WATER RESOURCES RESEARCH, VOL. 19, NO. 1, PAGES 269-276, FEBRUARY 1983 Uniqueness and Observabilityof Conceptual Rainfall-Runoff Model Parameters: The Percolation Process Examined VIJAI KUMAR GUPTA Department of Systems Engineering, CaseInstituteof Technology, CaseWestern Reserve University Cleveland,Ohio 44106 SOROOSH SOROOSHIAN ! Departments of Systems and Civil Engineering, CaseInstitute of Technology, Case Western Reserve University Cleveland, Ohio 44106 Many researchers have expressed concerns regarding the uniqueness of parameter estimates for concep- tual rainfall-runoff (R-R) models obtained throughcalibration. Recent studies (Sorooshian et al., thisissue; Sorooshianand Gupta, this issue)have revealed that even though stochastic parameter estimation techniques can help,the problems are not all due to inefficiencies in the calibration techniques used but are caused by the manner in which the model is structurally formulated. Thus even when calibratedunder idealconditions (simulation studies), it is oftenimpossible to obtainunique estimates for the parameters. It is possible to resolve this problem,at least in part, by appropriatereparameterizations of the pertinent model equations. In this paper the percolationequation of the soil moisture accounting model of the National Weather Service River Forecast System (SMA-NWSRFS) will be discussed. It is shown that a logical reparameterization of this equation can result in conditions thatimprove thechances of obtaining unique parameter estimates. It is believed that these results have implications for otherconceptual R-R models in which similar approaches are used in therepresentation ofthe percolation/infiltration process. INTRODUCTION The mostimportant problemin the automaticcalibrationof conceptual rainfall-runoff (R-R) models is the inability to obtain uniqueand conceptually realistic parameter estimates [see, for example,lbbitt, 1970; Johnston and Pilgrim, 1976; Pickup, 1977; Mein and Brown, 1978; Brazil and Hudlow, 1981]. This paperis the third in a three-paper sequence in whichthe main causes of this problem are discussed. In the first paper, Sorooshian et al. [this issue]demonstrated that part of this problemis caused by the use of an objective functionwhich doesnot properly account for the nature of the uncertainties present in both the data and the model. However, as discussed by Sorooshian andGupta [this issue] in the second paper, there are often factors inherent to the structure of the model that makethedetermination of a unique parameter set impossible. One of the most often-reported problems has been the in- ability to identify the parameters associated with the infiltra- tion or percolation process [Johnston andPilgrim,1976;lbbitt, 1970].Sorooshian andGupta [this issue], working with the soil moisture accounting model of the U.S. National Weather Ser- viceriver forecast system (SMA-NWSRFS),reported the exis- tence of an extended valley in the response surface of the parameters of thepercolation equation. Theydemonstrated the existence of a structuralproblem that prevents accurate cali- brationof the model. As shown by Gupta[1982], the perco- lation equation of the SMA-NWSRFS is similarin structure to the Horton infiltration equation,variationsof which are used in manymodels. In thispaperthe properties of the percolation process equa- • Now at theDepartment of Hydrology andWaterResources, Uni- versity of Arizona,Tucson, Arizona 85721. Copyright1983 by the American Geophysical Union. Paper number 2W 1892. 0043-1397/83/002W- 1892505.00 tion of the SMA-NWSRFS are examined in detail and the reasons for the high degree of interaction between its parame- ters are exposed. It is shownthat a suitable reparameterization of the equation can result in improved identifiability of the model. The effectiveness of the proposedapproachis demon- strated using simulation studies. TI• PERCOLATION PROCESS The SMA-NWSRFS model hasbeendiscussed extensively in the literature.A full description of the model may be found in the works of Peck [1976] and/or Brazil and Hudlow [1981]. The model recognizes the presence of differentverticallystrati- fied zones of soil in the ground (as do most conceptual R-R models). In brief, the stratification is represented by an upper zone which extendsfrom the surfaceto the depth of short rooted plants and a lower zone representing ground water storage.Each zone has both tension and free water storage representations. The percolation process connects the upper and lower zones,simulatingthe effects of gravity and down- ward suction. To quoteBrazil andHudlow [1981], The percolationfunction of the SMA-NWSRFS (developed by Burnashet al., 1973) is considered to be the key elementin the transfer of water within the model, as it affects water movement throughout the soil profile and is itself dependent on the current stateof the storage system. It is a complexnonlinear function relating the capacities and contents of both the upper and lower zones and their free water depletion coefficients. The percolation mechanism has been designed to coincide with observed characteristics of the motion of moisture through the soil mantle, including the formationand transmission characteristics of the wettingfront, asreported by Hanks et al. [1969] and Green et al. [1970]. It is generally considered to be a reasonably good representation of the true process. Because the percolation process helps decide on how muchof the water in the upperzonecontributes to the 269