Filtering problems in high dimensional geophysical applications often require spatially continuous models to interpolate spatially and temporally sparse data. Many applications in numerical weather and ocean state prediction are concerned with tracking and assessing the uncertainty in the position of large scale vorticity features, such as storm fronts, jets streams, and hurricanes. Quantifying the amplitude variance in these features is complicated by the fact that both height and lateral perturbations in the feature geometry are represented in the same covariance estimate. However, when there are sufficient observations to detect feature information like spatial gradients, the positions of these features can be used to further constrain the filter, as long as the statistical model (cost function) has provisions for both height perturbations and lateral displacements. Several authors since the 1990s have proposed various formalisms for the simultaneous modeling of position and amplitude errors, and the typical approaches to computing the generalized solutions in these applications are variational or direct optimization. The ensemble Kalman filter is often employed in large scale nonlinear filtering problems, but its predication on Gaussian statistics causes its estimators suffer from analysis deflation or collapse, as well as the usual curse of dimensionality in high dimensional Monte Carlo simulations. Moreover, there is no theoretical guarantee of the performance of the ensemble Kalman filter with nonlinear models. Particle filters which employ importance sampling to focus attention on the important regions of the likelihood have shown promise in recent studies on the control of particle size. Consider an ensemble forecast of a system with prominent feature information. The correction of displacements in these features, by pushing them into better agreement with observations, is an application of importance sampling, and Monte Carlo methods, including particle filters, and possibly the ensemble Kalman filter as well, are well suited to applications of feature displacement correction. In the present work, we show that the ensemble Kalman filter performs well in problems where large features are displaced both in amplitude and position, as long as it is used on a statistical model which includes both function height and local position displacement in the model state. In a toy model, we characterize the performance-degrading effect that untracked displacements have on filters when large features are present. We then employ tools from classical physics and fluid dynamics to statistically model displacements by area-preserving coordinate transformations. These maps preserve the area of contours in the displaced function, and using strain measures from continuum mechanics, we regularize the statistics on these maps to ensure they model smooth, feature-preserving displacements. The position correction techniques are incorporated into the statistical model, and this modified ensemble Kalman filter is tested on a system of vortices driven by a stochastically forced barotropic vorticity equation. We find that when the position correction term is included in the statistical model, the modified filter provides estimates which exhibit substantial reduction in analysis error variance, using a much smaller ensemble than what is required when the position correction term is removed from the model.
| Identifer | oai:union.ndltd.org:arizona.edu/oai:arizona.openrepository.com:10150/339055 |
| Date | January 2014 |
| Creators | Rosenthal, William Steven |
| Contributors | Restrepo, Juan M., Venkataramani, Shankar, Restrepo, Juan M., Venkataramani, Shankar, Arellano, Avelino |
| Publisher | The University of Arizona. |
| Source Sets | University of Arizona |
| Language | en_US |
| Detected Language | English |
| Type | text, Electronic Dissertation |
| Rights | Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. |
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