Carbonate reservoirs are usually diffcult to map and identify in seismic sections due to their complex structure, lithology and diagenetic frabrics. The Midland Basin, located in the Permian Basin of West Texas, is an excellent example of these complex carbonate structures. In order to obtain a better characterization and imaging of the carbonate buildups, an acoustic impedance inversion is proposed here. The resolution of the acoustic impedance is the same as the input seismic data, which is greatly improved with the addition of the low frequency content extracted from well data. From the broadband volume, high resolution maps of acoustic impedance distributions were obtained, and therefore the locations of carbonate buildups were easily determined. A correlation between acoustic impedance and porosity extracted from well data shows that areas with high acoustic impedance were correlated with low porosity values, whereas high porosities were located in areas of low acoustic impedance. Theoretical analyses were performed using the time-average equation and the Gassmann equation. These theoretical models helped to understand how porosity distributions affect acoustic impedance. Both equations predicted a decrease in acoustic impedance as porosity increases. Inversion results showed that average porosity values are 5% [plus or minus] 5%, typical for densely cemented rocks. Previous studies done in the study area indicate that grains are moderately to well-sorted. This suggests that time-average approximation will overestimate porosity values and the Gassmann approach better predicts the measured data. A comparison between measured data and the Gassmann equation suggests that rocks with low porosities (less than 5%) tend to have high acoustic impedance values. On the other hand, rocks with higher porosities (5% to 10%) have lower acoustic impedance values. The inversion performed on well data also shows that the fluid bulk modulus for currently producing wells is lower than in non-productive wells, (wells with low production rates for brine and hydrocarbons), which is consistent with pore fluids containing a larger concentration of oil. The acoustic impedance inversion was demonstrated to be a robust technique for mapping complex structures and estimating porosities as well. However, it is not capable of differentiating different types of carbonate buildups and their origin.

Carbonate reservoirs are usually diffcult to map and identify in seismic sections due to their complex structure, lithology and diagenetic frabrics. The Midland Basin, located in the Permian Basin of West Texas, is an excellent example of these complex carbonate structures. In order to obtain a better characterization and imaging of the carbonate buildups, an acoustic impedance inversion is proposed here. The resolution of the acoustic impedance is the same as the input seismic data, which is greatly improved with the addition of the low frequency content extracted from well data. From the broadband volume, high resolution maps of acoustic impedance distributions were obtained, and therefore the locations of carbonate buildups were easily determined. A correlation between acoustic impedance and porosity extracted from well data shows that areas with high acoustic impedance were correlated with low porosity values, whereas high porosities were located in areas of low acoustic impedance. Theoretical analyses were performed using the time-average equation and the Gassmann equation. These theoretical models helped to understand how porosity distributions affect acoustic impedance. Both equations predicted a decrease in acoustic impedance as porosity increases. Inversion results showed that average porosity values are 5% [plus or minus] 5%, typical for densely cemented rocks. Previous studies done in the study area indicate that grains are moderately to well-sorted. This suggests that time-average approximation will overestimate porosity values and the Gassmann approach better predicts the measured data. A comparison between measured data and the Gassmann equation suggests that rocks with low porosities (less than 5%) tend to have high acoustic impedance values. On the other hand, rocks with higher porosities (5% to 10%) have lower acoustic impedance values. The inversion performed on well data also shows that the fluid bulk modulus for currently producing wells is lower than in non-productive wells, (wells with low production rates for brine and hydrocarbons), which is consistent with pore fluids containing a larger concentration of oil. The acoustic impedance inversion was demonstrated to be a robust technique for mapping complex structures and estimating porosities as well. However, it is not capable of differentiating different types of carbonate buildups and their origin.

In simulation, time averages are important for estimating equilibrium parameters. In particular, we would like to have the variance, bias and mean-square error for time averages. First, we will discuss various factors and their effect on the bias, the variance and the mean-square error. We will use the Markovian Event System to model various systems, including M/M/1 queues, M/E_k/1 queues, M/M/c queues, sequential queues, inventory systems and queueing networks. We use a numerical method for the computation of the variance, the bias and the mean-square error of the time average. The effectiveness of the method is tested by experimenting with models of various stochastic systems. The contribution of this thesis is to use numerical and graphical interpretations to study the general characteristics of the measures. The important characteristics included in our study are decomposability and periodicity.

In simulation, time averages are important for estimating equilibrium parameters. In particular, we would like to have the variance, bias and mean-square error for time averages. First, we will discuss various factors and their effect on the bias, the variance and the mean-square error. We will use the Markovian Event System to model various systems, including M/M/1 queues, M/E_k/1 queues, M/M/c queues, sequential queues, inventory systems and queueing networks. We use a numerical method for the computation of the variance, the bias and the mean-square error of the time average. The effectiveness of the method is tested by experimenting with models of various stochastic systems. The contribution of this thesis is to use numerical and graphical interpretations to study the general characteristics of the measures. The important characteristics included in our study are decomposability and periodicity.

No description available.

Le consensus des systèmes multi-agents a eu une attention considérable au cours de la dernière décennie. Le consensus est un processus coopératif dans lequel les agents interagissent afin de parvenir à un accord. La plupart des études se sont engagés à l'analyse de l'état d'équilibre du comportement de ce processus. Toutefois, au cours de la transitoire de ce processus une énorme quantité de données est produite. Dans cette thèse, notre objectif est d'exploiter les données produites au cours de la transitoire d'algorithmes de consensus moyenne asymptotique afin de concevoir des protocoles de consensus moyenne en temps fini, évaluer la robustesse du graphique, et éventuellement récupérer la topologie du graphe de manière distribuée. Le consensus de moyenne en temps fini garantit un temps d'exécution minimal qui peut assurer l'efficacité et la précision des algorithmes distribués complexes dans lesquels il est impliqué. Nous nous concentrons d'abord sur l'étape de configuration consacrée à la conception de protocoles de consensus qui garantissent la convergence de la moyenne exacte dans un nombre donné d'étapes. En considérant des réseaux d'agents modélisés avec des graphes non orientés connectés, nous formulons le problème de la factorisation de la matrice de moyenne et étudions des solutions distribuées à ce problème. Puisque, les appareils communicants doivent apprendre leur environnement avant d'établir des liens de communication, nous suggérons l'utilisation de séquences d'apprentissage afin de résoudre le problème de la factorisation. Ensuite, un algorithme semblable à l'algorithme de rétro-propagation du gradient est proposé pour résoudre un problème d'optimisation non convexe sous contrainte. Nous montrons que tout minimum local de la fonction de coût donne une factorisation exacte de la matrice de moyenne. En contraignant les matrices de facteur à être comme les matrices de consensus basées sur la matrice laplacienne, il est maintenant bien connu que la factorisation de la matrice de moyenne est entièrement caractérisé par les valeurs propres non nulles du laplacien. Par conséquent, la résolution de la factorisation de la matrice de la moyenne de manière distribuée avec une telle contrainte sur la matrice laplacienne, permet d'estimer le spectre de la matrice laplacienne. Depuis le spectre peut être utilisé pour calculer des indices de la robustesse (Nombre d'arbres couvrant et la résistance effective du graphe), la deuxième partie de cette thèse est consacrée à l'évaluation de la robustesse du réseau à travers l'estimation distribuée du spectre du Laplacien. Le problème est posé comme un problème de consensus sous contrainte formulé de deux façons différentes. La première formulation (approche directe) cède à un problème d'optimisation non-convexe résolu de manière distribuée au moyen de la méthode des multiplicateurs de Lagrange. La seconde formulation (approche indirecte) est obtenue après une reparamétrisation adéquate. Le problème est alors convexe et résolu en utilisant l'algorithme du sous-gradient distribué et la méthode de direction alternée de multiplicateurs. En outre, trois cas sont considérés: la valeur moyenne finale est parfaitement connue, bruyant, ou complètement inconnue. Nous fournissons également une façon pour calculer les multiplicités des valeurs propres estimées au moyen d'une programmation linéaire en nombres entiers. L'efficacité des solutions proposées est évaluée au moyen de simulations. Cependant, dans plusieurs cas, la convergence des algorithmes proposés est lente et doit être améliorée dans les travaux futurs. En outre, l'approche indirecte n'est pas évolutive pour des graphes de taille importante car elle implique le calcul des racines d'un polynôme de degré égal à la taille du réseau. Cependant, au lieu d'estimer tout le spectre, il peut être possible de récupérer seulement un petit nombre des valeurs propres, puis déduire des limites significatives sur les indices de la robustesse. / Consensus of Multi-agent systems has received tremendous attention during the last decade. Consensus is a cooperative process in which agents interact in order to reach an agreement. Most of studies are committed to analysis of the steady-state behavior of this process. However, during the transient of this process a huge amount of data is produced. In this thesis, our aim is to exploit data produced during the transient of asymptotic average consensus algorithms in order to design finite-time average consensus protocols, assess the robustness of the graph, and eventually recover the topology of the graph in a distributed way. Finite-time Average Consensus guarantees a minimal execution time that can ensure the efficiency and the accuracy of sophisticated distributed algorithms in which it is involved. We first focus on the configuration step devoted to the design of consensus protocols that guarantee convergence to the exact average in a given number of steps. By considering networks of agents modelled with connected undirected graphs, we formulate the problem as the factorization of the averaging matrix and investigate distributed solutions to this problem. Since, communicating devices have to learn their environment before establishing communication links, we suggest the usage of learning sequences in order to solve the factorization problem. Then a gradient backpropagation-like algorithm is proposed to solve a non-convex constrained optimization problem. We show that any local minimum of the cost function provides an accurate factorization of the averaging matrix. By constraining the factor matrices to be as Laplacian-based consensus matrices, it is now well known that the factorization of the averaging matrix is fully characterized by the nonzero Laplacian eigenvalues. Therefore, solving the factorization of the averaging matrix in a distributed way with such Laplacian matrix constraint allows estimating the spectrum of the Laplacian matrix. Since that spectrum can be used to compute some robustness indices (Number of spanning trees and Effective graph Resistance also known as Kirchoff index), the second part of this dissertation is dedicated to Network Robustness Assessment through distributed estimation of the Laplacian spectrum. The problem is posed as a constrained consensus problem formulated in two ways. The first formulation (direct approach) yields a non-convex optimization problem solved in a distributed way by means of the method of Lagrange multipliers. The second formulation (indirect approach) is obtained after an adequate re-parameterization. The problem is then convex and solved by using the distributed subgradient algorithm and the alternating direction method of multipliers. Furthermore, three cases are considered: the final average value is perfectly known, noisy, or completely unknown. We also provide a way for computing the multiplicities of the estimated eigenvalues by means of an Integer programming. In this spectral approach, given the Laplacian spectrum, the network topology can be reconstructed through estimation of Laplacian eigenvector. The efficiency of the proposed solutions is evaluated by means of simulations. However, in several cases, convergence of the proposed algorithms is slow and needs to be improved in future works. In addition, the indirect approach is not scalable to very large graphs since it involves the computation of roots of a polynomial with degree equal to the size of the network. However, instead of estimating all the spectrum, it can be possible to recover only a few number of eigenvalues and then deduce some significant bounds on robustness indices.

We pursue topics in optics that follow three major themes; time averaged dynamics with the associated Effective Hamiltonian theory, quantification and transformation of polarization, and periodicity within quantum circuits. Within the first theme, we develop a technique for finding the dynamical evolution in time of a time averaged density matrix. The result is an equation of evolution that includes an Effective Hamiltonian, as well as decoherence terms that sometimes manifest in a Lindblad-like form. We also apply the theory to examples of the AC Stark Shift and Three-Level Raman Transitions. In the theme of polarization, the most general physical transformation on the polarization state has been represented as an ensemble of Jones matrix transformations, equivalent to a completely positive map on the polarization matrix. This has been directly assumed without proof by most authors. We follow a novel approach to derive this expression from simple physical principles, basic coherence optics and the matrix theory of positive maps. Addressing polarization measurement, we first establish the equivalence of classical polarization and quantum purity, which leads to the identical structure of the Poincar\' and Bloch spheres. We analyze and compare various measures of polarization / purity for general dimensionality proposed in the literature, with a focus on the three dimensional case. % entanglement? In pursuit of the final theme of periodic quantum circuits, we introduce a procedure that synthesizes the circuit for the simplest periodic function that is one-to-one within a single period, of a given period p. Applying this procedure, we synthesize these circuits for p up to five bits. We conjecture that such a circuit will need at most n Toffoli gates, where p is an n-bit number. Moreover, we apply our circuit synthesis to compiled versions of Shor's algorithm, showing that it can create more efficient circuits than ones previously proposed. We provide some new compiled circuits for experimentalists to use in the near future. A layer of "classical compilation" is pointed out as a method to further simplify circuits. Periodic and compiled circuits should be helpful for creating experimental milestones, and for the purposes of validation.

We pursue topics in optics that follow three major themes; time averaged dynamics with the associated Effective Hamiltonian theory, quantification and transformation of polarization, and periodicity within quantum circuits. Within the first theme, we develop a technique for finding the dynamical evolution in time of a time averaged density matrix. The result is an equation of evolution that includes an Effective Hamiltonian, as well as decoherence terms that sometimes manifest in a Lindblad-like form. We also apply the theory to examples of the AC Stark Shift and Three-Level Raman Transitions. In the theme of polarization, the most general physical transformation on the polarization state has been represented as an ensemble of Jones matrix transformations, equivalent to a completely positive map on the polarization matrix. This has been directly assumed without proof by most authors. We follow a novel approach to derive this expression from simple physical principles, basic coherence optics and the matrix theory of positive maps. Addressing polarization measurement, we first establish the equivalence of classical polarization and quantum purity, which leads to the identical structure of the Poincar\' and Bloch spheres. We analyze and compare various measures of polarization / purity for general dimensionality proposed in the literature, with a focus on the three dimensional case. % entanglement? In pursuit of the final theme of periodic quantum circuits, we introduce a procedure that synthesizes the circuit for the simplest periodic function that is one-to-one within a single period, of a given period p. Applying this procedure, we synthesize these circuits for p up to five bits. We conjecture that such a circuit will need at most n Toffoli gates, where p is an n-bit number. Moreover, we apply our circuit synthesis to compiled versions of Shor's algorithm, showing that it can create more efficient circuits than ones previously proposed. We provide some new compiled circuits for experimentalists to use in the near future. A layer of "classical compilation" is pointed out as a method to further simplify circuits. Periodic and compiled circuits should be helpful for creating experimental milestones, and for the purposes of validation.

In a large number of applications arising in various fields of study, time series are approximated using linear MMSE estimates. Such approximations include finite order moving average and autoregressive approximations as well as the causal Wiener filter. In this dissertation, we study two topics related to the estimation of wide sense stationary (WSS) time series using linear MMSE estimates. In the first part of this dissertation, we study the asymptotic behaviour of autoregressive (AR) and moving average (MA) approximations. Our objective is to investigate how faithfully such approximations replicate the original sequence, as the model order as well as the number of samples approach infinity. We consider two aspects: convergence of spectral density of MA and AR approximations when the covariances are known and when they are estimated. Under certain mild conditions on the spectral density and the covariance sequence, it is shown that the spectral densities of both approximations converge in L2 as the order of approximation increases. It is also shown that the spectral density of AR approximations converges at the origin under the same conditions. Under additional regularity assumptions, we show that similar results hold for approximations from empirical covariance estimates. In the second part of this dissertation, we address the problem of detecting interdependence relations within a group of time series. Ideally, in order to infer the complete interdependence structure of a complex system, dynamic behaviour of all the processes involved should be considered simultaneously. However, for large systems, use of such a method may be infeasible and computationally intensive, and pairwise estimation techniques may be used to obtain sub-optimal results. Here, we investigate the problem of determining Granger-causality in an interdependent group of jointly WSS time series by using pairwise causal Wiener filters. Analytical results are presented, along with simulations that compare the performance of a method based on finite impulse response Wiener filters to another using directed information, a tool widely used in literature. The problem is studied in the context of cyclostationary processes as well. Finally, a new technique is proposed that allows the determination of causal connections under certain sparsity conditions.