WATER RESOURCES RESEARCH, VOL. 17, NO. $, PAGES 1261-1272, OCTOBER 1981 The Mathematical Structure of Rainfall Representations 1. A Review of the Stochastic Rainfall Models ED WAYMIRE Department of Mathematics, University of Mississippi, University, Mississippi 38677 VIJAY K. GUPTA Department of CivilEngineering, University of Mississippi, University, Mississippi 38677 This is the firstof a three-part series on the mathematical structure of rainfall models. Several impor- tant attempts at modeling rainfallare reviewed. Special attention is given to the mathematical structures that arise in the rainfalldescriptions. A general overview of the three-part series is given aspreface to this part. THE SCOPE AND THE ORGANIZATION OF THE SERIES The newly emerging areas of physically based statistical hy- drology have created a need for exploring new mathematical toolsapplicable to the analysis of hydrologic processes from a physical viewpoint. Some examples of these areas includethe theory of soluteand water transport in porous media, sedi- ment transport, the rainfall process, the rainfall-runoff proc- ess, and the streamflowprocess. Within the context of the rainfall process and other hydrologicprocesses 'driven by' rainfall inputs the developing role of the mathematical theory of point processes and randommeasures is noteworthy. Un- fortunatelythe mathematics literature doesnot seemto con- tain a current expositorytreatment of the theory of point processes which is immediately ac•ssible to an applied au- dience. Because of thevoidbetween thecurrent mathematical knowledge of the point process theory. The notion of a proba- bility generating functional (pgfl) as a natural generalization of the concept of a probabilitygenerating function(pgf) plays a central role in the theory of point processes. We have tried to unfold the theory of point processes in sucha way that its relevance toward various applications will be clear to the reader, although only the mathematical techniques are em- phasized in this part. The third section of this part concerns operations on point processes which are particularly germane to the studyof rainfall and other hydrologic processes derived from rainfall (e.g., streamflows, groundwater recharge, etc.). In the third part we demonstrate applications of the point process theory to the modeling of a variety of hydrologic processes. For example, some of the rainfallmodels reviewed in part 1 are castinto the point process mold. The readerwill developments of the point process theory and its practicality see that this setting increases the-analytic capabilities for a in hydrology, we consider itto be worthwhile that this area of further in-depth study of these processes. Inaddition to the mathematics be exposed to the hydrologic community with a rainfall models, the examples of streamflows and ground wa- demonstration of how it fits into the analysis of the structure of rainfall and rainfall based processes (e.g., streamflows). Al- though the title of this articleis 'The Mathematical Structure of Rainfall Representations,' the scope of this article is not confined to rainfall alone. This will be clear to the reader from the organization of the paper.This series is dividedinto three parts: (1) A review of the stochastic rainfallmodels, (2) a re- view of the theory of point processes, and (3) some appli- ter level fluctuations are usedto showhow the smoothing op- eration on point processes enables one to couplea determinis- tic response function with a statistical input description. This couplingprovides the completestochastic description of the outputprocess (in the form of its pgfi) asopposed to its second order structureonly. The necessity of requirements beyond the second order analysisstemsfrom the fact that typically thesehydrologicprocesses are non-Gaussian. The power and scope of themathematical techniques exposed in part2 to cations of the point process theoryto rainfall processes. In the first part we will present the key mathematical ideas modeling hydrologic processes should become more apparent (in our judgment) that have been introduced within the lastto the reader in this part. However, our intent is 0nly to dem- two decades or so inmodeling some of the empirically ob- onstrate an application of the mathematical ideas developed in part 2 to various hydrologic processes. No part of this paper servedfeaturesof rainfall. This presentation is certainly not completely comprehensive nor does itinclude references to all is all-exhaustive, but if it contributes to a basis for longer the papers that have been published on this subject. The limi- strides in the developmen t of hydrologic theory, then our goal tations of space and our '!halted competence preclude such an in writing this series will be accomplished. undertaking. For example, part 1does not go deeply into the INTRODUCTION meteorological literature dealing with precipitation. Similarly, the second-order timeseries analysis of rainfall is not re- Rainfall modeling represents anarea which cuts across the viewed. boundaries of hydrology. Meteorology, atmospheric physics, In the second part the theory ofpoint processes is unveiled. climatology, and hydrology represent some of the disciplines This part can beread independently of the first part. Once which encompass one aspect or another of the rainfall phe- again, inthis part we sketch out the main mathematical ideas nomenon. Inview of the breadth of scope occupied by rainfall which are central to the development of a solid workinganalysis, it would bepresumptuous onour part to attempt to review it entirely in one article. In selecting material for the Copyright ¸ 1981 by theAmerican Geophysical Union. review we were guided by a searchfor the most basic in- Paper number 1W0657. 0043-1397/81/001W-0657 $01.00 1261