Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In Brazil over recent years the interest increased on the sunflower cultivation. Sunflower yield can be decreased by water deficit. To solve this problem, is necessary to calculate the probable water deficit in critical sunflower sub-phases and in the whole development cycle at different sowing dates. The objective of this study was to determine the probable duration values of sub-phases and the developmental cycle and get sowing dates with lower risk of water deficit and the occurrence probability in different levels of water deficit during the developmental sub-phases of sunflower crop sown at different sowing dates, considering the water storage capacity in different soils in the Central Region of Rio Grande do Sul. Also determine the probability of occurrence of water stress for different years classified as the El Niño Southern Oscilation (ENSO). Crop development was simulated using the thermal time method, for 14 sowing dates, from August until mid-February, for every year during the period from 1968 to 2011, covered by database of Meteorological Station of Santa Maria, RS. For calculating the water deficit, the 13 soils were grouped into six groups with similar water storage capacity (CAD) and infiltration capacity. The water deficit was calculated from daily water balance. Data analysis consisted of analysis of variance, means comparison tests and analysis of probability distribution for the variables: duration of crop developmental sub-phases and the whole developmental cycle of the sunflower, water deficit in the sub-phases and whole developmental cycle. The length of the sub-phases and the development cycle of the sunflower crop are variable depending on sowing date. The length of the developmental sub-phases that occur from sowing until flower bud visible of sunflower are higher in the earliest sowing date (01/08). After anthesis, the longer length of developmental sub-phases occurs in the latest sowing (16/02). The lognormal, normal and gamma distributions represent better the development of sunflower to estimate the length of the phases and the whole cycle. At sowing date of 16/12, for 90% probability level, sunflower has the shortest length of the developmental cycle ending the cycle in a maximum of 96 days. The longer length of the sunflower cycle occurs at sowing date of 01/08, which reaches 132 days, at 90% level of occurrence probability. The sowing dates from early October until early November are the ones with the highest water deficit, considering the whole development cycle of the sunflower regardless of soil, a different choice on sowing date reduces the risk and the level of water deficit in sunflower cycle. In the soils in which the water storage capacity is lower, water deficit is greater in sub-phases as in the full cycle of the sunflower compared to other soils and is little variable among the sowing dates. Sunflower Sowings in the first half of August and since December are the ones with the lowest risk occurrence of water deficit during the more critical sub-phase of sunflower crop, at least there are favorable conditions for sowing and initial establishment of plants. / No Brasil nos últimos anos elevou-se o interesse pelo cultivo do girassol. Quando submetida à deficiência hídrica a cultura do girassol apresenta redução na produtividade. Para contornar esse problema, é necessário calcular a provável deficiência hídrica nos subperíodos críticos e no ciclo de desenvolvimento do girassol para cada uma das diferentes datas de semeadura. O objetivo desse trabalho foi determinar os valores prováveis de duração dos subperíodos e do ciclo de desenvolvimento e obter as datas de semeadura com menor risco de deficiência hídrica e a probabilidade de ocorrência de diferentes níveis de déficit hídrico durante os subperíodos de desenvolvimento do girassol semeado em datas de semeadura distintas, considerando a capacidade de armazenamento de água nos diferentes solos da região central do RS. Também determinar a probabilidade de ocorrência de deficiência hídrica para os diferentes anos classificados conforme o fenômeno El Niño Oscilação Sul (ENOS). O desenvolvimento da cultura foi simulado por meio do método da soma térmica, para 14 datas de semeadura, do início do mês de agosto até meados de fevereiro, para cada ano do banco de dados da Estação Meteorológica Principal de Santa Maria, RS, utilizando o período de 1968 a 2011. Para calcular a deficiência hídrica, os 13 solos da região foram agrupados em seis grupos que apresentam características semelhantes de capacidade de armazenamento de água disponível (CAD) e capacidade de infiltração. As deficiências hídricas foram determinadas a partir do balanço hídrico diário. A análise dos dados consistiu na análise da variância, teste de comparação de médias e análise de distribuição de probabilidade para as variáveis: duração dos subperíodos e do ciclo de desenvolvimento do girassol, deficiência hídrica nos subperíodos e no ciclo do girassol. A duração dos subperíodos e do ciclo de desenvolvimento do girassol é variável conforme a data de semeadura. A duração dos subperíodos que ocorrem da semeadura até o botão floral visível do girassol são maiores na primeira data de semeadura (01/08). Após a antese a maior duração dos subperíodos ocorre na semeadura mais tardia (16/02). As distribuições lognormal, normal e gama representam melhor o desenvolvimento do girassol para estimar a duração dos subperíodos e do ciclo. Na data de semeadura de 16/12, ao nível de 90% de probabilidade de ocorrência, o girassol tem a menor duração do ciclo, completando o ciclo em no máximo de 96 dias. A maior duração do ciclo do girassol ocorre na data de semeadura de 01/08, na qual alcança 132 dias, em nível de 90% de probabilidade de ocorrência. As datas de semeadura de início de outubro até o início de novembro são as que apresentam a maior deficiência hídrica, considerando todo o ciclo de desenvolvimento do girassol independente do solo; a escolha de outra data de semeadura reduz o risco e o nível de deficiência hídrica durante o ciclo do girassol. Nos solos em que a capacidade de armazenamento de água disponível é menor, a deficiência hídrica é maior tanto nos subperíodos quanto no ciclo do girassol em relação aos demais solos e é pouco variável ao longo das datas de semeadura. Semeaduras de girassol na primeira quinzena de agosto e a partir do mês de dezembro são as que apresentam os menores riscos de ocorrer deficiência hídrica no transcorrer do subperíodo mais crítico do girassol, desde que se tenham condições favoráveis para a semeadura e o estabelecimento inicial das plantas.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The objective of this study was to determine the probable duration of the developmental phases of the cycle for the soybean at different sowing dates, the average values of water depth and number of irrigations needed and also its relationship with El Niño Oscillation South (ENSO), considering the capacity of water storage available (CAD) of the main soils of Central region of Rio Grande do Sul. This study was conducted by means of mathematical models of development of soybean and data published in the literature meteorological, climatological station collected in Santa Maria, RS (latitude: 29°43'23 "S, longitude: 53° 43'15" W and altitude: 95 m), from October 1968 to July 2012, totaling 44 years of daily observations. The simulation of crop development was carried out for different sowing dates every ten days, considering three maturity groups (GM): 5.9-6.8 6.9-7.3 and 7.8-8.0. To simulate irrigation, the 13 soils of the region covered in this study into five groups that have similar characteristics of water storage capacity available (CAD) and infiltration capacity. The daily water balance determined the variation of water availability and timing of irrigation. The timing of irrigation was determined when the soil reached a minimum fraction of available water to be maintained. For this, we used four handlings representing the condition when the water withdrawn from the soil fraction reached 20%, 30%, 40% and 50% of the CAD. Data analysis consisted of analysis of variance test for comparison of means and analysis of the probability distribution for the variables: duration of subperiods and development cycle of soybeans, water depth, number of irrigation and water depth associated to ENSO. The average duration of subperiods and soybean development cycle varies according to the date of sowing. The duration of the development cycle of the soybean crop is higher in the early sowing dates (October) decreasing until the last sowing dates (December). Early sowing dates require more water depth than the latest sowing dates. It was found that higher water depth necessary to soybean is associated with neutral years, while the lowest water depth is related to El Niño events. / O objetivo deste trabalho foi determinar os valores prováveis de duração dos subperíodos do ciclo para a cultura da soja semeada em diferentes datas, os valores Médios de lâmina de irrigação e do número de irrigações necessários e também sua relação com o fenômeno El Niño Oscilação Sul (ENOS), considerando-se a capacidade de armazenamento de água disponível (CAD) dos principais solos da região Central do Rio Grande do Sul. Este estudo foi realizado por meio de modelos matemáticos de desenvolvimento da cultura da soja publicados na literatura e dados meteorológicos, coletados na estação climatológica principal de Santa Maria, RS (latitude: 29°43 23‖ S, longitude: 53°43 15‖ W e altitude: 95 m), desde outubro de 1968 até julho de 2012, totalizando 44 anos de observações diárias. A simulação do desenvolvimento da cultura foi realizada para diferentes datas de semeadura, aproximadamente a cada dez dias, de acordo com os três grupos de maturação (GM) avaliados: 5.9 6.8 (Ciclo precoce/semiprecoce,), 6.9 7.3 (Ciclo médio,) e 7.8 8.0 (Ciclo semitardio/tardio). Para simular a irrigação, os 13 solos da região de abrangência do estudo, foram agrupados em cinco grupos que apresentam características semelhantes de capacidade de armazenamento de água disponível (CAD) e capacidade de infiltração. O balanço hídrico sequencial diário determinou a variação da água disponível e o momento da irrigação. O momento da irrigação foi determinado quando os solos alcançavam a fração mínima de água disponível a ser mantida. Para isso, foram utilizados quatro manejos que representaram a condição de quando a água retirada do solo alcançava a fração 20%, 30%, 40% e 50% da CAD. A análise dos dados consistiu de análise de variância, teste de comparação de médias e análise de distribuição de probabilidade para as variáveis: duração dos subperíodos e do ciclo de desenvolvimento da cultura da soja, lâmina de irrigação, número de irrigações a lâmina de irrigação associada o fenômeno ENOS. A duração média dos subperíodos e do ciclo de desenvolvimento da soja é variável conforme a data de semeadura. A duração do ciclo de desenvolvimento da cultura da soja é maior nas primeiras datas de semeadura (outubro) decrescendo até as últimas datas de semeadura (dezembro). As primeiras datas de semeadura necessitam de maior lâmina de irrigação do que as últimas datas de semeadura. Constatou-se que maior lâmina de irrigação necessária à cultura da soja está associada a anos neutros, enquanto que a menor lâmina de irrigação está relacionada a eventos de El Niño.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The expansion of soybean production area in Planosols is rather limited by the high frequency of occurrence of excess water, leading to reduced availability of oxygen in the root zone, reduced photosynthesis, as well as productivity, depending on its duration and developmental phase of plants it occurs. The aim of this study was to identify sowing dates with smaller risk of excess water to the subperiods and crop cycle, taking into account three relative maturity groups of soybean cultivars and water storage capacity of Planosols in the central region of Rio Grande do Sul State. The simulation of soybean development and the calculation of crop daily sequential water balance were performed at different sowing dates in each year from August 1968 to July 2012. Thus, the change of soil water storage and the water surpluses in the different soybean developmental phases were quantified for each sowing date. Data from days with excess water were submitted to analysis of variance and Scott-Knott test at 5% probability, and the sources of variation were sowing dates, soils and their interaction. These data also were submitted to the probability distribution analysis, using the chi-square and Kolmogorov-Smirnov tests to verify the probability density function that best fit the data distribution. The greatest number of fittings for the development cycle and subperiods were obtained by the Gamma and Weibull functions, respectively. October's sowings have the highest risk of excess water during the crop cycle. Subperiod sowing-emergence shows up as the most limiting to define the sowing date. Due to the lowest risk of excess water in this sub-period, the sowing carried out after November 1st are the most favorable for soybean sowing in Planosols. / A expansão da área de produção de soja em Planossolos é bastante limitada pela elevada frequência de ocorrência de excesso hídrico, ocasionando redução na disponibilidade de oxigênio na zona radicular, redução da fotossíntese, assim como da produtividade, dependendo da duração do excesso e do subperíodo de desenvolvimento das plantas em que ocorre. O objetivo deste trabalho foi identificar datas de semeadura com menor risco de ocorrência de excesso hídrico para os subperíodos e ciclo da cultura, considerando três grupos de maturidade relativa de cultivares de soja e a capacidade de armazenamento de água dos Planossolos da região central do Rio Grande do Sul. A simulação do desenvolvimento da soja e o cálculo do balanço hídrico sequencial diário da cultura foram realizados em diferentes datas de semeadura de cada ano do período de agosto de 1968 a julho de 2012. Assim, a variação do armazenamento hídrico no solo e a ocorrência de excedentes hídricos nos diferentes subperíodos de desenvolvimento da soja foram quantificadas para cada data de semeadura. Os dados de dias de excesso hídrico foram submetidos à análise de variância e teste de Scott-Knott, a 5% de probabilidade de erro, sendo que as fontes de variação constaram das datas de semeadura, os solos e a sua interação. Os dados também foram submetidos à análise de distribuição de probabilidades, utilizando-se os testes qui-quadrado e Kolmogorov-Smirnov para verificar a função densidade probabilidade que melhor se ajustou à distribuição dos dados. O maior número de ajustes para o ciclo de desenvolvimento e para os subperíodos foram obtidos para as funções gama e weibull, respectivamente. As semeaduras realizadas no mês de outubro são as de maior risco de ocorrência de excesso hídrico ao longo do ciclo da cultura. O subperíodo semeadura-emergência mostra-se como o mais limitante para a definição da data de semeadura. Devido ao menor risco de ocorrência de excesso hídrico neste subperíodo as semeaduras realizadas após o dia primeiro de novembro são as mais favoráveis para a semeadura da soja em Planossolos.

Magister Scientiae - MSc / In recent years, two estimators have been proposed to correct the bias exhibited by the leastsquares (LS) estimator of the lagged dependent variable (LDV) coefficient in dynamic regression models when the sample is finite. They have been termed as ‘mean-unbiased’ and ‘medianunbiased’ estimators. Relative to other similar procedures in the literature, the two locationbased estimators have the advantage that they offer an exact and uniform methodology for LS estimation of the LDV coefficient in a first order autoregressive model with or without exogenous regressors i.e. ARX(1). However, no attempt has been made to accurately establish and/or compare the statistical properties among these estimators, or relative to those of the LS estimator when the LDV coefficient is restricted to realistic values. Neither has there been an attempt to  compare their performance in terms of their mean squared error (MSE) when various forms of the exogenous regressors are considered. Furthermore, only implicit confidence intervals have been given for the ‘medianunbiased’ estimator. Explicit confidence bounds that are directly usable for inference are not available for either estimator. In this study a new estimator of the LDV coefficient is proposed; the ‘most-probably-unbiased’ estimator. Its performance properties vis-a-vis the existing estimators are determined and compared when the parameter space of the LDV coefficient is restricted. In addition, the following new results are established: (1) an explicit computable form for the density of the LS estimator is derived for the first time and an efficient method for its numerical evaluation is proposed; (2) the exact bias, mean, median and mode of the distribution of the LS estimator are determined in three specifications of the ARX(1) model; (3) the exact variance and MSE of LS estimator is determined; (4) the standard error associated with the determination of same quantities when simulation rather than numerical integration method is used are established and the methods are compared in terms of computational time and effort; (5) an exact method of evaluating the density of the three estimators is described; (6) their exact bias, mean, variance and MSE are determined and analysed; and finally, (7) a method of obtaining the explicit exact confidence intervals from the distribution functions of the estimators is proposed. The discussion and results show that the estimators are still biased in the usual sense: ‘in expectation’. However the bias is substantially reduced compared to that of the LS estimator. The findings are important in the specification of time-series regression models, point and interval estimation, decision theory, and simulation. / South Africa

Dans ce mémoire de thèse, nous nous intéressons à la modélisation non paramétrique de données spatiales et/ou fonctionnelles, plus particulièrement basée sur la méthode à noyau. En général, les échantillons que nous avons considérés pour établir les propriétés asymptotiques des estimateurs proposés sont constitués de variables dépendantes. La spécificité des méthodes étudiées réside dans le fait que les estimateurs prennent en compte la structure de dépendance des données considérées.Dans une première partie, nous appréhendons l’étude de variables réelles spatialement dépendantes. Nous proposons une nouvelle approche à noyau pour estimer les fonctions de densité de probabilité et de régression spatiales ainsi que le mode. La particularité de cette approche est qu’elle permet de tenir compte à la fois de la proximité entre les observations et de celle entre les sites. Nous étudions les comportements asymptotiques des estimateurs proposés ainsi que leurs applications à des données simulées et réelles.Dans une seconde partie, nous nous intéressons à la modélisation de données à valeurs dans un espace de dimension infinie ou dites "données fonctionnelles". Dans un premier temps, nous adaptons le modèle de régression non paramétrique introduit en première partie au cadre de données fonctionnelles spatialement dépendantes. Nous donnons des résultats asymptotiques ainsi que numériques. Puis, dans un second temps, nous étudions un modèle de régression de séries temporelles dont les variables explicatives sont fonctionnelles et le processus des innovations est autorégressif. Nous proposons une procédure permettant de tenir compte de l’information contenue dans le processus des erreurs. Après avoir étudié le comportement asymptotique de l’estimateur à noyau proposé, nous analysons ses performances sur des données simulées puis réelles.La troisième partie est consacrée aux applications. Tout d’abord, nous présentons des résultats de classification non supervisée de données spatiales (multivariées), simulées et réelles. La méthode de classification considérée est basée sur l’estimation du mode spatial, obtenu à partir de l’estimateur de la fonction de densité spatiale introduit dans le cadre de la première partie de cette thèse. Puis, nous appliquons cette méthode de classification basée sur le mode ainsi que d’autres méthodes de classification non supervisée de la littérature sur des données hydrologiques de nature fonctionnelle. Enfin, cette classification des données hydrologiques nous a amené à appliquer des outils de détection de rupture sur ces données fonctionnelles. / In this dissertation, we are interested in nonparametric modeling of spatial and/or functional data, more specifically based on kernel method. Generally, the samples we have considered for establishing asymptotic properties of the proposed estimators are constituted of dependent variables. The specificity of the studied methods lies in the fact that the estimators take into account the structure of the dependence of the considered data.In a first part, we study real variables spatially dependent. We propose a new kernel approach to estimating spatial probability density of the mode and regression functions. The distinctive feature of this approach is that it allows taking into account both the proximity between observations and that between sites. We study the asymptotic behaviors of the proposed estimates as well as their applications to simulated and real data. In a second part, we are interested in modeling data valued in a space of infinite dimension or so-called "functional data". As a first step, we adapt the nonparametric regression model, introduced in the first part, to spatially functional dependent data framework. We get convergence results as well as numerical results. Then, later, we study time series regression model in which explanatory variables are functional and the innovation process is autoregressive. We propose a procedure which allows us to take into account information contained in the error process. After showing asymptotic behavior of the proposed kernel estimate, we study its performance on simulated and real data.The third part is devoted to applications. First of all, we present unsupervised classificationresults of simulated and real spatial data (multivariate). The considered classification method is based on the estimation of spatial mode, obtained from the spatial density function introduced in the first part of this thesis. Then, we apply this classification method based on the mode as well as other unsupervised classification methods of the literature on hydrological data of functional nature. Lastly, this classification of hydrological data has led us to apply change point detection tools on these functional data.

В работе реализован численный алгоритм для решения уравнения Фоккера-Планка, позволяющий получать значения первой и третьей гармоники ансамбля взаимодействующих частиц для различных амплитуд переменного поля. В формулы первой и третьей гармоники вводились функции, зависящие от параметра и восприимчивости Ланжевена, с неопределенными коэффициентами. Неопределенные коэффициенты находились методом наименьших квадратов. Выражения для функций приближались данными из численного решения уравнения Фоккера-Планка и затем минимизировались относительно неопределённых коэффициентов. Получившиеся формулы сравнивались с численным решением и с известными теориями. / In this work, a numerical algorithm for solving the Fokker-Planck equation, which allows to obtain the values of the first and third harmonics of the ensemble of interacting particles for different amplitudes of the alternating field, was implemented. The functions depending on the parameter and Langevin susceptibility with uncertain coefficients were introduced into the formulas for the first and third harmonics. The uncertain coefficients were found by the least-squares method. Expressions for the functions were approximated with data from the numerical solution of the Fokker-Planck equation and then minimized with respect to the uncertain coefficients. The resulting formulas were compared with the numerical solution and with known theories.

[EN] This thesis concerns the analysis of differential equations with uncertain input parameters, in the form of random variables or stochastic processes with any type of probability distributions. In modeling, the input coefficients are set from experimental data, which often involve uncertainties from measurement errors. Moreover, the behavior of the physical phenomenon under study does not follow strict deterministic laws. It is thus more realistic to consider mathematical models with randomness in their formulation. The solution, considered in the sample-path or the mean square sense, is a smooth stochastic process, whose uncertainty has to be quantified. Uncertainty quantification is usually performed by computing the main statistics (expectation and variance) and, if possible, the probability density function. In this dissertation, we study random linear models, based on ordinary differential equations with and without delay and on partial differential equations. The linear structure of the models makes it possible to seek for certain probabilistic solutions and even approximate their probability density functions, which is a difficult goal in general. A very important part of the dissertation is devoted to random second-order linear differential equations, where the coefficients of the equation are stochastic processes and the initial conditions are random variables. The study of this class of differential equations in the random setting is mainly motivated because of their important role in Mathematical Physics. We start by solving the randomized Legendre differential equation in the mean square sense, which allows the approximation of the expectation and the variance of the stochastic solution. The methodology is extended to general random second-order linear differential equations with analytic (expressible as random power series) coefficients, by means of the so-called Fröbenius method. A comparative case study is performed with spectral methods based on polynomial chaos expansions. On the other hand, the Fröbenius method together with Monte Carlo simulation are used to approximate the probability density function of the solution. Several variance reduction methods based on quadrature rules and multilevel strategies are proposed to speed up the Monte Carlo procedure. The last part on random second-order linear differential equations is devoted to a random diffusion-reaction Poisson-type problem, where the probability density function is approximated using a finite difference numerical scheme. The thesis also studies random ordinary differential equations with discrete constant delay. We study the linear autonomous case, when the coefficient of the non-delay component and the parameter of the delay term are both random variables while the initial condition is a stochastic process. It is proved that the deterministic solution constructed with the method of steps that involves the delayed exponential function is a probabilistic solution in the Lebesgue sense. Finally, the last chapter is devoted to the linear advection partial differential equation, subject to stochastic velocity field and initial condition. We solve the equation in the mean square sense and provide new expressions for the probability density function of the solution, even in the non-Gaussian velocity case. / [ES] Esta tesis trata el análisis de ecuaciones diferenciales con parámetros de entrada aleatorios, en la forma de variables aleatorias o procesos estocásticos con cualquier tipo de distribución de probabilidad. En modelización, los coeficientes de entrada se fijan a partir de datos experimentales, los cuales suelen acarrear incertidumbre por los errores de medición. Además, el comportamiento del fenómeno físico bajo estudio no sigue patrones estrictamente deterministas. Es por tanto más realista trabajar con modelos matemáticos con aleatoriedad en su formulación. La solución, considerada en el sentido de caminos aleatorios o en el sentido de media cuadrática, es un proceso estocástico suave, cuya incertidumbre se tiene que cuantificar. La cuantificación de la incertidumbre es a menudo llevada a cabo calculando los principales estadísticos (esperanza y varianza) y, si es posible, la función de densidad de probabilidad. En este trabajo, estudiamos modelos aleatorios lineales, basados en ecuaciones diferenciales ordinarias con y sin retardo, y en ecuaciones en derivadas parciales. La estructura lineal de los modelos nos permite buscar ciertas soluciones probabilísticas e incluso aproximar su función de densidad de probabilidad, lo cual es un objetivo complicado en general. Una parte muy importante de la disertación se dedica a las ecuaciones diferenciales lineales de segundo orden aleatorias, donde los coeficientes de la ecuación son procesos estocásticos y las condiciones iniciales son variables aleatorias. El estudio de esta clase de ecuaciones diferenciales en el contexto aleatorio está motivado principalmente por su importante papel en la Física Matemática. Empezamos resolviendo la ecuación diferencial de Legendre aleatorizada en el sentido de media cuadrática, lo que permite la aproximación de la esperanza y la varianza de la solución estocástica. La metodología se extiende al caso general de ecuaciones diferenciales lineales de segundo orden aleatorias con coeficientes analíticos (expresables como series de potencias), mediante el conocido método de Fröbenius. Se lleva a cabo un estudio comparativo con métodos espectrales basados en expansiones de caos polinomial. Por otro lado, el método de Fröbenius junto con la simulación de Monte Carlo se utilizan para aproximar la función de densidad de probabilidad de la solución. Para acelerar el procedimiento de Monte Carlo, se proponen varios métodos de reducción de la varianza basados en reglas de cuadratura y estrategias multinivel. La última parte sobre ecuaciones diferenciales lineales de segundo orden aleatorias estudia un problema aleatorio de tipo Poisson de difusión-reacción, en el que la función de densidad de probabilidad es aproximada mediante un esquema numérico de diferencias finitas. En la tesis también se tratan ecuaciones diferenciales ordinarias aleatorias con retardo discreto y constante. Estudiamos el caso lineal y autónomo, cuando el coeficiente de la componente no retardada i el parámetro del término retardado son ambos variables aleatorias mientras que la condición inicial es un proceso estocástico. Se demuestra que la solución determinista construida con el método de los pasos y que involucra la función exponencial retardada es una solución probabilística en el sentido de Lebesgue. Finalmente, el último capítulo lo dedicamos a la ecuación en derivadas parciales lineal de advección, sujeta a velocidad y condición inicial estocásticas. Resolvemos la ecuación en el sentido de media cuadrática y damos nuevas expresiones para la función de densidad de probabilidad de la solución, incluso en el caso de velocidad no Gaussiana. / [CA] Aquesta tesi tracta l'anàlisi d'equacions diferencials amb paràmetres d'entrada aleatoris, en la forma de variables aleatòries o processos estocàstics amb qualsevol mena de distribució de probabilitat. En modelització, els coeficients d'entrada són fixats a partir de dades experimentals, les quals solen comportar incertesa pels errors de mesurament. A més a més, el comportament del fenomen físic sota estudi no segueix patrons estrictament deterministes. És per tant més realista treballar amb models matemàtics amb aleatorietat en la seua formulació. La solució, considerada en el sentit de camins aleatoris o en el sentit de mitjana quadràtica, és un procés estocàstic suau, la incertesa del qual s'ha de quantificar. La quantificació de la incertesa és sovint duta a terme calculant els principals estadístics (esperança i variància) i, si es pot, la funció de densitat de probabilitat. En aquest treball, estudiem models aleatoris lineals, basats en equacions diferencials ordinàries amb retard i sense, i en equacions en derivades parcials. L'estructura lineal dels models ens fa possible cercar certes solucions probabilístiques i inclús aproximar la seua funció de densitat de probabilitat, el qual és un objectiu complicat en general. Una part molt important de la dissertació es dedica a les equacions diferencials lineals de segon ordre aleatòries, on els coeficients de l'equació són processos estocàstics i les condicions inicials són variables aleatòries. L'estudi d'aquesta classe d'equacions diferencials en el context aleatori està motivat principalment pel seu important paper en Física Matemàtica. Comencem resolent l'equació diferencial de Legendre aleatoritzada en el sentit de mitjana quadràtica, el que permet l'aproximació de l'esperança i la variància de la solució estocàstica. La metodologia s'estén al cas general d'equacions diferencials lineals de segon ordre aleatòries amb coeficients analítics (expressables com a sèries de potències), per mitjà del conegut mètode de Fröbenius. Es duu a terme un estudi comparatiu amb mètodes espectrals basats en expansions de caos polinomial. Per altra banda, el mètode de Fröbenius juntament amb la simulació de Monte Carlo són emprats per a aproximar la funció de densitat de probabilitat de la solució. Per a accelerar el procediment de Monte Carlo, es proposen diversos mètodes de reducció de la variància basats en regles de quadratura i estratègies multinivell. L'última part sobre equacions diferencials lineals de segon ordre aleatòries estudia un problema aleatori de tipus Poisson de difusió-reacció, en què la funció de densitat de probabilitat és aproximada mitjançant un esquema numèric de diferències finites. En la tesi també es tracten equacions diferencials ordinàries aleatòries amb retard discret i constant. Estudiem el cas lineal i autònom, quan el coeficient del component no retardat i el paràmetre del terme retardat són ambdós variables aleatòries mentre que la condició inicial és un procés estocàstic. Es prova que la solució determinista construïda amb el mètode dels passos i que involucra la funció exponencial retardada és una solució probabilística en el sentit de Lebesgue. Finalment, el darrer capítol el dediquem a l'equació en derivades parcials lineal d'advecció, subjecta a velocitat i condició inicial estocàstiques. Resolem l'equació en el sentit de mitjana quadràtica i donem noves expressions per a la funció de densitat de probabilitat de la solució, inclús en el cas de velocitat no Gaussiana. / This work has been supported by the Spanish Ministerio de Economía y Competitividad grant MTM2017–89664–P. I acknowledge the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de València. / Jornet Sanz, M. (2020). Mean square solutions of random linear models and computation of their probability density function [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/138394

Cette thèse a pour objet la simulation d'une zone de mélange turbulente issue de l'instabilité de Richtmyer-Meshkov à l'aide d'un modèle à fonction de densité de probabilité (PDF). Nous analysons plus particulièrement la prise en charge par le modèle PDF du transport de l'énergie cinétique turbulente dans la zone de mélange.Dans cette optique, nous commençons par mettre en avant le lien existant entre les statistiques en un point de l'écoulement et ses conditions initiales aux grandes échelles. Ce lien s'exprime à travers le principe de permanence des grandes échelles, et permet d'établir des prédictions pour certaines grandeurs de la zone de mélange, telles que son taux de croissance ou son anisotropie.Nous dérivons ensuite un modèle PDF de Langevin capable de restituer cette dépendance aux conditions initiales. Ce modèle est ensuite validé en le comparant à des résultats issus de simulations aux grandes échelles (LES).Enfin, une analyse asymptotique du modèle proposé permet d'éclairer notre compréhension du transport turbulent. Un régime de diffusion est mis en évidence, et l'expression du coefficient de diffusion associé à ce régime atteste l'influence de la permanence des grandes échelles sur le transport turbulent.Tout au long de cette thèse, nous nous sommes appuyés sur des résultats issus de simulations de Monte Carlo du modèle de Langevin. A cet effet, nous avons développé une méthode spécifique eulérienne et à l'avons comparé à des alternatives lagrangiennes. / The aim of the thesis is to simulate a turbulent mixing zone resulting from the Richtmyer-Meshkov instability using a probability density function (PDF) model. An emphasis is put on the analysis of the turbulent kinetic energy transport.To this end, we first highlight the link existing between the one-point statistics of the flow and its initial conditions at large scales. This link is expressed through the principle of permanence of large eddies, and allows to establish predictions for quantities of the mixing zone, such as its growth rate or its anisotropy.We then derive a Langevin PDF model which is able to reproduce this dependency of the statistics on the initial conditions. This model is then validated by comparing it against large eddy simulations (LES).Finally, an asymptotic analysis of the derived model helps to improve our understanding of the turbulent transport. A diffusion regime is identified, and the expression of the diffusion coefficient associated with this regime confirms the influence of the permanence of large eddies on the turbulent transport.Throughout this thesis, our numerical results were based on Monte Carlo simulations for the Langevin model. In this regard, we proceeded to the development of a specific Eulerian method and its comparison with Lagrangian counterparts.

Le principale objectif de l'imagerie sismique pétrolière telle qu'elle est réalisée de nos jours est de fournir une image représentative des quelques premiers kilomètres du sous-sol. Cette image permettra la localisation des structures géologiques formant les réservoirs où sont piégées les ressources en hydrocarbures. Pour pouvoir caractériser ces réservoirs et permettre la production des hydrocarbures, le géophysicien utilise la migration-profondeur qui est un outil d'imagerie sismique qui sert à convertir des données-temps enregistrées lors des campagnes d'acquisition sismique en des images-profondeur qui seront exploitées par l'ingénieur-réservoir avec l'aide de l'interprète sismique et du géologue. Lors de la migration profondeur, les évènements sismiques (réflecteurs,…) sont replacés à leurs positions spatiales correctes. Une migration-profondeur pertinente requiert une évaluation précise modèle de vitesse. La précision du modèle de vitesse utilisé pour une migration est jugée au travers l'alignement horizontal des évènements présents sur les Common Image Gather (CIG). Les évènements non horizontaux (Residual Move Out) présents sur les CIG sont dus au ratio du modèle de vitesse de migration par la vitesse effective du milieu. L'analyse du Residual Move Out (RMO) a pour but d'évaluer ce ratio pour juger de la pertinence du modèle de vitesse et permettre sa mise à jour. Les CIG qui servent de données pour l'analyse du RMO sont solutions de problèmes inverses mal posés, et sont corrompues par du bruit. Une analyse de l'incertitude s'avère nécessaire pour améliorer l'évaluation des résultats obtenus. Le manque d'outils d'analyse de l'incertitude dans l'analyse du RMO en fait sa faiblesse. L'analyse et la quantification de l'incertitude pourrait aider à la prise de décisions qui auront des impacts socio-économiques importantes. Ce travail de thèse a pour but de contribuer à l'analyse et à la quantification de l'incertitude dans l'analyse des paramètres calculés pendant le traitement des données sismiques et particulièrement dans l'analyse du RMO. Pour atteindre ces objectifs plusieurs étapes ont été nécessaires. Elles sont entre autres :- L’appropriation des différents concepts géophysiques nécessaires à la compréhension du problème (organisation des données de sismique réflexion, outils mathématiques et méthodologiques utilisés);- Présentations des méthodes et outils pour l'analyse classique du RMO;- Interprétation statistique de l’analyse classique;- Proposition d’une approche stochastique;Cette approche stochastique consiste en un modèle statistique hiérarchique dont les paramètres sont :- la variance traduisant le niveau de bruit dans les données estimée par une méthode basée sur les ondelettes, - une fonction qui traduit la cohérence des amplitudes le long des évènements estimée par des méthodes de lissages de données,- le ratio qui est considéré comme une variable aléatoire et non comme un paramètre fixe inconnue comme c'est le cas dans l'approche classique de l'analyse du RMO. Il est estimé par des méthodes de simulations de Monte Carlo par Chaîne de Markov.L'approche proposée dans cette thèse permet d'obtenir autant de cartes de valeurs du paramètre qu'on le désire par le biais des quantiles. La méthodologie proposée est validée par l'application à des données synthétiques et à des données réelles. Une étude de sensibilité de l'estimation du paramètre a été réalisée. L'utilisation de l'incertitude de ce paramètre pour quantifier l'incertitude des positions spatiales des réflecteurs est présentée dans ce travail de thèse. / The main goal of the seismic imaging for oil exploration and production as it is done nowadays is to provide an image of the first kilometers of the subsurface to allow the localization and an accurate estimation of hydrocarbon resources. The reservoirs where these hydrocarbons are trapped are structures which have a more or less complex geology. To characterize these reservoirs and allow the production of hydrocarbons, the geophysicist uses the depth migration which is a seismic imaging tool which serves to convert time data recorded during seismic surveys into depth images which will be exploited by the reservoir engineer with the help of the seismic interpreter and the geologist. During the depth migration, seismic events (reflectors, diffractions, faults …) are moved to their correct locations in space. Relevant depth migration requires an accurate knowledge of vertical and horizontal seismic velocity variations (velocity model). Usually the so-called Common-Image-Gathers (CIGs) serve as a tool to verify correctness of the velocity model. Often the CIGs are computed in the surface offset (distance between shot point and receiver) domain and their flatness serve as criteria of the velocity model correctness. Residual moveout (RMO) of the events on CIGs due to the ratio of migration velocity model and effective velocity model indicates incorrectness of the velocity model and is used for the velocity model updating. The post-stacked images forming the CIGs which are used as data for the RMO analysis are the results of an inverse problem and are corrupt by noises. An uncertainty analysis is necessary to improve evaluation of the results. Dealing with the uncertainty is a major issue, which supposes to help in decisions that have important social and commercial implications. The goal of this thesis is to contribute to the uncertainty analysis and its quantification in the analysis of various parameters computed during the seismic processing and particularly in RMO analysis. To reach these goals several stages were necessary. We began by appropriating the various geophysical concepts necessary for the understanding of:- the organization of the seismic data ;- the various processing ;- the various mathematical and methodological tools which are used (chapters 2 and 3). In the chapter 4, we present different tools used for the conventional RMO analysis. In the fifth one, we give a statistical interpretation of the conventional RMO analysis and we propose a stochastic approach of this analysis. This approach consists in hierarchical statistical model where the parameters are: - the variance which express the noise level in the data ;- a functional parameter which express coherency of the amplitudes along events ; - the ratio which is assume to be a random variable and not an unknown fixed parameter as it is the case in conventional approach. The adjustment of data to the model done by using smoothing methods of data, combined with the using of the wavelets for the estimation of allow to compute the posterior distribution of given the data by the empirical Bayes methods. An estimation of the parameter is obtained by using Markov Chain Monte Carlo simulations of its posterior distribution. The various quantiles of these simulations provide different estimations of . The proposed methodology is validated in the sixth chapter by its application on synthetic data and real data. A sensitivity analysis of the estimation of the parameter was done. The using of the uncertainty of this parameter to quantify the uncertainty of the spatial positions of reflectors is presented in this thesis.

La capa límit atmosfèrica és l'àrea directament influenciada per la presència de la superfície de la terra i la seva alçada és d'uns centenars de metres a uns pocs quilòmetres. Durant el vespre, el refredament radiatiu estratifica establement l'aire prop del sòl i es forma el que es coneix com a Capa Límit Estable (CLE). D'avui en dia, la CLE és un règim que encara no està prou ben caracteritzat. La turbulència, que no és homogènia ni isòtropa, i la gran importància dels efectes locals com l'orografia, entre d'altres factors, dificulten l'estudi d'aquest règim. Per aquest motiu, la CLE és objecte d'especial atenció, sobretot a l'hora de millorar la seva representació en models tant de temps com de clima.Aquest treball es centra en l'estudi de la CLE mitjançant 3 eines diferents: 1) simulacions explícites de grans remolins (més conegudes com a simulacions LES), per determinar el comportament dels moviments turbulents, on les resolucions són de l'ordre de metres; 2) simulacions mesoscalars, per caracteritzar els efectes locals, on les resolucions són de l'ordre de kilòmetres; 3) anàlisi de les observacions sota aquestes condicions per tal de caracteritzar i entendre millor els fenòmens observats.En primer lloc s'estudia el rang d'estabilitats a on el model LES, que considera la teoria de Kolmogorov per la dissipació de l'energia, funciona correctament. Els resultats del model són realistes tal com mostra la seva comparació amb les mesures de dues campanyes experimentals (SABLES-98 i CASES-99). Per explorar més a fons els resultats LES i per comparar-los amb les mesures s'han utilitzat les Funcions de Distribució de Probabilitat (PDF). Aquests resultats LES són també comparables als obtinguts amb altres models LES, tal com mostra la intercomparació de models LES, més coneguda com a GABLS.Un cop desenvolupades totes les eines necessàries es fa un LES d'un cas més realista, basat en les observacions d'un màxim de vent de capes baixes (més conegut com a Low-Level Jet, LLJ). L'anàlisi combinat dels resultats LES i les mesures permet entendre millor els processos de barreja que tenen lloc a través de la inversió. Finalment, la contribució dels efectes locals s'estudia mitjançant les simulacions mesoscalars, en aquest cas centrades a l'illa de Mallorca. Durant el vespre es veu com les circulacions locals es desenvolupen a les conques (de longitud al voltant de 25km), formant-se, per exemple, vents catabàtics o LLJ com l'estudiat anteriorment. En aquest cas les simulacions es verifiquen amb imatges de satèl·lit NOAA i observacions de les estacions automàtiques de mesures, donant resultats semblants. / The atmospheric boundary layer is the area directly influenced by the presence of the Earth's surface and its height is from hundreds of meters to few kilometres. During the night, the radiative cooling stratifies the layer close to the surface and it forms the Stably-stratified Atmospheric Boundary Layer (SBL). Nowadays, the SBL is a regime not well enough characterized, yet. Turbulence, which is not homogeneous either isotropic, and the great importance of the local effects, like the orography, among other factors, make the SBL be a difficult regime to study. Even so, the SBL is an object of special attention, especially when improving its representation in numerical prediction models or climate models.This work focuses on the study of the SBL through 3 different tools: 1) Large-Eddy Simulations (LES), to determine the turbulent motions, where the resolutions are about 1m; 2) Mesoscale simulations, to characterize the local effects, where resolutions are about 1km; 3) Analysis of the observations under these conditions in order to better characterize and understand the observed phenomena.In first place, it is studied the range of stabilities where the LES model, that considers the Kolmogorov theory for the dissipation of the energy, works correctly. The results are realistic as the comparison with measures from two experimental campaigns (SABLES-98 and CASES-99) shows. To explore the results more thoroughly, and to compare the LES results to the measurements, the Probability Density Functions (PDF) have been used. The LES results are also comparable to the ones obtained with other LES models, as the intercomparison of different LES models show, better known as GABLS.Then, a more realistic case is performed using the LES model, based on observations of a Low-Level Jet (LLJ). The combined inspection of the LES results and the observations allow to better understand the mixing processes that take place through the inversion layer. Finally, the contribution of the local effects is studied through a mesoscale simulation. Here the attention is focused on the Mallorca Island. During the night, the model is able to reproduce the local circulations is a basin of a characteristic size of 25km. The main features obtained previously from the LES of the LLJ are also reproduced by the mesoscale model. These runs are verified with NOAA satellite images and observations from the automatic surface weather stations, giving that the model is able to reproduce realistic results.