Martin Verlaan

Tidal flow forecasting using reduced rank square root filters

The Kalman filter algorithm can be used for many data assimilation problems. For large systems, that arise from discretizing partial differential equations, the standard algorithm has huge computational and storage requirements. This makes direct use infeasible for many applications. In addition numerical difficulties may arise if due to finite precision computations or approximations of the error covariance the requirement that the error covariance should be positive semi-definite is violated.In this paper an approximation to the Kalman filter algorithm is suggested that solves these problems for many applications. The algorithm is based on a reduced rank approximation of the error covariance using a square root factorization. The use of the factorization ensures that the error covariance matrix remains positive semi-definite at all times, while the smaller rank reduces the number of computations and storage requirements. The number of computations and storage required depend on the problem at hand, but will typically be orders of magnitude smaller than for the full Kalman filter without significant loss of accuracy.The algorithm is applied to a model based on a linearized version of the two-dimensional shallow water equations for the prediction of tides and storm surges.For non-linear models the reduced rank square root algorithm can be extended in a similar way as the extended Kalman filter approach. Moreover, by introducing a finite difference approximation to the Reduced Rank Square Root algorithm it is possible to prevent the use of a tangent linear model for the propagation of the error covariance, which poses a large implementational effort in case an extended kalman filter is used.

Variance Reduced Ensemble Kalman Filtering

A number of algorithms to solve large-scale Kalman filtering problems have been introduced recently. The ensemble Kalman filter represents the probability density of the state estimate by a finite number of randomly generated system states. Another algorithm uses a singular value decomposition to select the leading eigenvectors of the covariance matrix of the state estimate and to approximate the full covariance matrix by a reduced-rank matrix. Both algorithms, however, still require a huge amount of computer resources. In this paper the authors propose to combine the two algorithms and to use a reduced-rank approximation of the covariance matrix as a variance reductor for the ensemble Kalman filter. If the leading eigenvectors explain most of the variance, which is the case for most applications, the computational burden to solve the filtering problem can be reduced significantly (up to an order of magnitude).

Nonlinearity in Data Assimilation Applications: A Practical Method for Analysis

A new method to quantify the nonlinearity of data assimilation problems is proposed. The method includes the effects of system errors, measurement errors, observational network, and sampling interval. It is based on computation of the first neglected term in a ‘‘Taylor’’ series expansion of the errors introduced by an extended Kalman filter, and can be computed at very little cost when one is already applying a second-order (or higher order) Kalman filter or an ensemble Kalman filter. The nonlinearity measure proposed here can be used to classify the ‘‘hardness’’ of the problem and predict the failure of data assimilation algorithms. In this manner it facilitates the comparison of data assimilation algorithms and applications. The method is applied to the well-known Lorenz model. A comparison is made between several data assimilation algorithms that are suitable for nonlinear problems. The results indicate significant differences in performance for more nonlinear problems. For low values of V, a measure of nonlinearity, the differences are negligible.

A global reanalysis of storm surges and extreme sea levels.

Extreme sea levels, caused by storm surges and high tides, can have devastating societal impacts. To effectively protect our coasts, global information on coastal flooding is needed. Here we present the first global reanalysis of storm surges and extreme sea levels (GTSR data set) based on hydrodynamic modelling. GTSR covers the entire worlds coastline and consists of time series of tides and surges, and estimates of extreme sea levels. Validation shows that there is good agreement between modelled and observed sea levels, and that the performance of GTSR is similar to that of many regional hydrodynamic models. Due to the limited resolution of the meteorological forcing, extremes are slightly underestimated. This particularly affects tropical cyclones, which requires further research. We foresee applications in assessing flood risk and impacts of climate change. As a first application of GTSR, we estimate that 1.3% of the global population is exposed to a 1 in 100-year flood.

Extreme sea levels on the rise along Europe's coasts

Future extreme sea levels (ESLs) and flood risk along European coasts will be strongly impacted by global warming. Yet, comprehensive projections of ESL that include mean sea level (MSL), tides, waves, and storm surges do not exist. Here, we show changes in all components of ESLs until 2100 in view of climate change. We find that by the end of this century, the 100-year ESL along Europes coastlines is on average projected to increase by 57 cm for Representative Concentration Pathways (RCP)4.5 and 81 cm for RCP8.5. The North Sea region is projected to face the highest increase in ESLs, amounting to nearly 1 m under RCP8.5 by 2100, followed by the Baltic Sea and Atlantic coasts of the UK and Ireland. Relative sea level rise (RSLR) is shown to be the main driver of the projected rise in ESL, with increasing dominance toward the end of the century and for the high-concentration pathway. Changes in storm surges and waves enhance the effects of RSLR along the majority of northern European coasts, locally with contributions up to 40%. In southern Europe, episodic extreme events tend to stay stable, except along the Portuguese coast and the Gulf of Cadiz where reductions in surge and wave extremes offset RSLR by 20–30%. By the end of this century, 5 million Europeans currently under threat of a 100-year ESL could be annually at risk from coastal flooding under high-end warming. The presented dataset is available through this link: http://data.jrc.ec.europa.eu/collection/LISCOAST. Plain Language Summary Future extreme sea levels and flood risk along European coasts will be strongly impacted by global warming. Here, we show changes in all acting components, i.e., sea level rise, tides, waves, and storm surges, until 2100 in view of climate change. We find that by the end of this century the 100-year event along Europe will on average increase between 57 and 81 cm. The North Sea region is projected to face the highest increase, amounting to nearly 1 m under a high emission scenario by 2100, followed by the Baltic Sea and Atlantic coasts of the UK and Ireland. Sea level rise is the main driver of the changes, but intensified climate extremes along most of northern Europe can have significant local effects. Little changes in climate extremes are shown along southern Europe, with the exception of a projected decrease along the Portuguese coast and the Gulf of Cadiz, offseting sea level rise by 20–30%. By the end of this century, 5 million Europeans currently under threat of a 100-year coastal flood event could be annually at risk from coastal flooding under high-end warming.

Improved water-level forecasting for the Northwest European Shelf and North Sea through direct modelling of tide, surge and non-linear interaction

In real-time operational coastal forecasting systems for the northwest European shelf, the representation accuracy of tide–surge models commonly suffers from insufficiently accurate tidal representation, especially in shallow near-shore areas with complex bathymetry and geometry. Therefore, in conventional operational systems, the surge component from numerical model simulations is used, while the harmonically predicted tide, accurately known from harmonic analysis of tide gauge measurements, is added to forecast the full water-level signal at tide gauge locations. Although there are errors associated with this so-called astronomical correction (e.g. because of the assumption of linearity of tide and surge), for current operational models, astronomical correction has nevertheless been shown to increase the representation accuracy of the full water-level signal. The simulated modulation of the surge through non-linear tide–surge interaction is affected by the poor representation of the tide signal in the tide–surge model, which astronomical correction does not improve. Furthermore, astronomical correction can only be applied to locations where the astronomic tide is known through a harmonic analysis of in situ measurements at tide gauge stations. This provides a strong motivation to improve both tide and surge representation of numerical models used in forecasting. In the present paper, we propose a new generation tide–surge model for the northwest European Shelf (DCSMv6). This is the first application on this scale in which the tidal representation is such that astronomical correction no longer improves the accuracy of the total water-level representation and where, consequently, the straightforward direct model forecasting of total water levels is better. The methodology applied to improve both tide and surge representation of the model is discussed, with emphasis on the use of satellite altimeter data and data assimilation techniques for reducing parameter uncertainty. Historic DCSMv6 model simulations are compared against shelf wide observations for a full calendar year. For a selection of stations, these results are compared to those with astronomical correction, which confirms that the tide representation in coastal regions has sufficient accuracy, and that forecasting total water levels directly yields superior results.

Non-uniqueness in probabilistic numerical identification of bacteria

In this note we point out an inherent difficulty in numerical identification of bacteria. The problem is that of uniqueness of the taxonomic structure or, in mathematical terms, the lack of statist ...

Conservation of Mass and Preservation of Positivity with Ensemble-Type Kalman Filter Algorithms

AbstractThis paper considers the incorporation of constraints to enforce physically based conservation laws in the ensemble Kalman filter. In particular, constraints are used to ensure that the ensemble members and the ensemble mean conserve mass and remain nonnegative through measurement updates. In certain situations filtering algorithms such as the ensemble Kalman filter (EnKF) and ensemble transform Kalman filter (ETKF) yield updated ensembles that conserve mass but are negative, even though the actual states must be nonnegative. In such situations if negative values are set to zero, or a log transform is introduced, the total mass will not be conserved. In this study, mass and positivity are both preserved by formulating the filter update as a set of quadratic programming problems that incorporate nonnegativity constraints. Simple numerical experiments indicate that this approach can have a significant positive impact on the posterior ensemble distribution, giving results that are more physically pla...

A modified RRSQRT-filter for assimilating data in atmospheric chemistry models

The RRSQRT-filter is a special formulation of the Kalman filter for assimilation of data in large scale models. In this formulation, the covariance matrix of the model state is expressed in a limited number of modes. Two modifications have been made to the filter such that it is more robust when applied in combination with an atmospheric chemistry model; both act on the reduction of the covariance matrix into modes. The first modification proposes a transformation of the state, which makes the reduction invariant for a change in units and helps to collect the most important covariance structures in the first modes. The second modification extracts additional information from the reduction algorithm to limit the formation of unphysical states by the filter

Classification of Enterobacteriaceae by minimization of stochastic complexity

A new method for classifying bacteria is presented and applied to a large set of biochemical data for the Enterobacteriaceae. The method minimizes the bits needed to encode the classes and the items or, equivalently, maximizes the information content of the classification. The resulting taxonomy of Enterobacteriaceae corresponds well to the general structure of earlier classifications. Minimization of stochastic complexity can be considered as a useful tool to create bacterial classifications that are optimal from the point of view of information theory.