Reinhard Furrer's Projects

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Ensemble Kalman Filter

Approximation of Forecast Covariances in Kalman Filter Variants
In collaboration with T. Bengtsson, Bell Labs, Lucent.

Many modern geophysical problems are characterized by extremely high-dimensional systems and pose difficult challenges for real-time assimilation of system information and observations. Recent focus in the atmospheric sciences has been on representing the knowledge of the atmospheric state using a probability density function, and various sample based techniques have been developed to address the problem of real-time updating and forecasting for high-dimensional systems. We study the effects of matrix sample variability for two Monte-Carlo based Kalman filter variants, the ensemble Kalman filter and the square root filter.

For the time being we obtained some good but theoretical results for which we do not know their impact in practice. We envision to proceed with the application and validation of the method in large scale problems such as operational numerical weather prediction.

Extreme value theory

U-Statistics and PWM in Modeling Extremes
In collaboration with P. Naveau, University of Colorado, Boulder/Laboratoire des Sciences du Climat et de l'Environnement (LSCE-IPSL), Gif-sur-Yvette, France

The generalized Pareto distribution is a key ingredient in modeling the distribution of the excess of observations over large thresholds. The parameters can be estimated with maximum likelihood, conventional methods of moments or with probability weighted moments (PWM). We discuss PWM as a particular U-estimator with which we can derive exact variances and covariances of the estimator and extend its limit distribution to alpha-stability.

Among other reasons PWMs are criticized by statisticians because they are not as easily adaptable to the non iid case. We are currently generalizing the theory to the dependent case for which we assume some type of mixing behavior and to bi- and multivariate random variates.

Robustness in geostatistics

Robust Prediction for Contaminated Random Fields
In collaboration with B. Fournier, EPF Lausanne, Switzerland.

Interpolation of a spatially correlated random processes is used in many scientific domains. The best unbiased linear predictor (BLUP), often called kriging predictor in geostatistical science, is sensitive to outliers. Although there have been a few attempts to robustify the kriging predictor, none of them is completely satisfactory. We introduce a new robust linear predictor for a substitutive error model. First, we derive a BLUP, which is computationally very expensive even for moderate sample sizes. The exact solution is approximated by a simple linear predictor, which is robust with respect to substitutive errors.

Model assumptions in the considered model could still be weakened resulting in more flexibility with respect to possible structures. Natural extensions of the method are to non-Gaussian fields and correlated contamination scenarios in the substitutive error model.