The probability-distributed principle in basin-scale hydrology considers the frequency of occurrence of hydrological variables (model inputs, parameters or elements) of certain magnitudes over the basin without regard to the location of a particular occurrence within the basin. The random assemblage of different parts is considered more important than the relation of the parts, one to another. Rainfall-runoff models based on probability-distributed infiltration capacity and storage capacity concepts, and which generate runoff according to Hortonian and saturation overland flow mechanisms respectively, are distinguished. Two types of probability-distributed storage capacity model are identified, one based on an assumption that storage elements at points in the basin respond independently of their neighbours, and the other where storage elements interact so as to equalize the depth of stored water over the basin. Allowing redistribution of water leads to simplification of the model equations.

[1] Rain gauge networks provide rainfall measurements with a high degree of accuracy at specific locations but, in most cases, the instruments are too sparsely distributed to accurately capture the high spatial and temporal variability of precipitation systems. Radar and satellite remote sensing of rainfall has become a viable approach to address this problem effectively. However, among other sources of uncertainties, the remote-sensing based rainfall products are unavoidably affected by sampling errors that need to be evaluated and characterized. Using a large data set (more than six years) of rainfall measurements from a dense network of 50 rain gauges deployed over an area of about 135 km 2 in the Brue catchment (south-western England), this study sheds some light on the temporal and spatial sampling uncertainties: the former are defined as the errors resulting from temporal gaps in rainfall observations, while the latter as the uncertainties due to the approximation of an areal estimate using point measurements. It is shown that the temporal sampling uncertainties increase with the sampling interval according to a scaling law and decrease with increasing averaging area with no strong dependence on local orography. On the other hand, the spatial sampling uncertainties tend to decrease for increasing accumulation time, with no strong dependence on location of the gauge within the pixel or on the gauge elevation. For the evaluation of high resolution satellite rainfall products, a simple rule is proposed for the number of rain gauges required to estimate areal rainfall with a prescribed accuracy. Additionally, a description is given of the characteristics of the rainfall process in the area in terms of spatial correlation.

The Probability Distributed Model, or PDM, has evolved as a toolkit of model functions that together constitute a lumped rainfallrunoff model capable of representing a variety of catchment-scale hydrological behaviours. Runoff production is represented as a saturation excess runoff process controlled by the absorption capacity (of the canopy, surface and soil) whose variability within the catchment is characterised by a probability density function of chosen form. Soil drainage to groundwater is controlled by the water content in excess of a tension threshold, optionally inhibited by the water content of the receiving groundwater store. Alternatively, a proportional split of runoff to fast (surface storage) and slow (groundwater) pathways can be invoked with no explicit soil drainage function. Recursive solutions to the HortonIzzard equation are provided for routing flows through these pathways, conveniently considered to yield the surface runoff and baseflow components of the total flow. An alternative routing function employs a transfer function that is discretely-coincident to a cascade of two linear reservoirs in series. For real-time flow forecasting applications, the PDM is complemented by updating methods based on error prediction and state-correction approaches. The PDM has been widely applied throughout the world, both for operational and design purposes. This experience has allowed the PDM to evolve to its current form as a practical toolkit for rainfall-runoff modelling and forecasting.

A grid-based approach to river flow modelling has been developed for regional assessments of the impact of environmental change on hydrologically sensitive systems. The approach also provides a means of assessing, and providing feedback on, the hydrological performance of the land-surface component of a regional climate model (RCM). When combined with information on the evolution of climate, the model can give estimates of the impact of future climate change on river flows and flooding. The high-resolution flow routing and runoff-production model is designed for use with RCM-derived rainfall and potential evaporation (PE), although other sources of gridded rainfall and PE can be employed. Called the Grid-to-Grid Model, or G2G, it can be configured on grids of different resolution and coverage (a 1 km grid over the UK is used here). The model can simulate flow on an area-wide basis as well as providing estimates of fluvial discharges for input to shelfsea and ocean models. Configuration of the flow routing model on a relatively high resolution 1 km grid allows modelled river flows to be compared with gauged observations for a variety of catchments across the UK. Modelled flows are also compared with those obtained from a catchment-based model, a parameter-generalised form of the Probability-Distributed Model (PDM) developed for assessing flood frequency. Using RCM re-analysis rainfall and PE as input, the G2G model performs well compared with measured flows at a daily time-step, particularly for high relief catchments. It performs less well for low-relief and groundwater-dominated regions because the dominant model control on runoff production is topography.

This paper begins with a critique of existing rainfall runoff models and proceeds to a largely new formulation in which the single store (representing, for example, interception of rainfall by vegetation, or retention of water in upper soil layers, or possibly both) is replaced by a statistical population of stores. The consequences of such an assumption are illustrated for the simplest, one‐parameter case in which the distribution of store depths is exponential. It is demonstrated that the use of a population of stores, even with but one parameter, can (1) afford a plausible description of the relation between actual evaporation and soil moisture deficit and (2) remove discontinuities of gradient in the objective function, optimization of which gives estimates of model parameters. The new formulation also permits observed runoff to be written down as a relatively simple function of past rainfall, potential evaporation, and the parameters in the statistical distribution of storages, with the consequence that gradient methods can be used to optimize the objective function in place of more time‐consuming direct search methods. An extension of the model to account for the translation of runoff to the basin outfall is accomplished by using a bivariate distribution of translation times and store depths. A simple recursive equation relating current flow to a proportion of the previous flow and an additive function of rainfall is obtained under the assumption that translation times and store depths are independent and exponentially distributed. More complex models are derived by relaxing the assumption of independence and by considering distributions other than exponential; expressions for two positively skewed density functions, the Weibull and gamma, are obtained. Series and parallel configurations of distribution function models are considered, and the relation of the models' elemental structure to different types of store commonly employed in conceptual modeling is discussed. The new formulation includes, as particular cases, all models based on linear systems theory. Application of the modeling approach to hourly values of flow, rainfall, and evapotranspiration from a number of the Institute of Hydrology's experimental basins results in very good model predictions of flows over the calibration period, with R2 values above 0.9. However, this level of performance as measured by the R2 statistic is not maintained over the test period, although quite reasonable predictions of the flood peaks are still obtained. The drop in performance is partly ascribed to the nature of the calibration period during which the basins were ‘wetting up’ after two years of relatively extreme drought. Model performance over the test period is improved by using a more realistic initial condition for the store contents but only at the expense of reduced R2 values in the calibration period. The need to assess the new model approach in a range of hydrological environments is recognised, especially where evapotranspiration forms an important com...