WATER RESOURCES RESEARCH, VOL. 21, NO. 4, PAGES 545-553, APRIL 1985 A Stochastic Streamflow Model Based on Physical Principles RoY W. Koc• Department of Civil Engineering, Portland State University, Oregon Based on hydrologic arguments, a stochastic model for discretized streamflow from rainfall is pro- posed in the form of an averaged filtered Poisson process (FPP). Usinga two-component description with excess precipitation and drainage as inputs to the surface and subsurface flow systems, simplified models are presented for the physical processes. The mean,variance, and covariance functionof dis- cretized streamflow are computed in terms of the firstand second moments of excess precipitation and drainage whichare derived analytically. The behavior of the stochastic properties of streamflow is evaluated for various soil types, basintravel times• and effective initial soil water contents. Where thereis an intuitive. feeling for the interaction of physical characteristics and stochastic properties, such as the mean, the modelappears to produce qualitatively correct results. Prediction of stochastic properties is limited by the selection of an effective initial soil moisturevalue. INTRODUCTION Although stochastic approaches, in particular the gener- ation of syntheticsequences of streamflow,have enhancedthe designand operation of water resource systems, the limited amount of data available for stochastic model development makes selection of the appropriate model difficult and results in a reasonably high degree of parameteruncertainty. In addi- tion, the statistical nature of the existingmethods limit their appncauon to ungagea areas. i ne .:nvesuganon aescnoea herein has been undertakento identify the proper form of a stochastic model of streamflow based on the relevant climatic and hydrologic processes. Given the model form expressed in hydrologicterms, an effort is made to incorporatephysical descriptions of these processes. The investigation is restricted to an evaluation of the extent to which simple, physically basedmodelscan be incorporated in sucha description, and the qualitativeand quantitative relationships between the sto- chastic properties of streamflow sequences and the parameters describing the physical hydrologic system. The development is restrictedto streamflowproducedfrom rainfall. Many differentstochastic modelshave been applied in the description of streamflow time series for the purpose of gener- ation of synthetic sequences. Examples includethe autoregres- sive moving average (ARMA) models [e.g., Box and Jenkins, 1976; Salas et al., 1980], fractional Gaussian noise (FGN) introduced to hydrology by Mandlebrot and Wallis [1968], .the broken line process presented by Mejia et al. [1972], the shot noisemodel presented by Weiss [1977], and the shifting level model developed by Boesand Salas [1978]. There have beenefforts to relatethe properties and parameters of some of thesemodels to the physicalhydrologicsystem[e.g., Yevje- vich, 1963; Fiering, 1967; Moss, 1972; Weiss, 1977; and Salas andSmith, 1981]; however, the correspondence hasbeen large- ly conceptual in nature. In addition to these conceptual relationships, therehas been some limited work in the area of direct derivation of sto- chastic properties of streamflow: the hydrodynamical level de- scribed by Klernes [1978]. Ea•lleson [1972] derived the distri- bution of flood peaks based on a relatively simple random precipitationmodel and physical description of the processes including a kinematic flood routing model. As a result, the Copyright 1985by the American Geophysical Union. Paper number 5W0042. 0043-1397/85/005W-0042505.00 545 distribution is a functibn of the parameters describing the physical system. Sometime later Eagleson [1978a, b, c, d, e,f, g) also applied this approach to derive the distribution and mean value of annual water yield. In this derivation each of the relevant processes was modeledphysically and the climate was considered stochastic. Each of the models was relatively simple and still many approximations were necessary along the way to arrive at a final analytical result. In addition, there estimation. Even so, reasonably good agreement with ob- served runoff was obtained.Sincethe initial appi'oach of look- ing at the long-term water balance was taken, however, the changes in storagein the system were assumed negligible over the period and thus any time dependence was not accounted for.Theworkof Eagleson has been based on thetraditional descriptionof surfaceflow generation via infiltration excess usuallytermed the Horton mechanism. Using a Monte Carlo simulation approach, Freeze [1979] described hillslope pro- cesses by partial differential equations solved by finite differ- encetechniques. In this work, runoff generated by subsurface flow and saturation overland flow were desi:ribed but, using this technique, no general relations weredeveloped. A STOCHASTIC STREAMFLOW MODEL To develop a stochastic model of streamflow from rainfall, the deriveddistributionapproachis adoptedwith the climate assumed to contain all of the uncertainty (randomness), while the terrestrial hydrologicprocesses transforming rainfall into streamflow are assumed to be completely deterministic. A rela- tively simpleconceptualization of the watershed is usedto describe the hydrologic processes which producestreamflow from precipitation. First, only the precipitation excess or Horton mechanism[e.g., Chorley, 1978] is assumed in the generation of overland flow with consideration for surface pondingand post ponding infiltration capacity. The watershed is represented as a two component system; one describing surface and the other subsurface runoff. Thesecomponents are gimply assumedto be additive to produce streamflow. The surfaceflow component has excess precipitation as its input while the subsurface component is driven by recharge out of the root zone due to infiltration and drainage. The streamflow process can be described on a continuous time basis as the sum of surface and subsurface flow compo- nents. The inputs to these components, excess precipitation and recharge,are random due to their direct dependence on