WATER RESOURCES RESEARCH, VOL. 22, NO. 4, PAGES 531-542, APRIL 1986

Continuous-Time Versus Discrete-Time 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 value of 1. Another important parame- groupsis called a cluster.Within eachcluster, the eventsoccur ter of a continuous-timestationary point processis the con- independentlyand their occurrenceis completelyspecified by the distribution of the number of events and the distribution ditional intensityfunction, h(t), definedas of their positions relative to their cluster center. Kavvas and Delleur [1981] assumed a geometric distribution, with param- h(t)= P(dNt = 1dtIN{0} = 1) (2) eter p, for the number of rainfall events in a cluster and an exponential distribution, with parameter 0, for the distancesof [Cox and Lewis, 1978, p. 73], where dNt is the limit of ANt as events from their cluster centers. For these distributions, the At--• 0, and N{0) denotes"event at time0." The interpretation observed processhas a rate of occurrencern = ho/p. To cope of h(t)dtis the probabilityof an eventat time t, givenan event with the long-term trends and within-year seasonality Kavvas at time 0. For a Poissonprocess h(t) = rn,¾ t. Inferencesabout and Delleur applied a homogenization scheme to the data. clusteringof a point processcan be made from the way h(t) This schemeconsisted of fitting a time-varying function 2(0 to approachesthe constantintensity rn of the process.For more the mean rate of occurrence and rescaling the original time information on all these functions see, for example, Cox and increments of one day, At, to time-varying increments Lewis [1978, chapter4]. For the spectralanalysis of a point Ar = 2(t)At. In this way a (i.e., a process processconsult Bartlett [1963]. independent of the time origin) was obtained to which a N-S A discrete-timepoint processis a processin which events model was fitted. However, as Kavvas and Delleur comment, can only occurat time valuesk = 1, 2, 3,.... Suchan oc- the fitted model cannot be usedfor simulation of daily rainfall currenceprocess can be viewedequivalently as a sequenceof occurrencesequences, since the inversetransformation, i.e., the binaryrandom variables (Z•,), whereZ•, takeson the values1 deterministic transformation that will give the nonhomoge- and 0 dependingupon whetheran eventdid or did not occur neous processfrom the homogeneousone is not valid for a at time k [Lewis, 1970]. Let rn' denotethe probabilityof oc- non-Poissonprocess [•'inlar, 1975,chapter 4]. currenceof an event at an arbitrary time value k. All the Smith and Karr [1983] introduced another point process, statisticalfunctions discussed previously can be extendedin a the renewal Cox processwith Markovian intensity (RCM pro- straightforwardway to the correspondingfunctions of a cess),which belongs to the class of doubly stochasticPoisson discrete-timepoint process.For example,F(x) will be the cu- processes(also called Cox processes).In an RCM process,the FOUFOULA-GEORGIOUAND LETTENMAIER' MODELS FOR RAINFALL OCCURRENCE SERIES 533

TABLE 1. StatisticalProperties of a Poissonand a

Poisson Bernoulli

Interarrivaltimes, X i f (x) = J.e-x,, J.> 0 p(x)= p(1- p)"-x 0 < p < 1 E(X) = 1/• E(X) = 1/p Var (X) = 1//L2 Var (X) = (1 - p)/p2 co= 1 co=(1-P)•/•< 1 2-p Cs= 2 Cs-(1 - p)•/:z> 2

Numberofevents, N, p(N,=k)-(•'t)•e-•'klß p(Nr = k)= (k)•,r p(1 - p)r-•' E(N,)= J.t E(N•)= pk Var (N,)= ;•t Var (Nk)= iv(1- p)k Conditionalintensity h(t)= 2 hk= p function Log survivorfunction In [R(x)] = -J.x In [R(x)] = -In (1 - p)x Variancetime curve V(t) = •t V• = p(1 - p)k Index of dispersion l(t) = 1 ¾t I• = 1-p < 1 ¾k function Spectrumof counts g +(•o)= •l/r• •o _>0 g +(•o) = p(1 - p)/n •o _>0 Normalizedspectrum • +'(•o)= 1 •o _>0 g +'(•o)= 1 - p < 1 •o _>0 of counts

Here •l, rate of occurrenceß p, probabilityof success'co, coefficientof variation;and %, skewness coefficient.The propertiesof the discretizedPoisson process can be obtainedfrom the right-handcolumn by simplysubstituting the valueof •l' = 1 - e-x for the valueof p. rate of occurrence•.(u) alternates between two states, one zero fitting the N-S model are due to the inappropriate use of and the otherpositive. Retaining the nomenclatureof Smith continuous-time point process models for daily rainfall oc- and Karr, let ax and a2 be the parameters oftthe exponential currences,rather than shortcomingsin the parameter fitting sojourndistributions of the intensity,•(u) in the states1 (dry) procedure. In that respect,even the maximum likelihood esti- and 2 (wet),respectively. This simply amounts to 1/axbeing mation (MLE) methods recently developedby Smith and Karr the mean duration of a dry period and 1/a2 the mean duration [1985] may causedifficulties when usedfor modeling discrete of a wetperiod. During periods When the intensity .is zero, no (e.g., daily) rainfall occurrences,since they use the likelihood eventscan occur; during periods with positive , intensity, events function of the continuous-time process. occur according to a Poisson process with rate of occurrence 9•,and the sequence of states •isited form a Markovchain. 4. INFERENCES ABOUT CLUSTERING OF DAILY RAINFALL OCCURRENCES

This processis a renewal one (i.e., interarrival times, are inde- pendent) and Smith and Karr found it an adequate model of In the theory of continuous-timepoint processes,the exis- thesummer season (July to October)daily rainfall occurrences tence and type of clusteringin an occurrenceprocess is often in the Potomac River basin. It should be noted that both the .. studied by comparing the processto an independent Poisson Neyman-Scott process and the renewal Cox process with process with the same rate of occurrence. A clustered point Markovian intensityare continuous-timepoint processesand process can be either overdispersed(i.e., more random oc- clustered(overdispersed)relative to the Poisson process. currences) or underdispersed (i.e., more regular occurrences) The problemof fitting and validatingdaily rainfalloc- relative to a Poissonprocess. However, it is important that the currence model s has not received as much attention as the clustering of a discrete-time point process,such as the daily specificationof the model form. Kavvas and Dellcur [1981] rainfall occurrenceprocess, be compared with the independent used an iterative least squaresestimation procedure, utilizing Bernoulli process(the discrete-time analogue of the Poisson the estimated normalized spectrum of counts and the esti- process).Most previous studieshave treated the daily rainfall matedlog-sUrv'ivor function, for the fitting of theN-S model occurrence process as a continuous-time point process and to daily rainfall occurrencesfrom Indiana. Problems were en- have modeled it in a continuous-time point process frame- countered in this procedure, however, since estimates of the work; i.e., clustering of daily rainfall has been inferred by implied variance of v, the number of events in a cluster, can comparingthe empiricalproperties of the observedoccurrence becomenegati:ve. To avoidthis probleman iterativeesti- seriesto the theoreticalproperties of the Poissonprocess. In mation procedureis requiredwhich involvesa somewhatarbi- this sectionit is shown that such an approach can result in trary assumptionabout the valueof E(v•)/E(v).The problem incorrectinferences about clusteringof the underlyingprocess. of negative variance for the number of eventsin a cluster was In particular, it is shown that if indeed the daily rainfall oc- also encounteredby Ramirez-Rodriguezand Bras [1982], who currenceswere an independentprocess, i.e., a Bernoulli pro- fitted .the N-S model to the daily rainfall occurrencesof cess,if modeledas a continuous-timepoint processthey would Denver, Colorado. Both of the above studies assessedthe va- be interpreted as underdispersed(relative to the Poisson pro- lidity of the fitted model by the agreement of the theoretical cess).On the other hand, daily rainfall occurrence series un- and empirical spectrum of counts and log-survivor function, derdispersed relative to the Poisson process are, in fact, all which were the samefunctions used for the fitting. However, shown to be overdispersed relative to Bernoulli. To illustrate the variancetime curvesinferred from the fitted parametersin these points, the statistical properties of a Bernoulli process both studies fail to reproduce the empirical ones, often dra- are first studied. matically. Consider a sequenceof independentrepeated trials with two We believethat the problemsencountered by both Kavvas possible outcomes, say, successand failure. Let p denote the and Dellcur [1981] and Ramirez-Rodriguezand Bras [1982] in probability of successat each trial and Nr the number of 534 FOUFOULA-GEORGIOUAND LETTENMAIER'MODELS FOR RAINFALL OCCURRENCE SERIES

2.0

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1oo 2 4 6 ,'o ,. ,. ,'o ;o ','o .'o INTERVRLLENGTH FREQUENCYFRCTOR

2.0 70.0

60.0

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.40.0

•.•o.o •20.o

10.0

i i i 0 0 10 20 30 40 50 60 90 80 90 100 øo ,'o ;o ,oo INTERVRL LENOTM INTERVRL LœNGTH Fig.2. Effectsofdiscretization ona Ncyman-Scott (N-S)process with parameters ho = 0.23, p = 0.67,and 0 = 0.75. Solidcurve corresponds to the continuous-time process and the broken curve to thediscrete-time process. successesin r trials. Then N, has a binomial probability distri- Considera sequenceof daily rainfall occurrences.If a Ber- bution noulli processis fit to the series,the maximumlikelihood estimate(MLE) of its probabilityof success,p, is i6= No/N, whereNo is the numberof dayson whichrainfall is recorded P(N,=k)=(•)pk(1-p)'-kk=0,1,2.... (4) and N is the total number of days. Similarly, if a Poisson and the number of trials between the nth and (n + 1)st success, processis fit (underthe interpretationthat rainfallevents rep- resent all of the eventsof the point process),the MLE of the Xn, has a geometricdistribution rate of occurrence)•, is • = No/N. Thus/5 = •. Notice,how- P(X. = k)= p(1- p)•'-' k = 1, 2.... (5) ever, from Table 1 how different the other propertiesof the two processesare. In particular,the Bernoulliprocess has a for all n. Note that for the daily rainfall occurrences,the coefficient of variation of interarrival times and an index of number of trials is interpretedas the number of discrete-time dispersionfunction always less than one,which imply under- units,i.e., days.As was mentionedearlier, in the discrete-time dispersionrelative to Poisson.This meansthat inferences point processterminology a successcorresponds to the oc- about over- and underdispersionof the daily rainfall oc- currence of an event (i.e., a rainy day), N• to the counting currenceswould be differentdepending on whetherthe empiri- process,that is the numberof eventsin {0..... r}, and X, to cal functionsof the processwere compared to thoseof a Pois- the time between events. son or to those of a Bernoulli process.This is a simplebut The Bernoulli processis the discrete-time analogue of the importantobservation and hasimmediate consequences in the Poissonprocess, in the sensethat it is characterizedby inde- interpretationof the statisticalfunctions of the daily rainfall pendent intervals and independent counting increments.This occurrenceprocess. We suggestthat a discrete-timepoint pro- lack of memory property is the result of the geometricdistri- cessmodel, such as daily rainfall occurrences,should be com- bution for the times between events, analogously to the ex- pared with the discrete-timeindependent Bernoulli process ponential distribution for the Poisson (see Feller [1968, p. (andnot with the continuous-timePoisson) if inferencesabout 329] for a proof). The statisticalproperties (i.e., mean, vari- independenceand clusteringare to be made. ance, and higher moments) of the geometric and binomial distributionsare well known (see,for example,Parzen [1967]). 5. EFFECTSOF DISCRETIZATIONON CONTINUOUS-TIME For this work, some additional properties of the Bernoulli counting processare of interest, such as the spectrum of In derivinga discrete-timeoccurrence series (binary series) counts,log survivorfunction, and variancetime curves.These from a continuous-timepoint process,two operationsare per- propertiescan be easilycomputed (see, for example,Foufoula- formed: discretizationand clipping.These operations are de- Georgiou[1985]) and are summarizedin Table 1 of this paper, fined as follows: discretization is the grouping of events oc- togetherwith the correspondingproperties of a Poissonpro- curring within intervals of length equal to the time scaleof cesswith rate of occurrence4. The significanceof comparing measurement,and clippingis the assignmentof the value of thesestatistical properties is illustratedbelow. zero (or one) to eachinterval dependingon whetheror not at FOUFOULA-GEORGIOUAND LETTENMAIER:MODELS FOR RAINFALL OCCURRENCE SERIES 535

5.0

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0 0 / i i 0 o Zo do do' do ioo INTERVIaL LENOTH

90-0 3.5

80.0 3.0 o•70.0 •o60.0 •so.o

•40.0 •30.0 zo .o

lo .o

o o øo ,'o z2' ;o do .'o 4o' ',oo

Fig. 3. Effectsof discretizationon a renewalCox process with Markovian intensity (RCM) withparameters at - 0.2, ae= 0.1,and 2 = 0.5.Solid curve corresponds to the continuous-time process and the broken curve to thediscrete-time process. least one event occurred within the interval. In this section, better illustrate this point, consider a Poisson process with the effects of discretization and clipping (usually referred to rate of occurrence;• = 0.5 (mean interarrival time of 2 days). If only as discretization) of the Poisson process,the Neyman- this processis used for simulation of daily rainfall occurrence, Scott process,and the renewal Cox processwith MarkovJan the synthetic arrival process will have a rate of occurrence intensity,all of which have previouslybeen usedfor modeling •'= 1- exp (-0.5)= 0.393 (mean interarrival time of 2.5 daily rainfall occurrences,are studied. days) and a coefficientof variation equal to 0.626 < 1. Fur- ther, if the simulated occurrencesequences is analyzed under 5.1. Poisson Process the continuous-time point processframework, they will indi- Let F(x) denote the cumulative distribution function of the cate a processunderdispersed relative to the Poissonprocess, exponentialpdf of the interarrival times in a Poissonprocess. where, in fact, it is an independent Bernoulli process.In the The abovediscretization scheme is equivalentto replacingthe above context, the improper use of the continuous-timepoint continuous exponential distribution,f(x), with a discretized processframework for the analysisand synthesisof daily rain- one,p(k), such that fall occurrencesbecomes apparent. p(k)= F(k) - F(k - 1) = e-(•-')•(1 - e-•) k = 1, 2.... (6) 5.2. Neyman-Scott Process Note that the resulting discrete pmf, p(k), is geometric with The statistical properties of intervals and counts of a parameter continuous-timeNeyman-Scott processcan be found in the work by Kavvas and Delleur [1981], while the corresponding ,•'= 1 -e -;t (7) propertiesof the resultingprocess after discretizationare given implying that the discretizedPoisson processis a Bernoulli in Guttorp [1985]. It is interestingto note that although the processwith a probability of occurrenceequal to ,•', a value rate of occurrenceof a Neyman-Scott processis m = ho/p, the alwaysless than ,L All the Otherproperties of the discretized probabilityof occurrenceof an eventin the discretizedprocess Poissonprocess can thereforebe obtainedby substitutingthe is [Guttorp, 1985] value of ,•' for p in the right-hand column of Table 1. It is seen,for example,that the discretizedPoisson process has a coefficientof variationc• = (1- •,)•/2 < 1, an indexof m'=l--e-no [1--1_(l_p)e p 0 (8) dispersion I•--1- g'< 1, and a normalized spectrum of Observe that m' is a function not only of the rate, of oc- counts g+'(ro)= 1- •'< 1 ¾ co. This observation illustrates currenceof the clustercenters (ho) and the clustersize (p), but that if a daily rainfall occurrenceprocess is modeled as a also of the dispersionof eventswithin each cluster(0). Poisson process,and if subsequentlythis Poisson processis Consider a Neyman-Scott process with parameters ho = usedfor simulationof rainfall,the resultingdiscrete-time rain- 0.23, p -- 0.67, and 0 -- 0.75. These parameterscorrespond to a fall occurrenceseries will haveproperties substantially differ- clustered(overdispersed) occurrence process and are approxi- ent from those of the inferred continuous-time process. To mately equal to thoseRamirez-Rodriguez and Bras [1982] 536 FOUFOULA-GEORGIOUANDLETTENMAIER.' MODELS FOR RAINFALL OCCURRENCE SERIES

TABLE 2. Information on the Six Daily Rainfall StationsAnalyzed

Station Station Identifi- Years Elevation, Observation Location cation Analyzed Latitude Longitude ft Time

Snoqualmie Falls 45-7773 1948-1977 47033' 121051' 440 5 P.M. Roosevelt 02-7281 1948-1977 33040 ' 111009 ' 2005 7 A.M. Austin 41-0428 1948-1977 30018 ' 97042 ' 597 midnight Miami 08-5663 1949-1978 25ø48 ' 80016 ' 12 midnight Philadelphia 36-6889 1948-1977 39053' 75015' 10 midnight Denver 05-2220 1949-1978 39ø46 ' 104052 ' 5286 midnight

One foot equals 30.48 cm.

found for daily rainfall occurrencesat Denver, Colorado. This Bernoulli process,it seems to be about as clustered as the arrival processhas a mean rate of occurrencem = 0.34 (mean continuousRCM process. interarrival time of approximately 3 days), while the arrival processresulting after discretization has a mean rate of oc- 6. STATISTICALANALYSIS OF DAILY RAINFALL OCCURRENCES currencem'= 0.260, that is, a mean interarrival time of ap- In the preceding sections, it has been shown that the proximately 4 days. Figure 2 shows the comparisonof the common practiceof testingthe independenceand degreeof conditionalinte•sity function, normalized spectrum of counts, clusteringof daily rainfall occurrencesby studyingdeviations variance time curve, and index of dispersionfor the continu- from a Poissonprocess can be highly misleading.The extent ous and discretizedN-S process.It should be noted that the of the differences that can result are demonstrated in this sec- differentlevels in the conditionalintensity function and spec- tion using daily rainfall occurrencesfrom SnoqualmieFalls, trum of counts are the result of the different rate of occurrence Washington,and Miami, Florida. A completestatistical analy- of the two processesand do not affect the inferencesabout sis of four other daily rainfall structures(from Arizona, Texas, clustering,which only dependson the rate of decay of these Colorado, and Pennsylvania)is given in the work by functions. These two functions, together with the variance Foufoula-Georgiou[1985]. (All six rainfall stations used for the time curve and index of dispersionfunction, indicate that the analysesabove are given in Table2 of thepresent papei'.) resulting discrete-timearrival processis less clusteredwhen Table 3 shows the mean, standard deviation, coefficientof comparedwith the continuous-timeprocess it originatedfrom. variation, and skewnesscoefficient of the interarrival times of Also note that severeestimation biases are expectedto resultif the daily rainfall occurrencesfor Snoqualmie Falls and the properties of the continuous(and not discretized)N-S Miami. It is observed that the coefficient of variation is not processare usedfor the fitting. always greater than one, and this was the casefor some of the other stations analyzed as well. In particular, the winter 5.3. RenewalCox ProcessWith Markovian Intensity (RCM Process) TABLE 3. of Interarrival Times The statisticalproperties of the RCM processare given in the work by Smithand Karr [1983], while the corresponding Number propertiesof the resultingprocess after discretizationare given Month • s,, co cs of Events co' in the work by Guttorp[1985]. The parametersselected to SnoqualmieFalls illustrate the effectsof discretizationon an RCM processare Jan. 1.40 1.17 0.84 3.82 667 0.53 a• = 0.2, a2 = 0.1, and 2 = 0.5. These parameterscorrespond Feb. 1.50 1.33 0.88 3.93 557 0.58 Mar. 1.54 1.55 1.00 5.54 607 0.60 to a daily rainfall occurrenceprocess in which the dry periods, Apr. 1.72 1.59 0.92 3.06 530 0.65 of an averageduration of 5 days (1/ax), are followed by wet May 2.21 2.50 1.13 3.19 429 0.74 periods,of an averageduration of 10 days (l/a2); during wet June 2.69 4.35 1.62 4.94 375 0.79 periodsrainfall eventsoccur on the averageevery 2 days(1/,•). July 4.26 6.08 1.43 2.91 205 0.87 The mean rate of occurrence of the continuous-time RCM Aug. 3.32 4.23 1.27 2.26 260 0.84 Sept. 2.71 3.76 1.39 4.27 332 0.79 processwith the above parametersis m = 0.333, while the rate Oct. 1.75 1.67 0.95 3.08 499 0.66 of occurrenceof the discretized process ism' = 0.264.Figure 3 Nov. 1.44 1.23 0.85 4.22 613 0.55 showsthe comparisonof the conditionalintensity function Dec. 1.34 0.98 0.73 4.17 694 0.50

(CIF), normalized spectrum of counts, variance time curve, Miami and index of dispersionfor the continuous and discretized Jan. 4.64 4.89 1.05 2.24 197 0.89 RCM processes.It is important to observethat while the CIF Feb. 6.16 6.37 1.03 1.63 131 0.92 of the continuous-timeprocess decreases monotonically to the Mar. 5.80 6.20 1.07 1.87 162 0191 intensity (this is true for all RCM processes'see smith and Apr. 4.99 5.75 1.15 1.96 174 0.89 May 2.26 2.61 1.16 3.33 375 0.75 Karr [1983]), the CIF of the discretizedprocess increases for June 1.99 1.97 0.99 3.25 444 0.71 lags up to 3 days and then startsdecreasing to the intensityof July 2.16 1.92 0.89 2.48 439 0.73 the processm'. This implies that the discretizedRCM process Aug. 1.83 i.47 0.80 2.40 502 0.67 has an autocorrelationfunction rk, which increasesup to lag 3 Sept. 1.84 1.62 0.88 2.64 486 0.68 Oct. 2.65 3.07 1.16 3.24 366 0.79 and then startsdecreasing. Such an autocorrelationfunction is Nov. 4.74 5.13 1.08 !.99 197 0.89 highly atypical for rainfall occurrence series and without Dec. 4.91 5.52 1.12 2.16 195 0.89 physicalbasis. Notice also from Figure 3 that the discrete-time occurrenceprocess is more clustered than the continuous-time Herei, samplemean; s,,, sample standard deviation' co, sample coefficientof variation' cs, sample coefficientof skewhess;and co', onewhen both are comparedwith the Poissonprocess. How- coefficientof variation of the Bernoulli processwith the same prob- ever, when the discretizedRCM processis comparedwith a ability of occurrenceas the rainfall series. FOUFOULA-GEORGIOU AND LETTENMAIER: MODELS FOR RAINFALL OCCURRENCE SERIES 537

I II III IV

2.5 0 80

-1.0 70 2.0 -• .0 5O 1.5 -3.0 4O JAN 1.0 ...... -4.0 -5.0 '5ø -6.0...... 10 o s'o15o 1;o •$o •;o •$o•;o 40o -•.o 1 • • lO øo 40 80 120 160 200 o 40 80 120 16o 200

2.5 1.4

140I 1.5 ' 120[

80 FEB 1.0 ......

ß 40 '1os'o1•o 1•o 2•o 2;0 3•o 3;0 400 '% 2 4 6 8 10 12 O0 40 80 120 160 200

.2

1.5 -3. 60 MAR 1.0 ...... -4.0 ß t

ß5 -6. ß 20 ß

o -7 o 0 50 100 150 200 250 300 350 400 0 2 4 6 8 10 12 14 16 18 20 0 40 80 120 160 200 0 40 80 120 160 200

2.5 0

-! .0 2.0

1.5 -3 .o APR -5 .o .5 -6 .o !.oo ...... -'7-4 .o o 50 lOO 15o 200 250 300 350 400 o 2 4 6 8 lO 12 14 0 40 80 120 160 20o o• ;o •'o •o 1•o 2oo

2.5 0 250 '.5 -1.o [ .o -2.0 MAY 1.5 -3.0 150[ 1.0 ...... -4.0 100[

o s'o15o 2;0 25o 2;0 35o3;0 400 -7.o 1'8 40 80 120 160 200 • 80 1EO 160 200

2.5

-1 • 300] 4.o 2.0 -2 250i 3.03.5 1.5 -3 2.5 2.0 JUNE 1 .o ...... -4 -5

-6 ' ß 500I ø0• 0 50 100 150 200 250 300 350 400 0 5 10 15 20 25 30 35 40 45 0 4O 8O 120 160 20O 0 40 8O 120 160 2O0

Fig. 4. Statistical properties of intervals and counts for daily rainfall occurrencesat Snoqualmie Falls, Washington. I, normalized spectrum of counts versus frequency factor; II, log-survivor function versus interarrival time (days); III, variance of countsversus interval length (days); and IV, index of dispersionversus interval length (days). months (October-February) for SnoqualmieFalls, the summer example, Table 3 shows that the coefficient of variation of months (May and June) for Roosevelt, the summer months daily rainfall interarrival times is always greater than that of (June-September) for Miami, and most of the months the Bernoulli process (the latter estimated as cv'=(1 (January-April, June, July, November, and December) for -1/•)•/2). It should be noted that for such structuresthe Philadelphia have a coefficientof variation lessthan one. Ob- discretizedNeyman-Scott or RCM processesmay be appro- serve that this underdispersionis consistentlyshown in the priate models, since both can admit coefficientof variations spectrumof counts,log survivor function, variance time curve, less than one. and index of dispersionfor these months and stations as well In referenceto the spectrumof counts (seeFigures 4 and 5), (seeFigures 4 and 5). Therefore for thesemonths comparison the following should be noted. For a continuous-timepoint with the Poisson processwould conclude that there is no processwhere theoretically,at least, events can occur arbi- clusteringin rainfall and that rainfall eventsoccur in a pattern trarily closeto each other, the spectrumof counts extendsto more regular than that of an independentPoisson process. In to- o•. For the daily rainfall occurrences,however, events fact, none of the available models would be able to accommo- cannot occur closerthan one day apart and this introducesa date such structures, since both the Neyman-Scott and re- cutoff frequency(Nyquist frequency)ton- •t, or equivalently, newal Cox processwith Markovian intensityhave a coefficient fN - 1/2 days-•. The valueplotted on the abscissaof the spec- of variation greater than one. Previous studieshave suggested trum of counts plots is called the frequencyfactor and is de- that deterministic explanation for such regular occurrences fined asj - toT/2•t, where T is the total length of observation. should be sought. However, under the suggestedapproach, all Therefore the frequency factor correspondingto the Nyquist these rainfall seriesappear to have a clusteredstructure. For frequency is JN = T/2, and this is the maximum value over 538 FOUFOULA-GEORGIOU AND LETTENMAIER: MODELS FOR RAINFALL OCCURRENCE SERIES

I II III IV

JULY

2.5 -1.øo 80 .0 2.0 70

1.$ -2.0 "•1 60 AUG -4-3. ! 5040 '51. • -6 10 -'7 0 '5øo5'0,3o,;o 23o 2;o 33o •o .oo ß 0 5 10 15 20 25 0 4O 8O 120 160 2OO 0 4O 8O

SEP 2.5 0•ß 3.5 ß5 -6.0 2040 1.00o 50 ...... 100150 2OO 25O 3OO 35O 4OO -•.o[-4.-? '0o0 5' 10' •'15% 20 ' 2•. 30 35'40 60oo0 0 ....•40 80 120 160 20O 10 40 80 120 160 2OO

2.5 -1.oo •so140[I •.•1. 2.0 120[ -2.0 lOO[ 1.o ...... OCT 1.5 -3.0 • -5.0 40 ß5 -6.0 20 ' 00 5O 100150 20O 25O 30O 35O 4OO -'7.O 2 4 6 8 10 12 0 4O 8O 120 160 2OO 0 4O 8O 120 160 2OO

-1 .o 40 2 .o 35 .8

1.5 -2-3.0.o 3025 ''76 NOV 1.0 ...... -4.0 I • 15 10 ß5 -6.0 00 5O 100150 2OO 25O 300 35O 4OO -'7.O 2 4 6 8 10 12 0 4O 8O 120 160 2OO 0 4O 8O 120 160 2OO

18 . 2.0 -1.0 16 . -2.0 14 1.5 -3.0 12 DEC

o •T ,•••.oo '700 1 2 3 4 5 6 '7 8 9 10 0...... 40 80 120 160 200 0 40 80 120 160 200

Fig. 4. (continued) which the spectrum of counts should be computed. Guttorp Poisson process,indicating underdispersionrelative to Pois- and Thompson [1983] discuss aliasing of the spectrum of son. However, all thesestructures are overdispersedrelative to counts estimated from discrete sampled counting processes, the Bernoulli, since the variance time curve of the Bernoulli and show that this can be severe,especially when the spectrum processhas a slopeequal to m(1 - m) < m. of counts does not decreaserapidly with respect to the sam- The log survivorfunction has beenplotted in sucha way as pling interval. For the daily rainfall occurrences,it was seen to emphasize the discretenessof the interarrival times. For that the spectrumof counts began to rise at high frequencies, example,for an interarrival time x - xo multiple points(trian- apparently due to aliasing, but the effects of aliasing intro- gles)are shown on the plot to illustrate the number of ties, i.e., duced into lower frequenciescannot be easily assessed.The number of intervalsof length x0. To interpret the log-survivor spectral estimatesshown in Figures 4 and 5 were obtained function, i.e., concavity or convexity and slope,only the lower- using a uniform averagingover 15 nonoverlappingintervals. most points (triangles) at each entry are needed. Also, the full Notice that the normalized spectra of counts for most of the length of interarrival times has been retained to illustrate ex- months decreasewith increasingfrequency to a value lessthan treme situations. These extreme points, however, are less reli- one and approximately equal to 1- m, where m is the esti- able and should be given less weight if the log-survivor func- mated rate of occurrence. For the months that have coef- tion is usedfor model fitting. ficients of variation less than one, the spectrumof counts is either approximatelyconstant (indicating an independentBer- 7. SAMPLING CONSIDERATIONS noulli process)or increasesslightly over a range of low fre- The previous discussion was oriented primarily toward quenciesand then decreases.Such spectra of counts are usu- daily rainfall sequences.One would naturally wonder if the ally consistent with variance time curves below the one for the problems addressedherein are associatedwith the large sam- FOUFOULA-GEORGIOUAND LETTENMAIER.'MODELS FOR RAINFALL OCCURRENCESERIES 539

I II III IV 2.5 o 40 ...... 1.4 1.2

1.0 ..... 1.5 -3.0 JAN 1.0 ...... -4.0 ß 15 . -5.0 ß 1 .! .5 -6.0 -?.0 0 •"•• • øo 50 100150 200 250 300 350 400 0 5 10 15 20 25 30 3 0 40 80 120 160 200 0 40 80 120 160 200

14 1.2 2.0 -1.0 12 -2.0 lO FEB 1.5 -3.0 ß

o so i oo •so •oo •so3oo 3so •oo 'o • •'o •'s •'o •'s 3'o 3s oo 0 40 80 120 160 200

2.5 -1.0 1.4 1.

1. -3.o 30 MAR 1.0 ...... -5 .o 20

_6.oi-.7 0 ...... 0 0 50 100 150 200 250 300 350 400 ß 0 5 10 15 20 25 30 35 0 40 80 120 160 200 0 40 80 120 160 200

2.0t -1-2.0ß ß""I 5040 1.41ß APR 30 -5 ß 20 ß -6 lO

0 50...... 100 150 200 250 300 350 400 .7'% • Fo F• io i• 3'o øo 4o 8o •2o •6o 200 o

-• .o 2.0 t -2.oo 18øF•2øI / 1I MAY ,.o ...... ot / : 40 ß5 -B.0 ß 20 % 50 100150 200 250 300 350 400 -.7 '00 5 10 15 20 25 00 40 80 120 160 200 0 40 80 120 160 200

2.0

-2 .o 1.4 JUNE :.s -3.0 lSøI • I 1.01.2 -5.0 ßß ß 50 -6.0 ß . 4 0 -'7.0 2 4 B 8 10 12 14 1 o0 40 80 120 1BO 200 øo ;T • • • 200

Fig. 5. Statisticalproperties of intervalsand countsfor daily rainfall occurrencesat Miami, Florida. I, normalized spectrumof countsversus frequency factor; II, log-survivorfunction versus interarrival time (days);III, varianceof counts versusinterval length(days); and IV, index of dispersionversus interval length (days). pling interval of 1 day and would not exist for, say, hourly function of a N-S processruNs(A) is shown for ho= 0.1, rainfall. In this section it is shown that a sampled rainfall p - 0.05,and 0 - 5.0, approximatelythe parametersobtained occurrence seriescannot be considered as a continuous point by Rodriguez-Iturbeet al. [1984] for Denver rainfall data process,even at samplingintervals as shortas a few minutes. duringthe periodof May 15 to June 16. This N-S processhas Let A denote the sampling time interval in days. Then the the samerate of occurrence(ho/O = 2 days-x)as the Poisson rate of a Poissonprocess (me) and NeymanoScottprocess (mss) process,but Figure 6 suggeststhat in this caseobservations are given as every30 s wouldbe requiredin orderto avoid seriousbias in me(A) = (1 - e- x•)/A the meanfunction. Again, hourly data are completelyunsatis- factorywhen regarded as observationsfrom this continuous- and time process.It is important also to observefrom Figure 6 that the effectsof discretizationstrongly depend on the degree of clustering.In general,the more clustereda process,the mNs(A) = 1 - e 1- (1- p)e-øA' larger the loss of informationthrough discretesampling. Thereforeit appearsto be inadvisableto considerrainfall oc- respectively[Guttorp, 1985], where all the parametershave currencesas the events of a point processand to use the beendefined previously. Figure 6 showsme(A ) for an arrival continuous-timemethodology for modeling,even when sam- rate ;•- 2 days-•. We see that observationsevery 10 min plingintervals are as smallas a fewminutes. It shouldalso be induceonly minor bias in the mean function,but hourly dis- notedthat the practicaldifficulties of measuringsmall rainfall cretization has serious effects. On the same figure, the mean accumulations,such as would occur over such short time in- 540 FOUFOULA-GEORGIOU AND LETTENMAIER: MODELS FOR RAINFALL OCCURRENCE SERIES

I II III IV

2.5 0 45 1.0

-2.0 35 30 '

JULY 1.0 ...... -4.0 20

-6.0

-'7 0 400 -•.o2 4 6 8 10 12 14 •0 40 80 120 160 200 i• • 00 120 160 200

2o5 0 140[ 1.2 2.0 1. ß...... -2.0 lOO

. 1.5 -3 .o 8o AUG -5 .o 40 1.0ß5 ...... -6.0-4.0 2060 ' ß øo 50 100150 200 250 300 350 400 -'7.0 0 2 4 6 8 10 1 0 40 80 120 160 200 00 40 80 120 160 200

-1 .o 30 ' 2.0

1.5 -3.0 20 . 65 SEP 1.0 ...... -4.0 15

ß5 -5.0-6.0 105 ' o• øo..... ' øo 50 100150 200 250 300:550 400 -'7.0 0 40 80 120 160 200 0 40 00 120 160 200

1401 1.4 2.o -1 .o 120t 1.2 -2.0 lOOt 1.0 1.5 -3.0 OCT 60

- . ß 40 .

ß5 ß 20 00 50 100150 200 250 300 350 400 '0 5 10 15 20 25 30 0 '0•

1.4

-1.0 18 1.2 2.0 16 -2.0 ß 14 1.0 ...... 1.5 -3.0 12 .8 NOV 2.51.0 ...... -4.00•. ßßß 2010

0 5O 100 150 2OO 25O 3OO 35O 400 10 15 2O 25 30 35 0 4O 80 120 160 2OO 0

1.8 2.0 -1.0 50 1.6 -2.0 40 1.4 1.5 -3.0 1-2 DEC t 30 1.0

-6.0 10 ' ...... '70 m ...... 0 .... ø0' .... 0 50 100 150 200 250 300 350 400 '0 5 10 15 20 25 30 3 40 80 120 160 200 0 40 80 120 160 200

Fig. 5. (continued) tervals, effectivelypreclude the application of the continuous- induce severebiases in estimation and can result in misleading time point process framework to any real rainfall series of interpretations regarding rainfall clustering. We suggest that hydrologic interest. rainfall occurrencesshould be comparedwith the discrete-time independent Bernoulli process(and not with the continuous- 8. SUMMARY AND CONCLUSIONS time Poisson) if inferencesabout clustering and dependencies Point process models for daily rainfall occurrencesat a are to be made. We have shown that interpretations regarding single station have been the subject of severalearlier studies. clustering made under the continuous-time versus discrete- Serious problems were encountered in fitting the Neyman- time framework differ, often critically. For example, daily rain- Scott model to daily rainfall occurrencesin two of thesestud- fall structures which are underdispersedrelative to Poisson ies. We suggestthat theseproblems may be the result of the (i.e., more regular occurrencesthan a Poisson process)are, in inappropriate use of continuous-timepoint processmethod- general, overdispersedrelative to Bernoulli (i.e., more random ology for the daily rainfall occurrences.If daily rainfall oc- occurrencesthan in a Bernoulli process). currencesare interpretedas (all of) the eventsof a point pro- The other issueaddressed in this paper is the fitting of point cess,they form a discrete-timepoint processin which events process models to daily rainfall occurrences. Fitting pro- can only occur at time marks integral multiples of the sam- cedureswhich directly use the parameters of the continuous- pling interval apart. Although previous studies have imple- time point processes(and which have been extensivelyused in mented the above interpretation of rainfall occurrences,they the hydrologic literature) have been shown to induce severe have failed to account for time discreteness and have followed biasesin the parameter estimates.Such proceduresshould not a continuous-time point processframework for. modeling. In be used even in modeling hourly rainfall, since the sampling this paper, it has been shown that such an approach can interval under which rainfall occurrence data may be con- FOUFOULA-GEORGIOUAND LETTENMAIER' MODELS FOR RAINFALL OCCURRENCESERIES 541

2.0

1.5 \

\ \ I.O \

0.5 4h'••, 6h •

I I I I o.o 0 '• I0 '4 I0 '$ I0 '• I0 '• I.

A (DAYS)

Fig. 6. Rate of occurrenceof a discretizedPoisson process with rate of occurrence• = 2 days-x (circles)and a discretizedNeyman-Scott process with parameters ho = 0.1days-•, p ---0.05, and 0 ---5.0 days- x (triangles)as function of the discretizationtime intervalA (in days).Here s, m, h, and d denoteseconds, minutes, hours, and days. sidered approximately continuous is on the order of a few Green,J. R., A modelfor rainfall occurrence,J. R. Stat. Soc.,Ser. B, minutes. We recommend instead that the appropriate ap- 26, 345-353, 1964. Guttorp,P., On binarytime series obtained from continuous-time proach to modelingrainfall data usingcontinuous-time point point processesdescribing rainfall, Tech. Rep. 9, Dep. of Stat., processesis the onefollowed by Rodriguez-lturbeet al. [1984] Univ. of Br. Columbia, Vancouver, 1985. and Foufoula-Georgiouand Guttorp [1985]. An alternate ap- Guttorp, P., and M. L. Thompson,Estimating point processparame- proach is to develop discrete-timepoint process models. ters from sampledcounting processes, Statistical Inference for Foufoula-Georgiou[1985] proposesthe use of a semi-Markov Point Processes,Tech. Rep. 35, Dep. of Stat., University of Wash- ington, Seattle, 1983. model in this context. It seems,however, that the development Kavvas,M. L., and J. W. Delleur,A stochasticcluster model of daily of other classesof discrete-timepoint processmodels, as, for rainfallsequences, Water Resour. Res., 17(4), 1151-1160, 1981. example,the discrete-timeanalogue of the Neyman-Scottpro- Lawrance, A. J., Some models for stationary series of univariate events,in StochasticPoint Processes'Statistical Analysis, Theory cess,will involve cumbersomeclosed form solutions,if they andApplications, edited by P. A. W. Lewis,John Wiley, New york, are feasible at all. 1972. LeCam, L., A stochasticdescription of precipitation,in Proceedingsof Acknowledgments.The authorsgratefully acknowledge helpful dis- the 4th BerkeleySymposium on MathematicalStatistics and Prob- cussionswith P. Guttorp of the StatisticsDepartment, University of ability,University of California,Berkeley, 1961. Washington,and, in particular,his commentson samplingconsider- Lewis,P. A. W., Remarkson the theory,computation and application ations(section 7). The criticalcomments of three anonymousreferees of the spectralanalysis of seriesof events,J. Sound.Vib., 12(3), are greatly appreciated. 353-375, 1970. Neyman,J. E., On a newclass of contagiousdistributions applicable REFERENCES in entomologyand bacteriology, Ann. Math. Star.,10, 35-57, 1939. Bartlett, M. S., The spectral analysisof point processes,J. R. Stat. Neyman,J. E., findE. L. Scott,A statisticalapproach to problemsof Soc., Set. B, 25, 264-296, 1963. cosmology,J. R. Stat. Soc., Ser. B, 20, 1-43, 1958. Chang, T. J., M. L. 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E. Foufoula-Georgiou, St. Anthony Falls Hydraulic Laboratory, (Received March 26, 1985; Department of Civil and Mineral Engineering, Mississippi River at revisedSeptember 18, 1985; 3rd Avenue S.E., Minneapolis, MN 55414. acceptedNovember 12, 1985.)