VOL. 12, NO. 6 WATER RESOURCES RESEARCH DECEMBER 1976 Rainfall Network Design for Runoff Prediction RAFAEL L. BRAS Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02159 IGNACIO RODRIGUEZ-ITURBE SimdnBolivar University, Caracas, Venezuela A multivariate state-space stochastic model of rainfallbased on a multidimensional rainfallgenerator suggested byBras and Rodriguez-lturbe (1976a) isused together with a runoff model to study theaccuracy of discharge prediction as a function of the rainfall-sampling network. Therunoff model used isa spatially distributed simulation based on a finite difference solution of thekinematic wave equations. Discharge prediction accuracy at any point in a basin can beobtained in terms of themean square error as a function of thenumber of rain-sampling stations, theirlocation, andtheerrors in sampling devices. The solution is also a function of thephysics of the basin at hand which is incorporated in the'rainfall-runoff model. The mean square error of discharge estimation is obtained by using linearestimation theory for dynamic systems. Particularly, thetechnique used istheKalman-Bucy filter, which permits filtering and extrapola- tion of noisy and incomplete observations. INTRODUCTION formationon the dissemination and forecast response systems Rainfall-runoff models arecommonly used for flood fore- which would become available once the forecasting procedure casting. In such models, catchment behavior as a low-pass, was in operation. filterattenuating highfrequency andhigh wave number input fluctuations is commonly accepted. Suchapparent behavior obviously leads to the question, How detailed temporally and spatially does the rainfall description need to be to keep the accuracy of floodforechsting models within a given range? At this point the reader may justifiably argue that flood forecasting has fairly well defined relatedbenefits and losses [Sniedovich et al., 1974; Robinson, 1970]. Therefore thedesign of data collection networks for usewith forecasting models should be attempted with the objective of maximizing net benefits accrued and not simply with the forecasting accuracy criteria. Loss functions, related to flood forecasting, areinher- ently nonsymmetric (i.e., the loss expected from over- prediction isnot the same as that from underprediction) and a function of forecast response. Maximizingnet benefits under suchconditions would requireknowledge of the conditional distribution of the predicted discharge given the true dis- charge. Similarly• statistical description of theforecasting sys- tem and population response to flood warningswould be needed. In this work, although acknowledging the ideal system de- sign approach described previously, the authors chose to deal onlywith the forecasting accuracy criteria in designing net- works. There are three main reasons for that decision.First, the idealapproach, briefly described .above, requires informa- tion anddata that aresimply notavailable. The onlywaysuch an approach could be attempted is throughthe useof exten- sivesimulation together with considerable historical informa- tion on discharges and system response to forecasts. Grayman and Eagleson [1971] usedthe simulation tool as suggested above. Second, it is the authors' opinion that faced with such lack of data the problem should be decomposed in various stages. The first stageshould optimize forecasting accuracy. The other stages would iteratively solve theproblem with accuracy constraints. The accuracy constraint would be imposed by in- Copyright ¸ 1976 bythe American Geophysical Union. The third and final, but important, reason for choosing the accuracy criteriais that they canbe expressed and handled by using first-andsecond-order moments, mean andmean square error of estimation. Such an approach hasthe analytical .ad- vantages of relativesimplicity, minimum data requirements, and no simulation needs, as can be realized from a similar approach usedby the authors for a static network problem [Bras andRodriguez-lturbe, 1976b]. Unfortunately, the method yieldsno informationwith respect to the shape of the prob- ability distributions or to thejoint distributions of the param- eters neededfor the full system approach. Eagleson and Shack [1966] and Harley et al. [1970] have studied the problem of determining the sampling time interval with due consideration to rainfall and catchment response characteristics. The first attempt,to the authors' knowledge, to determine the sensitivity of catchment peak discharge to sam- pling densitywas made by Eagleson [1967], Using harmonic analysis conceptswith a distributed'One-dimensional linear catchmentsystem. Eagleson's [1967] resultsyielded relations between rain gagedensity and forecast peak 'error.' His ap- proach preventedany statementas to the station locations. In this work a methodwill be suggested that will.providethe mean square error (as a measure of accuracy) of the'estimated discharge, as a function of not only the number of rain gages but also their location, inherent measurementerror, and, natu- rally, the spatial variability of the rainfall process. The solu- tion is dynamic (time varying) and so gives a measure of accuracy at everytime stepof the hydrograph, not only at the peak value. Rainfall is acknowledged to be a nonstationary dynamicmultidimensional stochastic process, which is a much more realistic representationthan the static stationary one usedby Eagleson [1967]. This type of rainfall representation will permit the study of the effectsof the multidimensional character of rainfall on runoff prediction. This has been a virgin problem in rainfall-runoffmodeling whereprecipitation is generally distributed in space according to weight factors or at most(when data are available)is simulated in a multivariate one-dimensional fashion,e.g., several stationsas functions of time. 1197