WATER RESOURCES RESEARCH, VOL. 16, NO. 6, PAGES 1025-1033, DECEMBER 1980 Real-Time Forecasting With a Conceptual HydrologicModel 1. Analysis of Uncertainty PETER K. KITANIDIS Institute of HydraulicResearch, University of lowa, Iowa City, Iowa 52242 RAFAEL L. BRAS Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 The optimal control of watershed systems requiresaccuratereal-time short-term forecasts of river flows. For the first time, this paper formulates a large, nonlinear conceptualmodel (the National Weather Service catchment model) in a mode amenable to analysis of uncertainty and the utilization of real-timeinformation(measurements, forecasts, guesses) to updatesystem states and improve streamflow predictions. The proposed methodology is based on the state space formulation of the equations describ- ing the hydrologic model and the assumption of sources of uncertaintyin the data and in the model structure. The first two moments of random variables are estimated in a computationally efficientway usingon-line linear estimation techniques. Linearizationof functionalrelationships is performedwith the uncommonbut powerful multiple-input describing function techniquefor the most strongly non- linear responses and Taylor expansion for the rest. The linear feedback rule developed is basedon the Kalman filter. 1. INTRODUCTION The costeffectiveand reliable operationof watershedsys- tems requires accurate real-time short-term forecasts of river flows. A large number of rainfall-runoff models have been de- velopedand are being used for on-line forecasting purposes. They range from linear hydrographs and cascades of linear reservoirs to conceptual Stanford-type models. In practice,operationalforecasting requires, in addition to the rainfall-runoff model, a method for the continuous correc- tion of forecasts from observed errors in earlier forecasts [Nashand Sutcliffe, 1970].This feedback informationis most valuable in improving the real-time forecasting performance of a rainfall-runoff model given the uncertainty in the mea- sured inputs and the complexity of all watersheds. Only re- cently has this real-time characteristicof operational river flow forecasting attractedthe attentionit deserves (see, for ex- ample, Rodriguez-Iturbe et al. [1978]). Suggested forecasting procedures rely mainly on methods of stochastic estimation and filtering developed in other areasof engineering. These methodologies have been applied to relatively simplesystems- orientedrainfall-runoff models(suchas autoregressive-mov- ing average modelsand linear reservoirO. Unfortunately, most of the proposed hydrologic examples of feedback utilization have been achievedat the expenseof employing oversimplified models. The continuous correction of the model basedon the output measurements has been em- phasized, while the useof models which can correctly project into the future has been overlooked.Furthermore, adaptive- hess (on-line estimation of as many modelparameters as pos- sible) hasoftenbeenconsidered as a substitute for a more so- phisticated modelstructure. This furtherincreases the reliance of the forecasting schemeupon the feedback. Such adaptive models may perform relatively well for forecastlead times which are short in comparisonto the response time of the catchment,under slowly varying hydrologicconditions,and when the error in the input measurements is large, while the Copyright¸ 1980 by the American Geophysical Union. Paper number 80W0999. 0043-1397/80/080W-0999501.00 error in the output measurements is small. However, as the forecast leadtime of interest gets longer and the qualityof in- put data gets better,the performance of the model is more and moredefined by its capability to simulate the input-output be- havior of the prototype rather than by its adaptiveness to measurements of the output. Ideally, the best modelwouldbe a deterministic description of the rainfall-runoff process based on well-known physical laws.Although, in principle, estimating the runoffgivenmete- orological inputsshould be based on the application of these laws, in practice, the complexity of natural catchments pre- cludes sucha rigorous approach. This has encouraged many hydrologists to simplify the representation of the elements of the catchment and the description of the physical processes. The developed conceptual models are not free of empiricism but can still incorporate informationabout the physical sys- tem which systems-oriented models cannot. Generally, this would imply the ability to forecast under conditions that did not exist in the calibration period of the model. Conceptual hydrologic models have been formulated in a deterministic way. The input (usuallyprecipitation and tem- perature)is sut•icient to describe the condition of the system, sothat measurements of the output(fiver flow) are considered redundant information. The abovemodeling philosophy ig- noresthe following sources of errors: 1. Model uncertainty. In modeling the various com- ponents of the rainfall-runoffprocess, several simplifications haveto be made.A spatially distributed process is represented as a lumped one. The various components of the hydrologic cycleare separated, and the interactions between elements are described by simplified functional relations. The imperfection of the modelintroduces an error whichis reflected in the pre- dictions and will be called model error. 2. Input uncertainty. The inputs requiredfor the opera- tion of conceptual hydrologic models usually are precipitation and temperature. These data are either measured or come from quantitativeprecipitationand weather forecasts. In the firstcase, thereare measurement preprocessing and averaging 1025