In this paper we present examples of ground roll attenuation for synthetic and real data gathers by using Contourlet and Curvelet transforms. These non-separable wavelet transforms are locoalized both (x,t)- and (k,f)-domains and allow for adaptive seperation of signal and ground roll. Both linear and non-linear filtering are discussed using the unique properties of these basis that allow for simultaneous localization in the both domains. Eventhough, the linear filtering techniques are encouraging the true added value of these basis-function techniques becomes apparent when we use these decompositions to adaptively substract modeled ground roll from data using a non-linear thesholding procedure. We show real and synthetic examples and the results suggest that these directional-selective basis functions provide a usefull tool for the removal of coherent noise such as ground roll

This dissertation presents solutions to two open problems in estimation theory. The first is a tractable analytical solution for problems in multi-target filtering which are too complex to solve using traditional techniques. The second explores a new approach to the nonlinear filtering problem for a general class of models. The approach to the multi-target filtering problem which involves jointly estimating a random process of the number of targets and their state, developed using the probability hypothesis density (PHD) filter alleviates the intractability of the problem by avoiding explicit data association. Moreover, the notion of linear jump Markov systems is generalized to the multiple target case to accommodate births, deaths and switching dynamics to derive a closed form solution to the PHD recursion for this so-called linear Gaussian jump Markov multi-target model. The proposed solution is general enough to accommodate a broad class of practical problems which are deemed intractable using traditional techniques. Based on this closed form solution, an efficient method is developed for tracking multiple maneuvering targets that switch between multiple models without the need for gating, track initiation and termination, or clustering for extracting state estimates. The approach to the nonlinear filtering problem explores the framework of the virtual linear fractional transformation (LFT) model which localizes the nonlinearity to the feedback with a simple and sparse structure. The LFT is an exact representation for any differentiable nonlinear mapping and therefore amenable to a general class of problems. An alternative analytical approximation method is presented which avoids linearization of the state space model. The uncorrelated structure of the feedback connection gives of the state space model. The uncorrelated structure of the feedback connection gives better second-order moment approximation of the nonlinearly mapped variables. By arranging the unscented transform in the feedback, the prediction and estimation steps are derived in closed form. The proposed filters for the discrete-time model and continuous-time dynamics with sampled-data measurements respectively are shown to be robust under highly nonlinear and uncertain conditions where standard analytical approximation based filters diverge. Moreover, the LFT based filters are efficient for online implementation. In addition, the LFT framework is applied to extend the closed form solution of the PHD recursion to the nonlinear jump Markov multi-target model.

Le problème de filtrage consiste à estimer l'état d'un système dynamique, appelé signal qui est souvent un processus markovien, à partir d'observation bruitées des états passés du système. Dans ce mémoire, nous considérons un modèle de filtrage en temps continu pour le processus de diffusion. Le but est d'étudier la stabilité du filtre optimal par rapport à sa condition initiale au-delà de l'hypothèse de mélange (fort) pour le noyau de transition en ignorant l'ergodicité du signal / The filtering problem consists of estimating the state of a dynamic, called signal which is often a Markov process, from the noisy observation of the past states. In this thesis, we consider a filtering model in continuous time for the diffusion process. The aim is to study the stability of the optimal filter with respect to its initial condition beyond the mixing (or quasi – mixing) hypothesis for the transition kernel

We are interested in a nonlinear filtering problem motivated by an information-based approach for modelling the dynamic evolution of a portfolio of credit risky securities. We solve this problem by `change of measure method\\\'' and show the existence of the density of the unnormalized conditional distribution which is a solution to the Zakai equation. Zakai equation is a linear SPDE which, in general, cannot be solved analytically. We apply Galerkin method to solve it numerically and show the convergence of Galerkin approximation in mean square. Lastly, we design an adaptive Galerkin filter with a basis of Hermite polynomials and we present numerical examples to illustrate the effectiveness of the proposed method. The work is closely related to the paper Frey and Schmidt (2010).

This dissertation consists of two main parts. One is about state estimation with two types of unconventional measurements and the other is about two types of network-induced state estimation problems. The two types of unconventional measurements considered are noise-free measurements and set measurements. State estimation with them has numerous real supports. For state estimation with noisy and noise-free measurements, two sequential forms of the batch linear minimum mean-squared error (LMMSE) estimator are obtained to reduce the computational complexity. Inspired by the estimation with quantized measurements developed by Curry [28], under a Gaussian assumption, the minimum mean-squared error (MMSE) state estimator with point measurements and set measurements of any shape is proposed by discretizing continuous set measurements. State estimation under constraints, which are special cases of the more general framework, has some interesting properties. It is found that under certain conditions, although constraints are indispensable in the evolution of the state, update by treating them as measurements is redundant in filtering. The two types of network-induced estimation problems considered are optimal state estimation in the presence of multiple packet dropouts and optimal distributed estimation fusion with transformed data. An alternative form of LMMSE estimation in the presence of multiple packet dropouts, which can overcome the shortcomings of two existing ones, is proposed first. Then under a Gaussian assumption, the MMSE estimation is also obtained based on a hard decision by comparing the measurements at two consecutive time instants. It is pointed out that if this comparison is legitimate, our simple MMSE solution largely nullifies existing work on this problem. By taking linear transformation of the raw measurements received by each sensor, two optimal distributed fusion algorithms are proposed. In terms of optimality, communication and computational requirements, three nice properties make them attractive.

Particle filtering has been a very popular method to solve nonlinear/non-Gaussian state estimation problems for more than twenty years. Particle filters (PFs) have found lots of applications in areas that include nonlinear filtering of noisy signals and data, especially in target tracking. However, implementation of high dimensional PFs in real-time for large-scale problems is a very challenging computational task. Parallel & distributed (P&D) computing is a promising way to deal with the computational challenges of PF methods. The main goal of this dissertation is to develop, implement and evaluate computationally efficient PF algorithms for target tracking, and thereby bring them closer to practical applications. To reach this goal, a number of parallel PF algorithms is designed and implemented using different parallel hardware architectures such as Computer Cluster, Graphics Processing Unit (GPU), and Field-Programmable Gate Array (FPGA). Proposed is an improved PF implementation for computer cluster - the Particle Transfer Algorithm (PTA), which takes advantage of the cluster architecture and outperforms significantly existing algorithms. Also, a novel GPU PF algorithm implementation is designed which is highly efficient for GPU architectures. The proposed algorithm implementations on different parallel computing environments are applied and tested for target tracking problems, such as space object tracking, ground multitarget tracking using image sensor, UAV-multisensor tracking. Comprehensive performance evaluation and comparison of the algorithms for both tracking and computational capabilities is performed. It is demonstrated by the obtained simulation results that the proposed implementations help greatly overcome the computational issues of particle filtering for realistic practical problems.

This dissertation develops a new concept for performance-based monitoring (PBM) of instrumented buildings subjected to earthquakes. This concept is achieved by simultaneously combining and advancing existing knowledge from structural mechanics, signal processing, and performance-based earthquake engineering paradigms. The PBM concept consists of 1) optimal sensor placement, 2) dynamic response reconstruction, 3) damage estimation, and 4) loss analysis. Within the proposed concept, the main theoretical contribution is the derivation of a nonlinear model-based observer (NMBO) for state estimation in nonlinear structural systems. The NMBO employs an efficient iterative algorithm to combine a nonlinear model and limited noise-contaminated response measurements to estimate the complete nonlinear dynamic response of the structural system of interest, in the particular case of this research, a building subject to an earthquake. The main advantage of the proposed observer over existing nonlinear recursive state estimators is that it is specifically designed to be physically realizable as a nonlinear structural model. This results in many desirable properties, such as improved stability and efficiency. Additionally, a practical methodology is presented to implement the proposed PBM concept in the case of instrumented steel, wood-frame, and reinforced concrete buildings as the three main types of structural systems used for construction in the United States. The proposed methodology is validated using three case studies of experimental and real-world large-scale instrumented buildings. The first case study is an extensively instrumented six-story wood frame building tested in a series of full-scale seismic tests in the final phase of the NEESWood project at the E-Defense facility in Japan. The second case study is a 6-story steel moment resisting frame building located in Burbank, CA, and uses the recorded acceleration data from the 1991 Sierra Madre and 1994 Northridge earthquakes. The third case is a seven-story reinforced concrete structure in Van Nuys, CA, which was severely damaged during the 1994 Northridge earthquake. The results presented in this dissertation constitute the most accurate and the highest resolution seismic response and damage measure estimates obtained for instrumented buildings. The proposed PBM concept will help structural engineers make more informed and swift decisions regarding post-earthquake assessment of critical instrumented building structures, thus improving earthquake resiliency of seismic-prone communities.

<p>Integrated tracking and detection, based on unthresholded measurements, also referred to as track before detect (TBD) is a hard nonlinear and non-Gaussian dynamical estimation and detection problem. However, it is a technique that enables the user to track and detect targets that would be extremely hard to track and detect, if possible at all with ''classical'' methods. TBD enables us to be better able to detect and track weak, stealthy or dim targets in noise and clutter and particles filter have shown to be very useful in the implementation of TBD algorithms. </p><p>This Master's thesis has investigated the use of particle filters on radar measurements, in a TBD approach.</p><p>The work has been divided into two major problems, a time efficient implementation and new functional features, as estimating the radar cross section (RCS) and the extension of the target. The later is of great importance when the resolution of the radar is such, that specific features of the target can be distinguished. Results will be illustrated by means of realistic examples.</p>

Integrated tracking and detection, based on unthresholded measurements, also referred to as track before detect (TBD) is a hard nonlinear and non-Gaussian dynamical estimation and detection problem. However, it is a technique that enables the user to track and detect targets that would be extremely hard to track and detect, if possible at all with ''classical'' methods. TBD enables us to be better able to detect and track weak, stealthy or dim targets in noise and clutter and particles filter have shown to be very useful in the implementation of TBD algorithms. This Master's thesis has investigated the use of particle filters on radar measurements, in a TBD approach. The work has been divided into two major problems, a time efficient implementation and new functional features, as estimating the radar cross section (RCS) and the extension of the target. The later is of great importance when the resolution of the radar is such, that specific features of the target can be distinguished. Results will be illustrated by means of realistic examples.

This dissertation addresses design and analysis aspects of stochastic dynamical systems using Fokker-Planck equation (FPE). A new numerical methodology based on the partition of unity meshless paradigm is developed to tackle the greatest hurdle in successful numerical solution of FPE, namely the curse of dimensionality. A local variational form of the Fokker-Planck operator is developed with provision for h- and p- refinement. The resulting high dimensional weak form integrals are evaluated using quasi Monte-Carlo techniques. Spectral analysis of the discretized Fokker- Planck operator, followed by spurious mode rejection is employed to construct a new semi-analytical algorithm to obtain near real-time approximations of transient FPE response of high dimensional nonlinear dynamical systems in terms of a reduced subset of admissible modes. Numerical evidence is provided showing that the curse of dimensionality associated with FPE is broken by the proposed technique, while providing problem size reduction of several orders of magnitude. In addition, a simple modification of norm in the variational formulation is shown to improve quality of approximation significantly while keeping the problem size fixed. Norm modification is also employed as part of a recursive methodology for tracking the optimal finite domain to solve FPE numerically. The basic tools developed to solve FPE are applied to solving problems in nonlinear stochastic optimal control and nonlinear filtering. A policy iteration algorithm for stochastic dynamical systems is implemented in which successive approximations of a forced backward Kolmogorov equation (BKE) is shown to converge to the solution of the corresponding Hamilton Jacobi Bellman (HJB) equation. Several examples, including a four-state missile autopilot design for pitch control, are considered. Application of the FPE solver to nonlinear filtering is considered with special emphasis on situations involving long durations of propagation in between measurement updates, which is implemented as a weak form of the Bayes rule. A nonlinear filter is formulated that provides complete probabilistic state information conditioned on measurements. Examples with long propagation times are considered to demonstrate benefits of using the FPE based approach to filtering.