Journal of Hydrologic Engineering

The point-location-scale conservation equations of hydrologic processes, when viewed at the scale of computational grid areas, become stochastic partial differential equations (PDEs). For the upscaling of the point-location-scale conservation equations to the scale of computational grid areas, a common approach is to develop the ensemble averages of these equations. Accordingly, in this study general ensemble average conservation equations for determining the probabilistic and mean behavior of nonlinear and linear hydrologic processes are developed to exact second order. From the derived equations it is seen that the evolution equation for the probabilistic behavior of a generally nonlinear hydrologic system becomes a Fokker–Planck equation (FPE). As such, the determination of the probabilistic behavior of a hydrologic system of processes reduces to the solution of a linear, deterministic PDE, the FPE, under appropriate initial and boundary conditions. The solution of the FPE yields the probability density function of the hydrologic system which can then be used to obtain the means of the state variables of the system by the expectation operation. One can also determine the mean behavior of nonlinear stochastic hydrologic processes by means of master key ensemble average conservation equations developed in this study. Upon examination of these generic deterministic equations, one may note that they are implicit integro-differential nonlinear equations in the mixed Eulerian–Lagrangian form. Meanwhile, the master key equations which were developed also for linear hydrologic processes, are explicit, linear PDEs.

In this paper, the second in a series of two, the theory developed in the companion paper is applied to the case of the stochastic overland flow equation, and a numerical solution method is presented for the resulting Fokker-Planck equation (FPE), which describes the evolution of the probability-density function (PDF) of overland flow depth at the downstream section of a hillslope. The derived FPE is evaluated for two different approximations to the diffusion coefficient of the FPE. The Monte Carlo analysis of stochastic overland flow equation is then performed using the random rainfall sequences, generated by a compound filtered Poisson process model for the stochastic rainfall, in order to provide a benchmark for the results obtained from the FPEs. When compared to the Monte Carlo simulation based PDFs and their ensemble average, the second approximation to the diffusion coefficient gives a good fit in terms of the shape of the PDF and the ensemble average of the overland flow depth. Therefore, the theory proposed here is quite promising for obtaining the ensemble averages of nonlinear hydrological processes.

Runoff data were analyzed from the semihumid 21.2 km² Goodwin Creek Experimental Watershed (GCEW) in northern Mississippi to examine watershed response over a range of scales. Runoff is monitored at the GCEW outlet and in 13 subcatchments, ranging in area from 0.06 to 17.6 km². Previous data-based studies have shown that simple scaling theory fails to describe scaling of flood quantiles in large watersheds, and there is a fundamental change in scaling behavior in semihumid watersheds at an area of approximately 100 km². It has been found that flood quantiles in nearly all subbasins in the GCEW are self-similar as described by simple scaling theory. It has also been found that expected values of peak flows during single runoff events are described by a power law function of catchment area. The primary reasons why flood quantiles are self-similar on Goodwin Creek are that precipitation is relatively uniform over the basin; peak discharges in smaller catchments are highly correlated with rainfall rates; nearly the entire watershed regularly contributes to runoff and; the groundwater table plays little role in runoff production.

Proposed in this study is an approach to estimate regional evapotranspiration from potential evaporation using a distributed hydrologic model. If water is freely available in the storage tank of interest, evapotranspiration is taken as Penman’s potential evaporation. Otherwise, the reduction of actual evapotranspiration from potential evaporation is estimated using the simplified Philip’s soil-moisture relationship with the combination of a physically based distributed tank (PDTank) model. The average annual evapotranspiration in the study area is estimated to be 716 mm. Results are also compared with the actual evapotranspiration estimated by using Morton’s complementary relationship areal evapotranspiration (CRAE) model and Brusaert-Stricker’s advection-aridity (AA) model. These three models give quite similar performance of goodness. The AA model gives the largest estimation for the annual evapotranspiration, the physically based distributed tank (PDTank) model gives the smallest estimation, and the CRAE model shows the amount in-between the AA model and the PDTank result. But the PDTank gives better results than both the CRAE and the AA models for winter evapotranspiration.

An entropy-based method is developed for estimating parameters of the four-parameter kappa distribution. Using four sets of data on annual maximum rainfall and on annual peak flow discharge, the entropy-based method is evaluated and compared with the methods of moments, L-moments, probability-weighted moments, and the maximum likelihood estimation. The results of estimation show that both the entropy method and the L-moment method enable the four-parameter kappa distribution to fit the data well and a combination of the two methods can further improve estimation.

Modeling of an event-based rainfall-runoff process has been of importance in hydrology. Historically, researchers have relied on conventional modeling techniques, either deterministic, which consider the physics of the underlying process, or systems theoretic/black box, which do not. This technical note investigates the suitability of some deterministic and statistical techniques along with the artificial neural networks (ANNs) technique to model an event-based rainfall-runoff process. Specifically, two unit hydrograph models, four regression models, and two ANN models were developed. Data derived from Salado Creek at Bitters Road, San Antonio were employed. It was found that the ANN models consistently outperformed conventional models, barring a few exceptions, and provided a better representation of an event-based rainfall-runoff process in general, and better prediction of peak discharge and time to peak discharge, in particular.

A numerical model is developed to determine the flow depth and discharge relationship for turbulent flows over vegetated flood plains. The model is based on the turbulent boundary layer similarity concepts developed for atmospheric and pipe flow hydraulics. Here, the Darcy–Weisbach formula is modified to account for vegetative resistance by application of surface roughness lengths developed from atmospheric methodologies. Such roughness lengths are well defined for many natural surfaces and can be determined from existing databases of vegetative surfaces or directly measured for specific sites. The model estimates of the flood flow discharges from a natural site are compared to observed data. The results show very good agreement with errors on the order of 7–8%. A sensitivity analysis was conducted to evaluate the effect of channel characteristics on discharge.