A stochastic model of water-table elevation

A second order stochastic differential equation is used for modeling of water-table elevation. The data were sampled at the Borden Aquifer as a part of a tracer experiment. The purpose of the water-table data collection was to determine presence of a water flow. We argue that the water-table surface is a simple plane oscillating up and down in time according to an equation for a stochastic oscillator. We derive the model, estimate its parameters and provide arguments for goodness-of-fit of the model.     

Higher-order approximation techniques for estimating stochastic parameter of a sediment transport model

Higher-order approximation techniques for estimating stochastic parameter of the non-homogeneous Poisson (NHP) model are presented. The NHP model is characterized by a two-parameter cumulative probability distribution function (CDF) of sediment displacement. Those two parameters are the temporal and spatial intensity functions, physically representing the inverse of the average rest period and step length of sediment particles, respectively. Difficulty of estimating the parameters has, however, restricted the applications of the NHP model. The approximation techniques are proposed to address such problem. The basic idea of the method is to approximate a model involving stochastic parameters by Taylor series expansion. The expansion preserves certain higher-order terms of interest. Using the experimental (laboratory or field) data, one can determine the model parameters through a system of equations that are simplified by the approximation technique. The parameters so determined are used to predict the cumulative distribution of sediment displacement. The second-order approximation leads to a significant reduction of the CDF error (of the order of 47%) compared to the first-order approximation. Error analysis is performed to evaluate the accuracy of the first- and second-order approximations with respect to the experimental data. The higher-order approximations provide better estimations of the sediment transport and deposition that are critical factors for such environment as spawning gravel-bed.     

Higher-order approximation techniques for estimating stochastic parameter of a sediment transport model

Higher-order approximation techniques for estimating stochastic parameter of the non-homogeneous Poisson (NHP) model are presented. The NHP model is characterized by a two-parameter cumulative probability distribution function (CDF) of sediment displacement. Those two parameters are the temporal and spatial intensity functions, physically representing the inverse of the average rest period and step length of sediment particles, respectively. Difficulty of estimating the parameters has, however, restricted the applications of the NHP model. The approximation techniques are proposed to address such problem. The basic idea of the method is to approximate a model involving stochastic parameters by Taylor series expansion. The expansion preserves certain higher-order terms of interest. Using the experimental (laboratory or field) data, one can determine the model parameters through a system of equations that are simplified by the approximation technique. The parameters so determined are used to predict the cumulative distribution of sediment displacement. The second-order approximation leads to a significant reduction of the CDF error (of the order of 47%) compared to the first-order approximation. Error analysis is performed to evaluate the accuracy of the first- and second-order approximations with respect to the experimental data. The higher-order approximations provide better estimations of the sediment transport and deposition that are critical factors for such environment as spawning gravel-bed.     

A survey of numerical methods for stochastic differential equations

The development of numerical methods for stochastic differential equations has intensified over the past decade. The earliest methods were usually heuristic adaptations of deterministic methods, but were found to have limited accuracy regardless of the order of the original scheme. A stochastic counterpart of the Taylor formula now provides a framework for the systematic investigation of numerical methods for stochastic differential equations. It suggests numerical schemes, which involve multiple stochastic integrals, of higher order of convergence. We shall survey the literature on these and on the earlier schemes in this paper. Our discussion will focus on diffusion processes, but we shall also indicate the extensions needed to handle processes with jump components. In particular, we shall classify the schemes according to strong or weak convergence criteria, depending on whether the approximation of the sample paths or of the probability distribution is of main interest.     

A survey of numerical methods for stochastic differential equations

The development of numerical methods for stochastic differential equations has intensified over the past decade. The earliest methods were usually heuristic adaptations of deterministic methods, but were found to have limited accuracy regardless of the order of the original scheme. A stochastic counterpart of the Taylor formula now provides a framework for the systematic investigation of numerical methods for stochastic differential equations. It suggests numerical schemes, which involve multiple stochastic integrals, of higher order of convergence. We shall survey the literature on these and on the earlier schemes in this paper. Our discussion will focus on diffusion processes, but we shall also indicate the extensions needed to handle processes with jump components. In particular, we shall classify the schemes according to strong or weak convergence criteria, depending on whether the approximation of the sample paths or of the probability distribution is of main interest.     

Stochastic partial differential equations in groundwater hydrology

Many problems in hydraulics and hydrology are described by linear, time dependent partial differential equations, linearity being, of course, an assumption based on necessity.Solutions to such equations have been obtained in the past based purely on deterministic consideration. The derivation of such a solution requires that the initial conditions, the boundary conditions, and the parameters contained within the equations be stipulated in exact terms. It is obvious that the solution so derived is a function of these specified, values.There are at least four ways in which randomness enters the problem. i) the random initial value problem; ii) the random boundary value problem; iii) the random forcing problem when the non-homogeneous part becomes random and iv) the random parameter problem.Such randomness is inherent in the environment surrounding the system, the environment being endowed with a large number of degrees of freedom.This paper considers the problem of groundwater flow in a phreatic aquifer fed by rainfall. The goveming equations are linear second order partial differential equations. Explicit form solutions to this randomly forced equation have been derived in well defined regular boundaries. The paper also provides a derivation of low order moment equations. It contains a discussion on the parameter estimation problem for stochastic partial differential equations.     

Development of the instantaneous unit hydrograph using stochastic differential equations

Recognizing that simple watershed conceptual models such as the Nash cascade ofn equal linear reservoirs continue to be reasonable means to approximate the Instantaneous Unit Hydrograph (IUH), it is natural to accept that random errors generated by climatological variability of data used in fitting an imprecise conceptual model will produce an IUH which is random itself. It is desirable to define the random properties of the IUH in a watershed in order to have a more realistic hydrologic application of this important function. Since in this case the IUH results from a series of differential equations where one or more of the uncertain parameters is treated in stochastic terms, then the statistical properties of the IUH are best described by the solution of the corresponding Stochastic Differential Equations (SDE's). This article attempts to present a methodology to derive the IUH in a small watershed by combining a classical conceptual model with the theory of SDE's. The procedure is illustrated with the application to the Middle Thames River, Ontario, Canada, and the model is verified by the comparison of the simulated statistical measures of the IUH with the corresponding observed ones with good agreement.     

Development of the instantaneous unit hydrograph using stochastic differential equations

Recognizing that simple watershed conceptual models such as the Nash cascade ofn equal linear reservoirs continue to be reasonable means to approximate the Instantaneous Unit Hydrograph (IUH), it is natural to accept that random errors generated by climatological variability of data used in fitting an imprecise conceptual model will produce an IUH which is random itself. It is desirable to define the random properties of the IUH in a watershed in order to have a more realistic hydrologic application of this important function. Since in this case the IUH results from a series of differential equations where one or more of the uncertain parameters is treated in stochastic terms, then the statistical properties of the IUH are best described by the solution of the corresponding Stochastic Differential Equations (SDE's). This article attempts to present a methodology to derive the IUH in a small watershed by combining a classical conceptual model with the theory of SDE's. The procedure is illustrated with the application to the Middle Thames River, Ontario, Canada, and the model is verified by the comparison of the simulated statistical measures of the IUH with the corresponding observed ones with good agreement.     

Evaluation of rainfall-runoff performance using the stochastic integral equation method

The stochastic integral equation method (S.I.E.M.) is used to evaluate the relative performance of a set of both calibrated and uncalibrated rainfall-runoff models with respect to prediction errors. The S.I.E.M. is also used to estimate confidence (prediction) interval values of a runoff criterion variable, given a prescribed rainfall-runoff model, and a similarity measure used to condition the storms that are utilized for model calibration purposes.Because of the increasing attention given to the issue of uncertainty in rainfall-runoff modeling estimates, the S.I.E.M. provides a promising tool for the hydrologist to consider in both research and design.     

Evaluation of rainfall-runoff performance using the stochastic integral equation method

The stochastic integral equation method (S.I.E.M.) is used to evaluate the relative performance of a set of both calibrated and uncalibrated rainfall-runoff models with respect to prediction errors. The S.I.E.M. is also used to estimate confidence (prediction) interval values of a runoff criterion variable, given a prescribed rainfall-runoff model, and a similarity measure used to condition the storms that are utilized for model calibration purposes.Because of the increasing attention given to the issue of uncertainty in rainfall-runoff modeling estimates, the S.I.E.M. provides a promising tool for the hydrologist to consider in both research and design.     

On a general class of stochastic partial differential equations

We shall consider in this article a general class of stochastic PDE which in particular covers the Zakai equation of nonlinear filtering and natural formulations of distributed systems involving control variables.We use only fixed point arguments, hence we get uniqueness results. In the case of the Zakai equation, Galerkin approximations have been considered by Pardoux (1979) to derive the existence of the solution     

Development of stochastic dynamic Nash game model for reservoir operation. I. The symmetric stochastic model with perfect information

Increasing water demands, higher standards of living, depletion of resources of acceptable quality and excessive water pollution due to agricultural and industrial expansions have caused intense social and political predicaments, and conflicting issues among water consumers. The available techniques commonly used in reservoir optimization/operation do not consider interaction, behavior and preferences of water users, reservoir operator and their associated modeling procedures, within the stochastic modeling framework. In this paper, game theory is used to present the associated conflicts among different consumers due to limited water. Considering the game theory fundamentals, the Stochastic Dynamic Nash Game with perfect information (PSDNG) model is developed, which assumes that the decision maker has sufficient (perfect) information regarding the associated randomness of reservoir operation parameters. The simulated annealing approach (SA) is applied as a part of the proposed stochastic framework, which makes the PSDNG solution conceivable. As a case study, the proposed model is applied to the Zayandeh-Rud river basin in Iran with conflicting demands. The results are compared with alternative reservoir operation models, i.e., Bayesian stochastic dynamic programming (BSDP), sequential genetic algorithm (SGA) and classical dynamic programming regression (DPR). Results show that the proposed model has the ability to generate reservoir operating policies, considering interactions of water users, reservoir operator and their preferences.     

A stochastic model describing the impact of daily rainfall depth distribution on the soil water balance

Simplified, vertically-averaged soil moisture models have been widely used to describe and study eco-hydrological processes in water-limited ecosystems. The principal aim of these models is to understand how the main physical and biological processes linking soil, vegetation, and climate impact on the statistical properties of soil moisture. A key component of these models is the stochastic nature of daily rainfall, which is mathematically described as a compound Poisson process with daily rainfall amounts drawn from an exponential distribution. Since measurements show that the exponential distribution is often not the best candidate to fit daily rainfall, we compare the soil moisture probability density functions obtained from a soil water balance model with daily rainfall depths assumed to be distributed as exponential, mixed-exponential, and gamma. This model with different daily rainfall distributions is applied to a catchment in New South Wales, Australia, in order to show that the estimation of the seasonal statistics of soil moisture might be improved when using the distribution that better fits daily rainfall data. This study also shows that the choice of the daily rainfall distributions might considerably affect the estimation of vegetation water-stress, leakage and runoff occurrence, and the whole water balance.     

Parameter estimation of the stochastic AMR model and its application to the study of several strong earthquakes

Introduction According to many published papers, seismicity in time-space domain shows some characteristics, such as doughnut epicenter distribution (Mogi, 1969) or quiescence of seismic activity before large earthquakes (WANG, et al, 2002), and aftershock decay (Ogata, 1998). In recent years, more and more seismologists (Lynnr, Steven, 1990) have found that many strong earthquakes are preceded by enhancing regional seismicity and accelerating strain energy release (ZHANG, et al, 2001). T…     

Clustering stochastic point process model for flood risk analysis

Since the introduction into flood risk analysis, the partial duration series method has gained increasing acceptance as an appealing alternative to the annual maximum series method. However, when the base flow is low, there is clustering in the flood peak or flow volume point process. In this case, the general stochastic point process model is not suitable to risk analysis. Therefore, two types of models for flood risk analysis are derived on the basis of clustering stochastic point process theory in this paper. The most remarkable characteristic of these models is that the flood risk is considered directly within the time domain. The acceptability of different models are also discussed with the combination of the flood peak counted process in twenty years at Yichang station on the Yangtze river. The result shows that the two kinds of models are suitable ones for flood risk analysis, which are more flexible compared with the traditional flood risk models derived on the basis of annual maximum series method or the general stochastic point process theory. Received: September 29, 1997     

Clustering stochastic point process model for flood risk analysis

Since the introduction into flood risk analysis, the partial duration series method has gained increasing acceptance as an appealing alternative to the annual maximum series method. However, when the base flow is low, there is clustering in the flood peak or flow volume point process. In this case, the general stochastic point process model is not suitable to risk analysis. Therefore, two types of models for flood risk analysis are derived on the basis of clustering stochastic point process theory in this paper. The most remarkable characteristic of these models is that the flood risk is considered directly within the time domain. The acceptability of different models are also discussed with the combination of the flood peak counted process in twenty years at Yichang station on the Yangtze river. The result shows that the two kinds of models are suitable ones for flood risk analysis, which are more flexible compared with the traditional flood risk models derived on the basis of annual maximum series method or the general stochastic point process theory. Received: September 29, 1997     

A review on stochastic differential equations for applications in hydrology

Fundamentals of the theory of stochastic calculus and stochastic differential equations (SDE's) which are finding increasing application in water resources engineering are reviewed. The basics of probability theory, mean square calculus and the Wiener, white Gaussian and compound Poisson processes are given in preparation for a discussion of the general Itô SDE with drift, diffusion and jump discontinuity terms driven by Gaussian white noise and compound Poissionian impulses. Also discussed are stochastic integration and the derivation of moment equations via the Itô differential rule. The lierature of SDE's is reviewed with an emphasis on the more accessible sources.     

A Monte Carlo study of rainfall forecasting with a stochastic model

A procedure for short-term rainfall forecasting in real-time is developed and a study of the role of sampling on forecast ability is conducted. Ground level rainfall fields are forecasted using a stochastic space-time rainfall model in state-space form. Updating of the rainfall field in real-time is accomplished using a distributed parameter Kalman filter to optimally combine measurement information and forecast model estimates. The influence of sampling density on forecast accuracy is evaluated using a series of a simulated rainfall events generated with the same stochastic rainfall model. Sampling was conducted at five different network spatial densities. The results quantify the influence of sampling network density on real-time rainfall field forecasting. Statistical analyses of the rainfall field residuals illustrate improvement in one hour lead time forecasts at higher measurement densities.     

Rainfall stochastic disaggregation models: Calibration and validation of a multiplicative cascade model

The simulation of long time series of rainfall rates at short time steps remains an important issue for various applications in hydrology. Among the various types of simulation models, random multiplicative cascade models (RMC models) appear as an appealing solution which displays the advantages to be parameter parsimonious and linked to the multifractal theory. This paper deals with the calibration and validation of RMC models. More precisely, it discusses the limits of the scaling exponent function method often used to calibrate RMC models, and presents an hydrological validation of calibrated RMC models. A 8-year time series of 1-min rainfall rates is used for the calibration and the validation of the tested models. The paper is organized in three parts. In the first part, the scaling invariance properties of the studied rainfall series is shown using various methods (q-moments, PDMS, autocovariance structure) and a RMC model is calibrated on the basis of the rainfall data scaling exponent function. A detailed analysis of the obtained results reveals that the shape of the scaling exponent function, and hence the values of the calibrated parameters of the RMC model, are highly sensitive to sampling fluctuation and may also be biased. In the second part, the origin of the sensivity to sampling fluctuation and of the bias is studied in detail and a modified Jackknife estimator is tested to reduce the bias. Finally, two hydrological applications are proposed to validate two candidate RMC models: a canonical model based on a log-Poisson random generator, and a basic micro-canonical model based on a uniform random generator. It is tested in this third part if the models reproduce faithfully the statistical distribution of rainfall characteristics on which they have not been calibrated. The results obtained for two validation tests are relatively satisfactory but also show that the temporal structure of the measured rainfall time series at small time steps is not well reproduced by the two selected simple random cascade models.     

Multisite generalization of a daily stochastic precipitation generation model

The familiar chain-dependent-process stochastic model of daily precipitation, consisting of a two-state, first-order Markov chain for occurrences and a mixed exponential distribution for nonzero amounts, is extended to simultaneous simulation at multiple locations by driving a collection of individual models with serially independent but spatially correlated random numbers. The procedure is illustrated for a network of 25 locations in New York state, with interstation separations ranging approximately from 10 to 500 km. The resulting process reasonably reproduces various aspects of the joint distribution of daily precipitation observations at the modeled locations. The mixed exponential distributions, in addition to providing substantially better fits than the more conventional gamma distributions, are convenient for representing the tendency for smaller amounts at locations near the edges of wet areas. Means, variances, and interstation correlations of monthly precipitation totals are also well reproduced. In addition, the use of mixed exponential rather than gamma distributions yields interannual variability in the synthetic series that is much closer to the observed.