A.W. Heemink

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).

A novel approach to multi-criteria inverse planning for IMRT

Treatment plan optimization is a multi-criteria process. Optimizing solely on one objective or on a sum of a priori weighted objectives may result in inferior treatment plans. Manually adjusting weights or constraints in a trial and error procedure is time consuming. In this paper we introduce a novel multi-criteria optimization approach to automatically optimize treatment constraints (dose-volume and maximum-dose). The algorithm tries to meet these constraints as well as possible, but in the case of conflicts it relaxes lower priority constraints so that higher priority constraints can be met. Afterwards, all constraints are tightened, starting with the highest priority constraints. Applied constraint priority lists can be used as class solutions for patients with similar tumour types. The presented algorithm does iteratively apply an underlying algorithm for beam profile optimization, based on a quadratic objective function with voxel-dependent importance factors. These voxel-dependent importance factors are automatically adjusted to reduce dose-volume and maximum-dose constraint violations.

Inverse 3D shallow water flow modelling of the continental shelf

The adjoint method has been derived and implemented for solving inverse 3D shallow water flow modelling problems. Attention is concentrated on estimating the harmonic constants in the open boundary conditions, the space varying friction parameter, the space varying viscosity parameter and the depth values in a 3D shallow sea model of the entire European Continental Shelf. Here the estimation problem is formulated as a large-scale optimization problem that is solved with a gradient-based optimization method. The gradient is determined efficiently by using the solution of the adjoint problem.

Model-Reduced Variational Data Assimilation

Abstract This paper describes a new approach to variational data assimilation that with a comparable computational efficiency does not require implementation of the adjoint of the tangent linear approximation of the original model. In classical variational data assimilation, the adjoint implementation is used to efficiently compute the gradient of the criterion to be minimized. Our approach is based on model reduction. Using an ensemble of forward model simulations, the leading EOFs are determined to define a subspace. The reduced model is created by projecting the original model onto this subspace. Once this reduced model is available, its adjoint can be implemented very easily and can be used to approximate the gradient of the criterion. The minimization process can now be solved completely in reduced space with negligible computational costs. If necessary, the procedure can be repeated a few times by generating new ensembles closer to the most recent estimate of the parameters. The reduced-model-based ...

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.

Stochastic modelling of dispersion in shallow water

A random walk model to describe the dispersion of pollutants in shallow water is developed. By deriving the Fokker-Planck equation, the model is shown to be consistent with the two-dimensional advection-diffusion equation with space-varying dispersion coefficient and water depth. To improve the behaviour of the model shortly after the deployment of the pollutant, a random flight model is developed too. It is shown that over long simulation periods, this model is again consistent with the advection-diffusion equation. The various numerical aspects of the implementation of the stochastic models are discussed and finally a realistic application to predict the dispersion of a pollutant in the Eastern Scheldt estuary is described.

Data assimilation of ground‐level ozone in Europe with a Kalman filter and chemistry transport model

[1] A Kalman filter coupled to the atmospheric chemistry transport model EUROS has been used to estimate the ozone concentrations in the boundary layer above Europe. Two Kalman filter algorithms, the reduced rank square root (RRSQRT) and the ensemble Kalman filter (ENKF), were implemented in this study. Both required, in general, a large number of EUROS model simulations for an assimilation. The observations consisted of hourly ozone data in a set of 135 ground-based stations in Europe for the period, June 1996. Half of these stations were used for the assimilation and the other half only for validation of the results. The combination between data assimilation (Kalman filter) and the atmospheric chemistry transport model, EUROS, gave more accurate results for boundary layer ozone than the EUROS model or measurements used separately. The average difference between assimilated and measured ozone concentrations decreased from 27.4 to 20.5 m gm � 3 for the average of the stations used for validation in Europe. Both algorithms tend to converge to about the same accuracy, with an increasing number of EUROS model runs. About 10–20 EUROS model calculations were found sufficient for a good assimilation. The results are supported by a number of simulations that also reveal a local character for the assimilation process. INDEX TERMS: 3337 Meteorology and Atmospheric Dynamics: Numerical modeling and data assimilation; 3307 Meteorology and Atmospheric Dynamics: Boundary layer processes; 0345 Atmospheric Composition and Structure: Pollution—urban and regional (0305); 0368 Atmospheric Composition and Structure: Troposphere—constituent transport and chemistry; KEYWORDS: atmospheric NOx, VOC

Inverse modeling of groundwater flow using model reduction

Numerical groundwater flow models often have a very high number of model cells (greater than a million). Such models are computationally very demanding, which is disadvantageous for inverse modeling. This paper describes a low?dimensional formulation for groundwater flow that reduces the computational burden necessary for inverse modeling. The formulation is a projection of the original groundwater flow equation on a set of orthogonal patterns (i.e., a Galerkin projection). The patterns (empirical orthogonal functions) are computed by a decomposition of the covariance matrix over an ensemble of model solutions. Those solutions represent the behavior of the model as a result of model impulses and the influence of a chosen set of parameter values. For an interchangeable set of parameter values the patterns yield a low?dimensional model, as the number of patterns is often small. An advantage of this model is that the adjoint is easily available and most accurate for inverse modeling. For several synthetical cases the low?dimensional model was able to find the global minimum efficiently, and the result was comparable to that of the original model. For several cases our model even converged where the original model failed. Our results demonstrate that the proposed procedure results in a 60% time reduction to solve the groundwater flow inverse problem. Greater efficiencies can be expected in practice for large?scale models with a large number of grid cells that are used to compute transient simulations.

Two-dimensional shallow water flow identification

Abstract A discrete time-invariant Kalman filter for the identification and prediction of two-dimensional shallow water flow using observations of the water level registered at some locations, has been developed. The filter is based on a set of difference equations derived from the linear two-dimensional shallow water equations using the finite difference scheme proposed by Sielecki. By introducing a system noise process, we can embed the difference equations inot a stochastic environment. This enables us to take into account the uncertainties of these equations. A Chandrasekhar-type algorithm is employed to obtain the steady-state filter. In this way the fact that the noise is less spatially variable than the underlying process can be exploited to reduce the computational burden. The capabilities of the filter are illustrated by applying it to the six-hours-ahead prediction of storm surges in the North Sea. The results show excellent filter performance, and, with respect to the results of the underlying deterministic model which were achieved without using the water-level measurements available, the improvement obtained by filtering the measurements is substantial.