WATER RESOURCES RESEARCH, VOL. 22, NO. 4, PAGES 531-542, APRIL 1986 Continuous-Time Versus Discrete-Time Point Process Models for Rainfall Occurrence Series EFI FOUFOULA-GEORGIOU x AND DENNIS P. LETTENMAIER Departmentof Civil Engineering,University of Washington,Seattle Several authors have had apparent successin applying continuous-time point processmodels to rainfall occurrencesequences. In this paper, it is shown that if rainfall occurrencesare interpretedas the eventsof a point process(and not as a censoredsample), the continuous-timepoint processmethodology and estimationprocedures are not directly applicablesince they fail to accountfor the time discreteness of the sample process.This is demonstratedanalytically by studying the effects of discretizationon selectedstatistical properties of a Poissonprocess, a Neyman-Scott process,and a renewal Cox process with Markovian intensity.In general,the study of rainfall occurrencesunder the continuous-timepoint processframework may result in misleadinginferences regarding clustering(dispersion), and conse- quently incorrectinterpretations of the underlyingrainfall generatingmechanisms. For example,daily rainfall occurrencestructures underdispersed relative to the Poissonprocess are usually overdispersed relativeto the Bernoulliprocess (the discrete-timeanalogue of the Poisson).These findings are confirmed by the statisticalanalysis of six daily rainfall recordsrepresentative of a range of U.S. climates,two of which are described in detail. 1. INTRODUCTION AND PROBLEM STATEMENT i.e., a stochasticprocess which is completely characterizedby The stochastic structure of daily rainfall occurrences has the position of its events, two interpretations of the sampled been extensively studied over the past two decades. The data are possible: (1) the occurrences represent all of the models suggested have evolved from the alternating renewal eventsof the point processand (2) the occurrencesrepresent a models [Green, 1964; Grace and Eagleson, 1966], to Poisson filtered sample of an underlying point processin which multi- models [Todorovic and Yevjevich, 1969; Duckstein et al., ple occurrencesduring a day are possible, but only one is 1972], Markov chains [Gabriel and Neumann, 1962; Todorovic recorded when one or more occur. If the first interpretation is and Woolhiser, 1974; Smith and Schreiber, 1973], discrete implemented, the event takes the meaning of a rainy day (see autoregressivemoving average (DARMA) models [Chang et Figure 1) and one has to deal with a discrete-timepoint pro- al., 1984], and finally, to point processmodels [Kavvas and cess,i.e., a point processin which events can occur only at Delleur, 1981; Smith and Karr, 1983]. This paper concentrates time marks integer multiples of the sampling interval. All pre- only on the point processmodeling approach. Other rainfall vious studies on point process modeling of daily rainfall occurrence models have been reviewed elsewhere [Waymire (exceptthe recent work of Rodri•iuez-lturbeet al. [1984]) have and Gupta, 1981a; Roldan and Woolhiser, 1982]. For the gen- implemented the first interpretation. It will be shown in this eral theory of point processesthe reader is referredto Cox and paper that this approach fails to account for the time dis- Lewis [1978], •inlar[1975], Lawrance [1972], and Daley and cretenessof the process. Vere-Jones [1972]. Waymire and Gupta [1981b, c] have pre- A related issuethat can presentserious difficulties is model sented a careful review of the theory of point processesand fitting. Under the first interpretation of rainfall occurrences, have illustrated their applicability to modeling rainfall and model parameters are estimated by fitting proceduresbased rainfall-driven hydrologicprocesses. on direct comparison of the empirical properties of the Rainfall is a continuous intermittent process,whose inten- discrete-time data with their theoretical continuous-time sity we denote as •(t). Rainfall measurementsrepresent cumu- counterparts. This approach, which has been used in several lative amounts over discrete time intervals such as minutes, previous studies [e.g., Kavvas and Delleur, 1981; Smith and hours, or days. Let {Y•(A)}, k = 1, 2, 3,... denote the discrete Karr, 1983; Ramirez-Rodriguez and Bras, 1982], will be shown to introduce severe estimation biases. This will be demon- sequenceof rainfall observations over an arbitrary time inter- val A. The continuous process •(t) is related to the discrete strated by studying the effects of discretization on selected process{ Y•(A)}by statistical properties of three commonly used point process models for daily rainfall occurrences: a Poisson process, a Neyman-Scott process,and a renewal Cox processwith Mar- Y•(A) - •(•) d• (1) -1 kovian intensity. where tk- tk-• = A is the time scale of measurement.Figure 2. STATISTICAL BACKGROUND AND TERMINOLOGY 1 illustrates this point: the continuous process•(t) is integrat- For the discussion which follows, it is necessaryto intro- ed over, say, daily time intervalsto give the sequenceof daily duce a few functions which describe the statistical properties data {Y•(A)},A= 1 day. of a point process.More details on these functionscan be In modeling daily rainfall occurrencesas a point process, found in statistical texts such as Cox and Lewis [1978] or in the work by Kavvas and Delleur 1-1981]. •Now at St. AnthonyFalls HydraulicLaboratory, Department of Let rn denote the rate of occurrence of a continuous-time Civil and Mineral Engineering,University of Minnesota, Minnea- polis. stationary point process,and F(x) denote the cumulative dis- tribution of the interarrival times. The log-survivor function, Copyright 1986 by the American GeophysicalUnion. In [R(x)], is defined as the logarithm of the probability of Paper number 5W4097. exceedance R(x)= 1--F(x). For a Poisson process In 0043-1397/86/005W-4097505.00 JR(x)] =--rnx. A concave log-survivor function indicates 531 532 FOUFOULA-GEORGIOUAND LETTENMAIER:MODELS FOR RAINFALL OCCURRENCESERIES RAINFALL mulative distribution of the discrete probability mass function I NTENSI TY (mm/sec) (pmf) of the interarrival times; N•, will be the number of events occurring within k time units; V• will be the variance of the countingprocess {Nk), etc. Similarly, the discrete-timeana- logueof the conditionalintensity function is a sequence{hk) 180 360 540 720 t(sec) ../ of conditional probabilities of occurrence,where h•,is defined as • {Yk(A)},A=,hour hn= e(zn = 1 I Zo- 1) (3) (mm) 1111,..1..... ...... .... Note that hn are probabilities, whereas h(t) is a probability 12 18 24 30 36 42 48 ..,/ t (hours) density. One major differencebetween continuous-timeand discrete-time function definitions is the spectrum of counts, RAINFALL DEPTH (mm) sincethe differentialcounting process {ANt} is not definedfor a discrete-timepoint process.The spectrum of counts is in .I, that case defined as the Fourier transform of the auto- I 2 3 4 5 6 7 8 9 IO II 12 t (days) covariancesequence (ck) of the binarytime series (Zn). Help- ful remarks on the spectralanalysis of continuous-timeversus Fig. 1. Continuous rainfall process •(t) and discrete hourly and discrete-timepoint processescan be found in Lewis [1970]. dailyrainfall sequences { Y•(A)). The reader is referred to Guttorp [1985] for more rigorous definitions of the statistical properties of a discrete-timepoint overdispersion(i.e., a tendencyfor clusteringof rainfall events) process. relative to Poisson, whereas a convex log-survivor function is indicative of a processunderdispersed relative to the Poisson. 3. REVIEW OF CONTINUOUS-TIME POINT PROCESS MODELS The countingprocess, {Nt), of a point processis definedas the number of eventsin (0, Fl. The variance of Nt is a continu- A Poisson cluster process,which has become known as the ous function called the variance-time curve, V(t)= Var (Nt). Neyman-Scott (N-S) process, was developed by Neyrnan When divided by the mean number of eventsin (0, t], M(t), a [1939] for entomology and bacteriology population growth function called the index of dispersion,I(t) = V(t)/M(t) results. modeling. Subsequently, it was used by Neyrnan and Scott For a Poisson process, M(t)--V(t)=rnt, and therefore [1958] to model the spatial distribution of galaxies,and later I(t) = 1, ¾ t. An index of dispersionI(t)> 1 (< 1) indicates by LeCarn [1961] to model the areal distribution of rainfall. overdispersion(underdispersion) relative to the Poisson pro- Based on the work of LeCam, Kavvas and Delleur [1981] cess.This property is analogousto the coefficientof variation applied a Neyman-Scott model to the rainfall occurrenceson cvfor the interarrivaltimes, where similarly, cv > 1 (< 1).indi- the time continuum, and found that such a processappeared cates overdispersion(underdispersion) relative to the Poisson to describe the clustering of daily rainfall occurrencesin In- diana. processfor which c,= 1. The spectrumof the countingpro- cessg+(c0), is definedas the Fourier transformof the covari- A N-S processis a two-level process.At the primary level, ancedensity of the differentialcounting process •ANt), where the rainfall generatingmechanisms (RGM) occur accordingto ANt is defined as the number of eventsin (t, t q-At]; i.e., a Poisson processwith rate of occurrenceho (i.e., mean in- Nt +at- Nt- For a Poissonprocess g +(co)= rn/•, and the nor- terarrival time 1/ho).Each RGM (also called a clustercenter) malized spectrum of counts, defined as g+'(co)= •rg+(co)/rn, gives rise to a group of rainfall events and each of these takes on the constant