Article
Ensemble average and ensemble variance behavior of unsteady, onedimensional groundwater flow in unconfined, heterogeneous aquifers: an exact secondorder model
Stochastic Environmental Research and Risk Assessment (Impact Factor: 1.96). 23(7):947956. DOI: 10.1007/s0047700802631

Article: A Subordinated Kinematic Wave Equation for HeavyTailed Flow Responses from Heterogeneous Hillslopes
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ABSTRACT: Simplified models of lateral storm flow through hillslopes have been derived from the Boussinesq equation, but as yet have been unable to account for the effect of spatially variable hillslope properties on the flow response. This work presents an approach based on time subordination that holds great promise in overcoming this obstacle. Time subordination alters the instantaneous unit hydrographs obtained for the homogeneous case by randomizing the timing of outflow according to a memory function that represents the effects of the heterogeneity. To explore the validity of this approach, numerical Boussinesq flow simulations are used to investigate the influence of hydraulic conductivity and porosity on lateral storm flow through idealized hillslopes. These simulations model the outflow of an instantaneous recharge impulse applied to random hydraulic conductivity fields with lognormal K and exponential correlation length scale(s). The selection of model parameters favors conditions that promote a hydraulic gradient within the simulated hillslope that is approximately equal to the slope of the underlying bedrock. This allows for the comparison of numerical solutions of outflow to a simplified linear kinematic wave equation proposed by Beven (1981,1982) that models lateral storm flow as a constant velocity piston. Preliminary results show that piston flow is not achieved in any heterogeneous cases as heterogeneity within the random hydraulic conductivity fields leads to a wide range of flow path configurations and velocities, and complex spatial and temporal distributions of soilwater storage. Hillslopes with minor heterogeneity (σlnK=1) exhibit outflow recessions with exponential tails, while hillslopes with moderate to high degrees of heterogeneity (σlnK≥5) exhibit outflow recessions with powerlaw tails. In the case of powerlaw outflow recessions, the incorporation of a stable subordinator to the linear kinematic wave equation adequately models the simulated outflow recession. The influence of σlnK, single and multiple of correlation length scales, and correlated porosity fields on outflow are examined.Journal of Geophysical Research 11/2010; 115:F00A08. · 3.17 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Forecasting of extreme events and phenomena that respond to nonGaussian heavytailed distributions (e.g., extreme environmental events, rock permeability, rock fracture intensity, earthquake magnitudes) is essential to environmental and geoscience risk analysis. In this paper, new parametric heavytailed distributions are devised starting from the exponential power probability density function (pdf) which is modified by explicitly including higherorder “cumulant parameters” into the pdf. Instead of dealing with whole power random variables, novel “residual” random variables are proposed to reconstruct the cumulant generating function. The expected value of a residual random variable with the corresponding pdf for order G, gives the input higherorder cumulant parameter. Thus, each parametric pdf is used to simulate a random variable containing residuals that yield, in average, the expected cumulant parameter. The cumulant parameters allow the formulation of heavytailed skewed pdfs beyond the lognormal to handle extreme events. Monte Carlo simulation of heavytailed distributions with higherorder parameters is demonstrated with a simple example for permeability.Stochastic Environmental Research and Risk Assessment 01/2012; 26(6). · 1.96 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Secondorder exact ensemble averaged equation for linear stochastic differential equations with multiplicative randomness and random forcing is obtained by using the cumulant expansion ensemble averaging method and by taking the time dependent sure part of the multiplicative operator into account. It is shown that the satisfaction of the commutativity and the reversibility requirements proposed earlier for linear stochastic differential equations without forcing are necessary for the linear stochastic differential equations with forcing when the cumulant expansion ensemble averaging method is used. It is shown that the applicability of the operator equality, which is used for the separation of operators in the literature, is also subjected to the satisfaction of the commutativity and the reversibility requirements. The van Kampen’s lemma, which is proposed for the analysis of nonlinear stochastic differential equations, is modified in order to make the probability density function obtained through the lemma depend on the forcing terms too. The secondorder exact ensemble averaged equation for linear stochastic differential equations with multiplicative randomness and random forcing is also obtained by using the modified van Kampen’s lemma in order to validate the correctness of the modified lemma. Secondorder exact ensemble averaged equation for one dimensional convection diffusion equation with reaction and source is obtained by using the cumulant expansion ensemble averaging method. It is shown that the van Kampen’s lemma can yield the cumulant expansion ensemble averaging result for linear stochastic differential equations when the lemma is applied to the interaction representation of the governing differential equation. It is found that the ensemble averaged equations given for one the dimensional convection diffusion equation with reaction and source in the literature obtained by applying the lemma to the original differential equation are restricted with small sure part of multiplicative operator. Secondorder exact differential equations for the evolution of the probability density function for the one dimensional convection diffusion equation with reaction and source and one dimensional nonlinear overland flow equation with source are obtained by using the modified van Kampen’s lemma. The equation for the evolution of the probability density function for one dimensional nonlinear overland flow equation with source given in the literature is found to be not secondorder exact. It is found that the differential equations for the evolution of the probability density functions for various hydrological processes given in the literature are not secondorder exact. The significance of the new terms found due to the secondorder exact ensemble averaging performed on the one dimensional convection diffusion equation with reaction and source and during the application of the van Kampen’s lemma to the one dimensional nonlinear overland flow equation with source is investigated.Stochastic Environmental Research and Risk Assessment 27(1). · 1.96 Impact Factor
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