Reduced rank filtering in chaotic systems with application in geophysical sciences

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 2008.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student submitted PDF version of thesis.Includes bibliographical references.Recent technological advancements have enabled us to collect large volumes of geophysical noisy measurements that need to be combined with the model forecasts, which capture all of the known properties of the underlying system. This problem is best formulated in a stochastic optimization framework, which when solved recursively is known as Filtering. Due to the large dimensions of geophysical models, optimal filtering algorithms cannot be implemented within the constraints of available computation resources. As a result, most applications use suboptimal reduced rank algorithms. Successful implementation of reduced rank filters depends on the dynamical properties of the underlying system. Here, the focus is on geophysical systems with chaotic behavior defined as extreme sensitivity of the dynamics to perturbations in the state or parameters of the system. In particular, uncertainties in a chaotic system experience growth and instability along a particular set of directions in the state space that are continually subject to large and abrupt state-dependent changes. Therefore, any successful reduced rank filter has to continually identify the important direction of uncertainty in order to properly estimate the true state of the system. In this thesis, we introduce two efficient reduced rank filtering algorithms for chaotic system, scalable to large geophysical applications. Firstly, a geometric approach is taken to identify the growing directions of uncertainty, which translate to the leading singular vectors of the state transition matrix over the forecast period, so long as the linear approximation of the dynamics is valid.The singular vectors are computed via iterations of the linear forward and adjoint models of the system and used in a filter with linear Kalman-based update. Secondly, the dynamical stability of the estimation error in a filter with linear update is analyzed, assuming that error propagation can be approximated using the state transition matrix of the system over the forecast period. The unstable directions of error dynamics are identified as the Floquet vectors of an auxiliary periodic system that is defined based on the forecast trajectory. These vectors are computed by iterations of the forward nonlinear model and used in a Kalman-based filter. Both of the filters are tested on a chaotic Lorenz 95 system with dynamic model error against the ensemble Kalman filter. Results show that when enough directions are considered, the filters perform at the optimal level, defined by an ensemble Kalman filter with a very large ensemble size. Additionally, both of the filters perform equally well when the dynamic model error is absence and ensemble filters fail. The number of iterations for computing the vectors can be set a priori based on the available computational resources and desired accuracy. To investigate scalability of the algorithms, they are implemented in a quasi-geostrophic ocean circulation model. The results are promising for future extensions to realistic geophysical applications, with large models.by Adel Ahanin.Ph.D

The role of model dynamics in ensemble Kalman filter performance for chaotic systems

The ensemble Kalman filter (EnKF) is susceptible to losing track of observations, or ‘diverging’, when applied to large chaotic systems such as atmospheric and ocean models. Past studies have demonstrated the adverse impact of sampling error during the filter’s update step. We examine how system dynamics affect EnKF performance, and whether the absence of certain dynamic features in the ensemble may lead to divergence. The EnKF is applied to a simple chaotic model, and ensembles are checked against singular vectors of the tangent linear model, corresponding to short-term growth and Lyapunov vectors, corresponding to long-term growth. Results show that the ensemble strongly aligns itself with the subspace spanned by unstable Lyapunov vectors. Furthermore, the filter avoids divergence only if the full linearized long-term unstable subspace is spanned. However, short-term dynamics also become important as nonlinearity in the system increases. Non-linear movement prevents errors in the long-term stable subspace from decaying indefinitely. If these errors then undergo linear intermittent growth, a small ensemble may fail to properly represent all important modes, causing filter divergence. A combination of long and short-term growth dynamics are thus critical to EnKF performance. These findings can help in developing practical robust filters based on model dynamics.National Science Foundation (U.S.) (CMF Program Grant 0530851)National Science Foundation (U.S.) (DDAS Program Grant 0540259)National Science Foundation (U.S.) (ITR/AP Program Grant 0121182

Modeling of nitrate loading and transport in the Plymouth aquifer

Thesis (M.Eng.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 2002.Includes bibliographical references (leaves 59-61).by Adel Ahanin.M.Eng