WATER RESOURCES RESEARCH, VOL. 20, NO. 1, PAGES 47-56, JANUARY 1984 Disaggregation Procedures for Generating Serially Correlated Flow Vectors JERY R. STEDINGER 1 AND RICHARD M. VOGEL Department of Environmental En•lineerin•], Cornell University The structure of disaggregation modelsplaces severe constraints on the feasible valuesof the lagged covariance of generated flow vectors. A new and simple class of disaggregation models is presented which employ the Valencia-Schaake disaggregation model structurebut allow the models'innovationsto be serially correlated. Thesemodels can reproduce (1) the covariance matrix of the disaggregated flows,(2) their covariance with the upperlevelflows, and (3) reasonable approximations to the lag one covariance of the disaggregated flow vectors given the constraints imposed by a disaggregation approach. The Mejia-Rousselle disaggregation modelis shown, in general, to fail to reproduce the anticipated variances and covariances of the disaggregated flows because the model and its parameter estimators are not self-consistent. The paper closes with a discussion of practicalmodelingconsiderations and of staged disaggregation procedures which reduce the size of multisite multiseason models. INTRODUCTION Since their introductionby Valenciaand Schaake [1972, 1973], disaggregation models have beenrecognized as a rea- sonable way to divideannualflows into seasonal flows[Mejia and Rousselle, 1976; Tao and Delleur,1976; Srikanthan, 1978; Lane, 1979; Todini, 1980; Salas et al., 1980] and to divide aggregate basin flows (monthly or annual) into flows at indi- vidual sites [Loucks et al., 1981; Lane, 1979, 1982; Salaset al., 1980]. An important contribution was the suggestion by Mejia and Rousselle [1976] that the Valencia and Schaake model structure could be extended to allow reproduction of the correlation among monthlyflowsin different water years, i.e., the correlationbetweendisaggregated flow volumesin different time units. The Mejia-Rousselle model has been employed by Lane [1979, 1982] and Salas et al. [1980] in their "staged" disaggre- gationprocedures. In the staged disaggregation procedure de- scribed by Louckset al. [1981], the Mejia-Rousselle model wasemployed to reproduce the period-to-period correlation of flows at individual sites obtained by disaggregation of the aggregate basinflow for eachperiod.Lane [1979, 1980, 1983] documents a general purpose computer programwhichcan be obtainedupon request and which implements most of these procedures. Other disaggregation models havebeen proposed by Svani- dize[1980], by Harmsand Campbell [1967], and by Hoshiand Burges [1979], though problemswere identified with the third model [Hoshi and Burges, 1980]. Lane [1979, 1982], Salaset al., [1980], and Pei and Stedinger[1982] consider models similar to that proposed by Valencia and Schaake but which have fewerparameters. The focus of this paper is on disaggregation techniques for the situation wherethe flow vectors X, generated by the dis- aggregation of annualflowsto seasonal flowsor of aggregate basin flows to those at individual sites, have serial correlation not capturedby the upper level flow model coupledwith the Valencia-Schaakedisaggregation procedure. As mentioned •Nowonleave at theU.S. Geological Survey, National Center. Copyright1984by the American Geophysical Union. Paper number 3W 1670. 0043-1397/84/003 W- 1670505.00 47 above, this problem was addressed by Mejia and Rousselle [1976]. However, as demonstratedin the first section of this paper and by Lane [1980, 1982], their model fails to perform as anticipated. Examination of the causes of that failure reveal the constraintsimposed by the disaggregation framework on the laggedcovariances of the disaggregated flows.The second section of the paper presents a class of disaggregation models which, within the disaggregation framework, can reproduce the covariances of the disaggregated flow vectors X, their co- variances with the upper level flows, and reasonable approxi- mationsto the lagged covariances of the Xt series. MEJI^ ^ND ROUSSELLE MODEL This sectiondiscusses, as does Lane [1980, pp. V-14, V-16, 1982], why the Mejia-Rousselle modification of the Valencia and Schaake disaggregation model generally fails to generate synthetic flows which have the desired or anticipated vari- ances and covariances. This illustrates the difficulties facedby an algorithm which attempts to generate disaggregated flow vectors with a specified lag one covariance matrix. First, some notation is necessary. The values of generated random variables such as the higher level annual or aggregate monthly flow vectorsZ, will be represented by upper caseletters,while the historical (ob- served) values will be represented by lowercase lettersz,; here t may refer to a year, season, or a month, depending on whether Z, representsan annual, seasonal,or monthly flow vector, respectively. Following the notation of Loucks et al. [1981, pp. 302-306], the generatedZ• will be an n x 1 vector repre- senting the higher level flows, while X, is the m x 1 vector of flows generated by the disaggregation procedure;x, are the historical values. All random variables are taken to have zero mean; the historical flow series are assumed to have had their averagesubtracted, perhapsafter some normalizing transfor- mation. Covariances such as E[XtXt T] (1) will represent the true variance and covariances of the gener- ated values, whereas 1 N E[x'x'T] - N - k ,• (x'x'r) (2)