B.J. Williams

Effect of rainfall errors on accuracy of design flood estimates

A procedure is presented to explicitly evaluate the effect of estimation errors in the temporal and spatial distribution of rainfall on the uncertainty of calibrated rainfall-runoff model parameters. The effect of this uncertainty on the reliability of design flood predictions is considered. A case study of the Hacking catchment, south of Sydney, Australia, is presented to illustrate the procedure. For a major storm event a simple stochastic transformed rainfall model is calibrated and validated using kriging and then used to infer the mean and covariance of subareal rainfall. The RORB model, a distributed nonlinear rainfall-runoff model, is calibrated to this storm event. When allowance is made for uncertainty in the calibration event rainfall, the results indicate that the uncertainty in the calibrated parameters increases, especially in the rainfall excess parameters, and the 90% prediction interval on the 100-year design flood increases by about 100%.

Parameter estimation in rainfall-runoff models

Abstract This paper presents techniques for the estimation of parameters in rainfall-runoff models. In the practical application of models a number of catchment parameters are not directly measureable and it is desirable to make the best possible estimate from known rainfall-runoff data. Three techniques are presented. Linear programming (LP) is used to minimize the sum of the absolute errors (MSAE) of the computed hydrograph. Quadratic programming (QP) is used in the two other techniques, namely ordinary least squares (OLS) and generalized least squares (GLS). OLS uses the traditional regression objective of minimizing the square of the deviation while GLS uses a weighted form of the OLS objective which can eliminate the effect of serially correlated errors (noise). The techniques are demonstrated using a hypothetical catchment for which rainfall-runoff series are generated using a conceptual model. Four parameters necessary for the models operation are then estimated, with varying levels of noise superimposed on the generated series, using the three techniques and then finally the techniques are applied to a real catchment. The covariance and the correlation matrices of the estimated parameters are computed.

A Stochastic Tokunaga Model for stream networks

The Tokunaga cyclic model describes average network topology. A stochastic generalization is proposed. The stochastic model assumes that actual tributary numbers are random realizations from a negative binomial distribution whose mean is defined by the Tokunaga parameters ϵ1 and K. These parameters can be interpreted as representing the effects of regional controls. Upon these regional controls is superimposed an inherent spatial variability in network topology. A third parameter α characterizes this spatial variability. When α becomes large, the negative binomial model approaches a Poisson model. A goodness-of-fit test based on a χ2 test statistic is developed, and an inference framework for estimation of parameters and stream-related statistics is described. This methodology is illustrated on tributary data from three catchments, one of the order of 5 and two of the order of 8. It is shown that the stochastic Tokunaga model using the negative binomial distribution is not inconsistent with the tributary data, whereas the Poisson model is unambiguously rejected by the data. Monte Carlo Bayesian methods are used to evaluate the uncertainty in the Tokunaga parameters and in stream number related statistics such as the bifurcation ratio. It is shown that tributary data from the order-5 network provide little power for discriminating between model hypotheses. The tributary data for the two order-8 basins are significantly different from the asymptotic stream number statistics predicted by Shreves random network model. Finally, the problem of space filling or preservation of nontopological properties is considered in the context of the stochastic Tokunaga model.

A generalised one-dimensional kinematic catchment model

Abstract A numerical model has been developed for the solution of the kinematic wave equation for a one-dimensional catchment, using the Lax-Wendroff technique. The model has been verified by comparing its solutions with some analytical solutions already available. Certain catchment parameters have been identified and the model applied with promising results to two actual catchments to ascertain their values.

Downstream characteristic Lagrangian hybrid method for flows in open channels

In this paper, we present a one step downstream characteristic Lagrangian hybrid method (DSCLH) for solution of the time-dependent open channel flow equations. It is a fixed-grid hybrid method, in which a numerically stable Lagrangian method is used to compute the nonlinear or linear convection process by convecting the grid downstream one step along the trajectory of a fluid particle. It significantly reduces numerical smoothing and simplifies Lagrangian advection since there is no need to determine the upstream interpolation points. It works well for both steady and unsteady state calculations where discontinuities are present. Solutions are found for the open channel flow equations using different initial and boundary conditions. Comparison with known results shows that the DSCLH method is both convergent and L ∞ stable. It can be applied to a wide range of shock problems and run for long times without oscillation. Certain fundamental conditions which are necessary for the successful application of the...

Input errors in rainfall-runoff modelling

Precipitation is one of the most variable of the hydrologic processes. This presents two particular problems when hydrologic process of basin scale are studied via mathematical models. The first is the extent to which areal rainfall distribution can be determined from point measurements and the second is the effect of precipitation uncertainty on the uncertainty of catchment model parameters. This paper presents a state-of-the-art survey of methods used for handling the first problem. Spatial interpolation models commonly in use are discussed. A Bayesian type optimisation procedure to estimate model parameters is presented and demonstrated for a real world example. The technique relaxes the usual least squares assumption of normally distributed residuals and estimates an error distribution which is a member of a family of distributions which includes the normal and Laplace distribution.

Rainfall runoff models in flood forecasting applications

Two conceptual rainfall-runoff models are described incorporating non-linear surface and groundwater storages in association with kinematic wave routing of channel flow. The models are shown to be suitable for design purposes and one of them, in preliminary investigations, appears to be suitable for flood forecasting applications where parameters and states are estimated using an iterated extended Kalman filter.