalman Smoother (EnKS) and the Singular Evolutive Extended Kalman (SEEK) Filter, which all in different ways resemble the original Kalman Filter. In the EnKF and EnKS an ensemble of model states is integrated forward in time according to the model dynamics, and statistical moments needed at analysis time are calculated from the ensemble of model states. The EnKS, as opposed to the EnKF, update the analysis also backward in time whenever new observations are available, thereby improving the estimated states at the previous analysis times. The SEEK filter reduces the computational burden of the error propagation by representing the errors in a subspace which is initially calculated from a truncated EOF analysis. A hindcast experiment, where sea-level anomaly and sea-surface temperature data are assimilated, has been conducted in the North Atlantic for the time period July until September 1996. In this paper, we describe the implementation of ensemble-based assimilation methods with a common theoretical framework, we present results from hindcast experiments achieved with the EnKF, EnKS and SEEK filter, and we discuss the relative merits of these methods from the perspective of operational marine monitoring and forecasting systems. We found that the three systems have similar performances, and they can be considered feasible technologically for building preoperational prototypes. (C) 2003 Elsevier Science B.V. All rights reserved.

We consider the inverse problem of recovery of unknown coefficient functions in differential equations. The set of PDEs constituting the current forward model describes a special case of two-phase porous-media flow. The focus of the paper is on the influence of different length scales on parameter estimation efficiency. The investigation into these issues is facilitated by applying a multiscale spline wavelet parametrization of the unknown function. Earlier investigations with an ODE forward model found that use of the multiscale Haar parametrization had a positive effect on the estimation efficiency of a quasi-Newton algorithm. Recently, a way to systematically enhance these effects has been suggested. In this paper, we further this approach with the Levenberg-Marquardt algorithm. This results in three variants of the Levenberg-Marquardt algorithm, each incorporating a possibility to enhance multiscale effects. Through numerical experiments with the PDE forward model, we assess the estimation efficiency of the variants when varying the enhancement of multiscale effects.

Operator splitting methods are often used to solve convection–diffusion problems of convection dominated nature. However, it is well known that such methods can produce significant (splitting) errors in regions containing self sharpening fronts. To amend this shortcoming, corrected operator splitting methods have been developed. These approaches use the wave structure from the convection step to identify the splitting error. This error is then compensated for in the diffusion step. The main purpose of the present work is to illustrate the importance of the correction step in the context of an inverse problem. The inverse problem will consist of estimating the fractional flow function in a one‐dimensional saturation equation.

So-called corrected operator splitting methods are applied to a 1-D scalar advection-diffusion equation of Buckley-Leverett type with general initial data. Front tracking and a 2nd order Godunov method are used to advance the solution in time. Diffusion is modelled by piece wise linear finite elements at each new time level. To obtain correct structure of shock fronts independently of the size of the time step, a dynamically defined residual flux term is grouped with diffusion. Different test problems are considered, and the methods are compared with respect to accuracy and runtime. Finally, we extend the corrected operator splitting to 2-D equations by means of dimensional splitting, and we apply it to a Buckley-Leverett type problem including gravitational effects.