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dc.rights.licenseAtribución-NoComercial 4.0 Internacional
dc.rights.licenseAtribución-NoComercial 4.0 Internacional
dc.rights.licenseAtribución-NoComercial 4.0 Internacional
dc.rights.licenseAtribución-NoComercial 4.0 Internacional
dc.contributor.advisorRamírez Osorio, Jorge Mario
dc.contributor.advisorPoveda Jaramillo, Germán
dc.contributor.authorVallejo Bernal, Sara María
dc.description.abstractWe derive and solve a linear stochastic model for the evolution of discharge and runoff in an order-one watershed. The system is forced by a statistically stationary compound Poisson process of instantaneous rainfall events. The relevant time scales are hourly or larger, and for large times, we show that the discharge approaches a limiting invariant distribution. Hence any of its properties are with regard to a rainfall-runoff system in hydrological equilibrium. We give an explicit formula for the Laplace transform of the invariant density of discharge in terms of the catchment area, the residence times of water in the channel and the hillslopes, and the mean frequency and the probability distribution of rainfall inputs. As a study case, we consider a watershed under a stationary rainfall regime in the tropical Andes and test the probability distribution predicted by the model against the corresponding seasonal statistics. A mathematical analysis of the invariant distribution is performed yielding formulas for the invariant moments of discharge in terms of those of the rainfall. The asymptotic behavior of probabilities of extreme events of discharge is explicitly derived for heavy-tailed and light-tailed families of distributions of rainfall inputs. The scaling structure of discharge is asymptotically characterized in terms of the parameters of the model and under the assumption of wide sense scaling for the precipitation amounts and the inverse of the residence time in the channel. The results give insights into the conversion of uncertainty inherent to the rainfall-runoff dynamics, and the roles played by different geophysical variables. The ratio between the mean frequency of rainfall events to the residence time along the hillslopes is shown to largely determine the qualitative properties of the distribution of discharge. Finally, a purely theoretical approach is proposed to reinterpret the hydrological concept of return period in the context of time-continuous Markov processes.
dc.description.abstractEn este trabajo derivamos y resolvemos un modelo estocástico lineal para la evolución del caudal y la escorrentía en una cuenca hidrográfica de orden uno. El sistema es forzado por un proceso de Poisson compuesto, estadísticamente estacionario, de eventos de lluvia instantáneos. Las escalas de tiempo relevantes son horarias o mayores, y cuando el tiempo tiende a infinito, mostramos que el caudal se acerca a una distribución invariante límite. Por tanto, cualquiera de sus propiedades está relacionada con un sistema de lluvia-escorrentía en equilibrio hidrológico. Damos una fórmula explícita para la transformada de Laplace de la densidad invariante del caudal en términos del área de la cuenca, los tiempos de residencia del agua en el canal y las laderas, y la frecuencia media y la distribución de probabilidad de los eventos de lluvia. Como caso de estudio, consideramos una cuenca bajo un régimen de lluvias estacionario en los Andes tropicales y evaluamos la distribución de probabilidad predicha por el modelo con las estadísticas estacionales correspondientes. Realizamos un análisis matemático de la distribución invariante obteniendo fórmulas para los momentos invariantes del caudal en términos de los de la precipitación. El comportamiento asintótico de las probabilidades de los eventos extremos del caudal se deriva explícitamente para familias de distribuciones de lluvia de cola pesada y cola ligera. La estructura de escalamiento del caudal se caracteriza asintóticamente en términos de los parámetros del modelo y bajo el supuesto de escalamiento simple para la precipitación y el inverso del tiempo de residencia en el canal. Los resultados dan una idea de la conversión de la incertidumbre inherente a la dinámica lluvia-escorrentía y los roles que juegan las diferentes variables geofísicas. Mostramos que la relación entre la frecuencia media de los eventos de lluvia y el tiempo de residencia en las laderas determina en gran medida las propiedades cualitativas de la distribución del caudal. Finalmente, proponemos un enfoque puramente teórico para reinterpretar el concepto hidrológico de período de retorno en el contexto de procesos de Markov continuos en el tiempo.
dc.rightsDerechos reservados - Universidad Nacional de Colombia
dc.subject.ddc510 - Matemáticas::519 - Probabilidades y matemáticas aplicadas
dc.subject.ddc550 - Ciencias de la tierra::551 - Geología, hidrología, meteorología
dc.titleA conceptual stochastic rainfall-runoff model applied to tropical watersheds
dc.rights.spaAcceso abierto
dc.description.additionalÁrea de investigación: Hidrología estocástica
dc.publisher.programMedellín - Ciencias - Maestría en Ciencias - Matemática Aplicada
dc.publisher.departmentEscuela de matemáticas
dc.publisher.branchUniversidad Nacional de Colombia - Sede Medellín
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dc.subject.proposalhidrología estocástica
dc.subject.proposalstochastic hydrology
dc.subject.proposalrainfall-runoff modeling
dc.subject.proposalmodelo de lluvia-escorrentía
dc.subject.proposalPrecipitación de Poisson
dc.subject.proposalPoisson precipitation

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