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.     

Asymptotic Rayleigh instantaneous unit hydrograph

The instantaneous unit hydrograph for a channel network under general linear routing and conditioned on the network magnitude,N, tends asymptotically, asN grows large, to a Rayleigh probability density function. This behavior is identical to that of the width function of the network, and is proven under the assumption that the network link configuration is topologically random and the link hydraulic and geometric properties are independent and identically distributed random variables. The asymptotic distribution depends only on a scale factor, , where is a mean link wave travel time.     

Asymptotic Rayleigh instantaneous unit hydrograph

The instantaneous unit hydrograph for a channel network under general linear routing and conditioned on the network magnitude,N, tends asymptotically, asN grows large, to a Rayleigh probability density function. This behavior is identical to that of the width function of the network, and is proven under the assumption that the network link configuration is topologically random and the link hydraulic and geometric properties are independent and identically distributed random variables. The asymptotic distribution depends only on a scale factor, , where is a mean link wave travel time.     

Nonlinear transformation of unit hydrograph

Unit hydrograph (UH) and its numerous derivatives have been popular for estimation of flood hydrographs. Two major assumptions still overshadow UH applications. One is the linearity and the other is time invariance. In theory, only peak discharge of an equilibrium hydrograph follows linear proportionality to excess rainfall intensity. In trying to relax the linearity constraint, this paper aims to propose a nonlinear way of transforming a given UH to other general hydrographs. The transformation or mapping technique relies on a simple rainfall ratio raised to a power less than unity. The case of nonlinear transformation is illustrated for a number of watershed geometries with either known kinematic wave analytic solutions or observed data. The nonlinear UH approach also relaxes the assumption of constant time base of the UH. The proposed nonlinear UH transformation may thus be viewed as a major step in closing the gap between physically based and traditional UH-based surface runoff simulation approaches.     

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.     

The geomorphological unit hydrograph from a historical‐critical perspective

In this paper we present a brief overview of geomorphological instantaneous unit hydrograph (GIUH) theories and analyze their successful path without hiding their limitations. The history of the GIUH is subdivided into three major sections. The first is based on the milestone works of Rodríguez‐Iturbe and Valdés (Water Resources Research 1979; 15 (6): 1409–1420) and Gupta et al. (Water Resources Research 1980; 16 (5): 855–862), which recognized that a treatment of water discharges with ‘travel times’ could provide a rich interpretation of the theory of the instantaneous unit hydrograph (IUH). We show how this was possible, what assumptions were made, which of these assumptions can be relaxed, and which have become obsolete and been discarded. The second section focuses on the width‐function‐based IUH (WFIUH) approach and its achievements in assessing the interplay of the topology and geometry of the network with water dynamics. The limitations of the WFIUH approach are described, and a way to work around them is suggested. Finally, a new formal approach to estimating the water budget by ‘travel times’, which derives from a suitable use of the water budget equation and some hypotheses, has been introduced and disentangled. Copyright © 2015 John Wiley & Sons, Ltd.     

Estimation of Nash's IUH parameters using stochastic search algorithms

In rainfall–runoff studies, it is often necessary to change the duration of a given unit hydrograph. Nash's Instantaneous Unit Hydrograph (IUH) is an ideal method that eliminates the hydrograph duration. This paper presents the results of the application of search algorithms, namely a genetic algorithm and hill climbing, to develop the IUH that minimizes the error between the observed and generated hydrographs. Also the performance of these methods has been compared with that of the classical method used for estimation of IUH, namely the method of moments. The genetic algorithm is a popular search procedure for function optimization that applies the mechanics of natural genetics and natural selection to explore a given search space. Hill climbing is an optimization technique that belongs to the family of local search and algorithms can be used to solve problems that have many solutions, with some solutions better than others. The results obtained from both the genetic algorithm and hill climbing algorithm for estimation of Nash's IUH parameters were compared with the results obtained by the method of moments for storms from two river basins that are located in different climatic regions. It was found that both the genetic algorithm and hill climbing provided improved and consistent results. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

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.     

沉积模型和储层随机建模

沉积模型是地层分析的重要工具，可以分为比例尺模型、概念模型和数学模型三大类型，其中数学模型又可分为确定性模型和随机模型。在实际地层分析及模拟工作中，特别是在小尺度问题的研究中，采用随机模型(或称统计学模型)往往更为有利。储层随机建模技术，作为这方面研究的典范，近年来成为储层预测和风险评价的一项较为有效的手段。然而，由于研究目标的复杂性，不同沉积模型之间的嵌套制约关系亦应引起重视。     

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     

A stochastic differential equation approach to soil moisture

The dynamics of water within the unsaturated root zone of the soil are represented by a pair of stochastic differential equations (SDE's), one representing the so-called surplus state of the moisture and the other the deficit condition. The inputs to the model are the climatically controlled random infiltration events and evapotranspiration which are modeled as a compound Poisson process and a Wiener (Brownian motion) process, respectively.The solutions to these SDE's are not in close-form but sample functions are obtained by numerical integration. The moment properties of the soil moisture evolution process have also been derived analytically including the mean, variance, covariance and autocorrelation functions.To illustrate the model, climatic parameters representing the surplus and deficit cases and properties of clay loam soil have been used to numerically derived the corresponding sample functions. With proper selection of all the parameters, physically realistic sample trajectories can be obtained for the model.     

Correlation of the velocity parameter of three geomorphological instantaneous unit hydrograph models

Three geomorphological instantaneous unit hydrograph (GIUH) models are investigated. These GIUHs were derived as a function of watershed geomorphological characteristics. The geomorphological input parameters of the models were determined for 10 basins in Indiana, USA. The three GIUH models were used to simulate 187 observed runoff hydrographs of these basins. The non-measurable velocity parameters of the GIUH models were optimized and the relationships between them were investigated. The results of the investigation show that the velocity parameters of the three models are correlated. © 1998 John Wiley & Sons, Ltd.     

Runoff estimation for an ungauged catchment using geomorphological instantaneous unit hydrograph (GIUH) models

A geomorphological instantaneous unit hydrograph (GIUH) is derived from the geomorphological characteristics of a catchment and it is related to the parameters of the Clark instantaneous unit hydrograph (IUH) model as well as the Nash IUH model for deriving its complete shape. The developed GIUH based Clark and Nash models are applied for simulation of the direct surface run‐off (DSRO) hydrographs for ten rainfall‐runoff events of the Ajay catchment up to the Sarath gauging site of eastern India. The geomorphological characteristics of the Ajay catchment are evaluated using the GIS package, Integrated Land and Water Information System (ILWIS). The performances of the GIUH based Clark and Nash models in simulating the DSRO hydrographs are compared with the Clark IUH model option of HEC‐1 package and the Nash IUH model, using some commonly used objective functions. The DSRO hydrographs are computed with reasonable accuracy by the GIUH based Clark and Nash models, which simulate the DSRO hydrographs of the catchment considering it to be ungauged. Inter comparison of the performances of the GIUH based Clark and Nash models shows that the DSRO hydrographs are estimated with comparable accuracy by both the models. Copyright © 2007 John Wiley & Sons, Ltd.     

Inverse analysis of stochastic moment equations for transient flow in randomly heterogeneous media

We present a nonlinear stochastic inverse algorithm that allows conditioning estimates of transient hydraulic heads, fluxes and their associated uncertainty on information about hydraulic conductivity (K) and hydraulic head (h  ) data collected in a randomly heterogeneous confined aquifer. Our algorithm is based on Laplace-transformed recursive finite-element approximations of exact nonlocal first and second conditional stochastic moment equations of transient flow. It makes it possible to estimate jointly spatial variations in natural log-conductivity (Y=lnK)$(Y=\ln K)$, the parameters of its underlying variogram, and the variance–covariance of these estimates. Log-conductivity is parameterized geostatistically based on measured values at discrete locations and unknown values at discrete “pilot points”. Whereas prior values of Y at pilot point are obtained by generalized kriging, posterior estimates at pilot points are obtained through a maximum likelihood fit of computed and measured transient heads. These posterior estimates are then projected onto the computational grid by kriging. Optionally, the maximum likelihood function may include a regularization term reflecting prior information about Y. The relative weight assigned to this term is evaluated separately from other model parameters to avoid bias and instability. We illustrate and explore our algorithm by means of a synthetic example involving a pumping well. We find that whereas Y and h can be reproduced quite well with parameters estimated on the basis of zero-order mean flow equations, all model quality criteria identify the second-order results as being superior to zero-order results. Identifying the weight of the regularization term and variogram parameters can be done with much lesser ambiguity based on second- than on zero-order results. A second-order model is required to compute predictive error variances of hydraulic head (and flux) a posteriori. Conditioning the inversion jointly on conductivity and hydraulic head data results in lesser predictive uncertainty than conditioning on conductivity or head data alone.     

Reliability analysis of hydraulic structures considering unit hydrograph uncertainty

Unit hydrographs (UHs), along with design rainfalls, are frequently used to determine the discharge hydrograph for design and evaluation of hydraulic structures. Due to the presence of various uncertainties in its derivation, the resulting UH is inevitably subject to uncertainty. Consequently, the performance of hydraulic structures under the design storm condition is uncertain. This paper integrates the linearly constrained Monte-Carlo simulation with the UH theory and routing techniques to evaluate the reliability of hydraulic structures. The linear constraint is considered because the water volume of each generated design direct runoff hydrograph should be equal to that of the design effective rainfall hyetograph or the water volume of each generated UH must be equal to one inch (or cm) over the watershed. For illustration, the proposed methodology is applied to evaluate the overtopping risk of a hypothetical flood detention reservoir downstream of Tong-Tou watershed in Taiwan.     

Reliability analysis of hydraulic structures considering unit hydrograph uncertainty

Unit hydrographs (UHs), along with design rainfalls, are frequently used to determine the discharge hydrograph for design and evaluation of hydraulic structures. Due to the presence of various uncertainties in its derivation, the resulting UH is inevitably subject to uncertainty. Consequently, the performance of hydraulic structures under the design storm condition is uncertain. This paper integrates the linearly constrained Monte-Carlo simulation with the UH theory and routing techniques to evaluate the reliability of hydraulic structures. The linear constraint is considered because the water volume of each generated design direct runoff hydrograph should be equal to that of the design effective rainfall hyetograph or the water volume of each generated UH must be equal to one inch (or cm) over the watershed. For illustration, the proposed methodology is applied to evaluate the overtopping risk of a hypothetical flood detention reservoir downstream of Tong-Tou watershed in Taiwan.     

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.