Reduced rank filtering in chaotic systems with application in geophysical sciences
Author(s)Ahanin, Adel, 1977-
Massachusetts Institute of Technology. Dept. of Civil and Environmental Engineering.
Dara Entekhabi and Dennis McLaughlin.
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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.
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.
DepartmentMassachusetts Institute of Technology. Department of Civil and Environmental Engineering
Massachusetts Institute of Technology
Civil and Environmental Engineering.