TEMPORAL AND SPATIAL CHARACTERIZATION OF RAINFALL OVER CENTRAL AND SOUTH FLORIDA

JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATIONVOL. 36, NO.4 AMERICAN WATER RESOURCES ASSOCIATION AUGUST 2000

TEMPORAL AND SPATIAL CHARACTERIZATION OFRAINFALL OVER CENTRAL AND SOUTH FLORIDA'

Alaa Au, Wossenu Abtew, Stuart Van Horn, and Nagendra Khanal2

ABSTRAC'I' Frequency evaluation and spatial characterization ofrainfall in Central and South Florida are presented. Point frequen-cy analysis performed at all available sites has shown that the2-parameter Gamma probability density function is the best modelfor monthly rainfall frequency over Central and South Florida. Themodel's parameters estimated at 145 stations were used to providemonthly rainfall estimates for 10- and 100-year dry and wet returnperiods. Experimental and theoretical variograms computed forthese estimates, as well as the Kriging estimation variance maps,show that the existing rain gage network is less capable of resolv-ing monthly rainfall variation in the wet season than the dry sea-son. May is the dry-to-wet transition month, while October is thewet-to-dry transition month with average rainfall of 4.5 inches.Monthly average rainfall is above 7 inches during the wet seasonand below 3 inches during the dry season. Two-thirds of the annualrainfall is accumulated in the wet season. Annual average rainfallis maximum (above 60 inches) in many areas along the east coast,and is minimum (below 45 inches) in many areas over Lake Okee-chobee and Central Florida. Rainfall maps show a changing patternbetween the wet and the dry seasons. Frontal rainfall occurs in thedry season, while convective rainfall, tropical depression, and hur-ricanes occur in the wet season. Average rainfall is higher along theeast coast area in the dry season and it is higher along the westcoast area in the wet season.(KEY TERMS: rainfall; Florida; frequency analysis; spatial analy-sis.)

INTRODUCTION

The South Florida Water Management District(SFWMD) area covers South Florida and part of Cen-tral Florida (henceforth called Central and SouthFlorida). In this area rainfall represents the mostimportant component of the water budget where rain-fall depth resulting from a storm event occurring witha given frequency is an essential variable for the

design and operation of water management struc-tures, flood control, consumptive use estimation,water supply planning, and total water resourcesmanagement. Therefore, maps of this variable evalu-ated for several storm event durations and frequen-cies are of great need. The construction of these mapsrequires (1) temporal characterization of rainfalldepth using a proper frequency analysis, and (2) spa-tial representation of this depth using a rigorousgeostatistical tool.

Point and regional frequency analysis are used toestimate rainfall depth of various durations and fre-quencies. Point frequency analysis deals with a singletime series representing a single location, whileregional frequency analysis deals with a combinedtime series from several locations representing anentire basin. Regional frequency analysis providesbetter information when a lumped basin-wide repre-sentation is desired (e.g., rainfall volume over theentire basin for a given duration and frequency).However, point frequency analysis is needed if mapsof rainfall depth for different durations and frequen-cies over the entire area are desired. To obtain thesemaps over Central and South Florida, frequency esti-mates are first obtained using point frequency analy-sis followed by geostatistical analysis to infer theseestimates at ungaged points.

Central and South Florida, in general, have a lowrelief topography and subtropical climate. They aresurrounded by the Gulf of Mexico on the west and bythe Atlantic Ocean on the east. This area has a highwater table area with patches of lakes and extensivewetland systems. The heaviest rains in Central andSouth Florida are produced by convective systems

iPaper No. 99021 of the Journal of the American Water Resources Association. Discussions are open until April 1, 2001.2Respectively, Senior Engineer, Senior Supervising Engineer, Senior Engineer, and Lead Engineer, Resources Assessment Division, South

Florida Water Management District, 3301 Gun Club Road, West Palm Beach, Florida 33406 (E-Mail/Aali: [email protected]).

JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION 833 JAWRA

Mi, Abtew, Van Horn, and Khanal

with the cooler (dry) season having an extratropicalnature and the warmer (wet) season having a tropicalorigin (Rosenthal, 1994). This region has high rainfallwith 52 inches of areal average annual rainfall andrelatively low annual variation with a standard devia-tion of 12 inches.

Rainfall frequency studies pertaining to the Cen-tral and South Florida area have been reported in theliterature. An earlier regional rainfall analysis studyhas produced isopluvial maps for two to ten-day pre-cipitation for return periods of two to 100 years(Miller, 1964). The isopluvial maps for the durationsand frequencies presented by that study indicate thatthe area of Central and South Florida is relativelywetter than the rest of the continental United States.A generalized rainfall-frequency study was conductedfor Central and South Florida by the U. S. ArmyCorps of Engineers for the purpose of developingflood-frequency curves. This study presented isoplu-vial maps for maximum one-day rainfall for returnperiods of 2, 5, 10, 20, 50 and 100 years. It also pre-sented isopluvials for maximum one-month, two-months, four-months, and six-months and annualrainfall for return periods of 2, 5, 10, 25, 50, and 100years (U. S. Army Corps of Engineers, 1953).

In a recent study conducted for the SFWMD,MacVicar (1981) developed isopluvial maps for theCentral and South Florida area for durations of one-day, two-days, three-days, and five-days for returnperiods of 2, 5, 10, 25, 50 and 100 years. Also, addi-tional isopluvial maps were produced for the dry sea-son (November through April), wet season (Maythrough October), and annual rainfall for the samereturn periods. In all cases, the Fisher Tippett Type Istatistical distribution (also referred to as Gumbeldistribution) was used for frequency analysis. Sculley(1986), using rainfall data available through 1985,produced areal rainfall tabular and graphic represen-tations of rainfall frequencies for 12 water manage-ment basins for the dry season (November throughMay), wet season (June through October), and annualrainfall. Normal and Log-normal distributions wereused depending on test of fitness for each durationand each basin. The SFWMD areal annual rainfallfrequency was fitted with a Log-normal distribution,and magnitudes for dry and wet extremes for 2, 5, 10,25, 50, 100, and 200 years were presented. Based on71 years of annual rainfall, the SFWMD areal aver-age rainfall was reported as 53 inches with a range of39 to 77 inches. Trimble (1990), applying the two-parameter Gumbel distribution, produced isopluvialsfor a one-day rainfall for return periods of 3, 5, 10, 25,and 100 years, and for a three-day rainfall for returnperiods of 10, 25, and 100 years. Although these stud-ies provided isopluvial maps, the main focus was the

rainfall frequency analysis with a limited emphasison the spatial analysis across the study area.

Spatial rainfall analyses are important for floodforecast, hydrometeorological network design, missingdata estimation, and many other hydrologic applica-tions. Geostatistical studies using Kriging for spatialcharacterization of maximum rainfall depth inCentral and South Florida have been reported.Wanielista et. al. (1996) used Ordinary Kriging todevelop storm event maps for one-day, two-day, 3-day,4-day, 7-day, and 10-day durations for return periodsof 2, 5, 10, 25, 50, and 100 years. Commercial soft-ware was used in the analysis, and basic steps suchas experimental variogram computation and modelfitting were not presented. Also, a measure for theadequacy of fit of Kriging (e.g., estimation error vari-ance maps) was not available.

Relevant studies with a stronger geostatisticalemphasis were also reported. These studies used oneor more statistical techniques to address the stochas-tic spatial structure of raw rainfall data within Cen-tral and South Florida. Abtew et al. (1993) used sixspatial models for monthly rainfall data in compara-tive analysis. They concluded that optimal interpola-tion and Kriging are the most appropriate methodsfor characterizing monthly rainfall data in this area.

In a recent study, Van Lent and Tracy (1994) con-ducted a geostatistical analysis for rainfall to investi-gate the adequacy of the rain gage network in Centraland South Florida. In this study, they conducted spa-tial characterization for annual, seasonal, monthly,and daily rainfall measurements. They used exponen-tial variogram models for annual and seasonal rain-fall. The range parameter of the vaJriogram modelswas interpreted as the total correlation length, where-as it is only one-third of that length. This explainswhy their interpreted correlation length is clearly andsignificantly lower than that observed in the van-ogram plots in that study. Also, the computed van-ograms for monthly and daily records exhibited nospatial coherence, indicating some concerns regardingthe computation method of such variograms. Themonthly variograms exhibited significant nuggeteffect, and the daily variograms are pure nuggets.The nugget effect is due to variability that the net-work density cannot resolve. The estimation errorvariance (monthly and daily if estimated) may havebeen high due to the nugget effect, or may have beendue to the way the variogram was computed. Theauthors suggested an increase of one order of magni-tude for rainfall gages to reduce monthly estimationvariance. In their study, they concluded that theannual and seasonal rainfall is isotropic and station-ary. Concerns about this study were provided by Moss(1996). He pointed out that a discrepancy existed

JAWRA 834 JOURNAL OF THE AMERICAN WATER RESOURCES ASSOCIATION

Temporal and Spatial Characterization of Rainfall Over Central and South Florida

between the annual variogram plot and the associatedequation. Also, he concluded that the residuals werenot stationary and were not normally distributed.This is clearly expected given the monthly variogramscomputed. Also, Moss (1996) concluded that there wasno significant temporal trend of rainfall for the pur-pose of Kriging analyses.

In a similar study to assess the evaporation andrain gage network, Zhao and Chin (1995) conducted aspatial analysis for monthly rainfall and evaporation.The experimental monthly variograms exhibitedstrong spatial coherence, and the exponential modelappeared to be a reasonable fit. An experimental van-ogram was computed as the average of all historicalrealizations of variograms. Such a computationrequires testing for the assumption of second orderstationarity for each realization and may not warrantthe ergodicity assumption. Furthermore, their inter-pretation for the variogram range (correlation scale)was similar to that reported by Van Lent and Tracy(1994). Therefore, any conclusion based on this inter-pretation should be re-evaluated. For example, theyidentified 82 circles of 30 km radius (half the mini-mum range) where additional stations are potentiallydesired. The number of additional stations should besignificantly reduced if the minimum range was inter-preted accurately.

Our discussion thus far suggests that a study thatcombines rainfall frequency analysis with a thoroughgeostatistical characterization is not available in thearea of Central and South Florida. The current inves-tigation is an attempt to provide such a study for aone-month duration event using the most recentavailable data. Temporal variation of monthly rainfallat each site was modeled with two-parameter gammadistribution. In addition to its average, monthly rain-fall at each site for dry and wet return periods of 5,10, 20, 50, and 100 years were estimated. Variogramanalysis and spatial mapping using Ordinary Krigingfor monthly rainfall of each return period were per-formed. The analyses of different rainfall durationsare deferred to another study.

RAINFALL DATA SELECTION

The SFWMD collects precipitation data from anetwork of recording and nonrecording precipitationgages throughout its 16-county water managementarea, which encompasses 18,000 square miles.In addition to the SFWMD precipitation gages, Feder-al, state, and local government agencies also maintaingages throughout the SFWMD region. For instance,the National Oceanic and Atmospheric Administra-tion (NOAA) maintains a network of gages primarily

located at airports and cities. The U.S. GeologicalSurvey (USGS) and the U.S. Army Corp of Engineers(COE) collect precipitation data from gages located atremote data collection platforms and water controlstructures throughout the Central and SouthernFlorida Flood Control Project. A precipitation gagenetwork is also maintained by the U.S. National ParkService in the Everglades National Park (ENP) locat-ed in the southern region of Florida. The FloridaForestry Service (FS) maintains gages at many of theForest Service tower locations. In order to select theappropriate records for this study, a number of stepswere followed. These steps included a comprehensivereview of the available precipitation records, the iden-tification of duplicate records, and the selection ofgage records with a minimum record length of 25years with 5 percent maximum of missing data forany year (Markovic, 1965). In addition, monthly pre-cipitation records were tested through a correlationanalysis to identify if any records were unsuitablebased on a poor correlation with neighboring sites.Based on this procedure, a final set of 145 gagerecords were selected for this study and are presentedin Figure 1.

Data Statistics

Table 1 presents lumped statistics for the data setsat the 145 stations depicted in Figure 1. There is ahigh dispersion for all monthly data. The standarddeviation, a, is relatively high compared to the arith-metic average, 31. The coefficient of variation, a/31, forthe wet months is significantly lower than the drymonths. The degree of skewness is high in the drymonths and relatively low in the wet months. In thedry months, the mode is significantly lower than theaverage (positive skewness). The Kurtosis values indi-cate that the dry months are Leptokurtic (Kurtosis >3) and the wet months are Platykurtic (Kurtosis < 3).Note that these statistics reflect both the spatial andtemporal variations. Table 1 also presents theThiessen weighted average for monthly rainfall withand without Florida Keys. Figure 2 shows plots forthese values and the corresponding cumulative val-ues. It is clear that dry season monthly rainfall is lessthan 3 inches and wet season monthly rainfall is morethan 4 inches. The slopes of the cumulative valuecurves increase in May, reflecting the beginning of thewet season, and decrease in October, reflecting thebeginning of the dry season. Two-thirds of the annualrainfall is accumulated in the wet season. The Dis-trict-wide Thiessen weighted annual average forthese data is 51.8 inches when the Florida Keys areconsidered and 52.3 inches when the Florida Keys are


• Rain Gages Includedin the Thiessen Network'.

1,18C 88,RrrrA2.3A-36 89,ROYAL PA3,ALVA FAR 90 RUNYON4.ARCHBO2 91 S135,AVON PRK 92S1316,BABSON P 93 S18C7,BARE BEA 94 S208BASSETE 95S20F9BELLE GI. 96 S310,BENBOW 97S30811 BROOKS P 98 S3612,CANAL P2 99:s5A13,CAPTIVA 100S614.CLERMONT 101 S6515,COCONUT 102 S6SA16,CORK.HQ 103 S65B1 7,DANIA 4 104 S65018,DEVILS 105 S65E19,DIXIE WA 106s6820,DIXIE 107:07021.EVERGL2 108 S7822,FLAMIN 2 io'so23,FORTPIE 110S8024.FT MEYER i'sg25,FT. PIER 112:scOTroG26G54 I13ÔSNIVELY27.GILL flEA 1 14,SOum SH28,HGS1 115.SOUTHBA29,HGS2 116 SR30.HOS4 117:sT.CLAI31,HGS5X 118,STUART 132,HGS6 119.TAMrTR4033,HIALEAH 120,TAYLC.S734.HOMES.ES 121 ,TOWNSITE35,HOMES.FS 1 22,USDAIMM36,HVPOLUXO 123,VENUS 4$37,IMMOKA 2 1 24.VERO 4W

125,WPB AIRP

40.JUDSON41,KISS 24.FS A Rain Gages Outside of441. ALF EX the Thiessen Network451. HART 126.ARCAD1A

127.BARTOW128.EUS11S 2

47. 2 1e,FuSMc48.LA.BELLE 130,HILiLSBOR49,LEHIGH I WE 250.LIBERTY 132KEY WEST51, 133,LAKELA552,LWD.Ei.3 134,UGNUMVI53.LWD.E2.2 135.USBON

136,MELBOURN137,00ALA56.LWD.HQ 138,OBLAN AP

57.LWD.L28 i*piAiiici4O,SANFORD59.1. I 141 ,SANFORX60. •142.TAVERNIE

61,LWD.RAN i43jnijsvi62,M1AMI BE 144.WAUCHUL145,WINIERHA

65,MIAMI.AP66.MIAMLFS.67,MOBLEY68.MOUN11N69.NAPLES70.NNRC.R271OKEE F272.OPAL73.ORLAN74.PANOKEEI75,PAHOKEE276,PALMDALE77.PEL3478,PEL LAK179,PEL LAK280.PEt'INSUCO81 ,PLANT IN82,POIIPANOB83.POMPANOF84,PRA1T AN85,PUNTA 286,PUNrA6487.RAULERS

Au, Abtew, Van Horn, and Khanal

M4Jrion£

Figure 1. General Layout for the Study Area Within Centraland South Florida and Rainfall Gage Locations.


137

A £128 .j4o 141(trs135 ... . 4 Atlantic

Semiiwile 143 Oceani,ake 138 ' AIlenwiujo ' .

•

132 131AS

ThIoSonaprResourceAssessrnent.Dlv.

OBS!lO97


TABLE 1. Lumped Statistics for Raw Monthly Rainfall Data and Weighted Averages.

Rainfall Measurements WeightedAverageArithmetic Standard Coefficient With Without

Month Average Deviation of Variation Skewness Kurtosis The Keys The Keys

January 2.20 2.05 0.93 1.69 3.92 2.05 2.05

February 2.36 1.85 0.78 1.52 3.72 2.15 2.18

March 2.94 2.56 0.87 1.79 4.85 2.63 2.68

April 2.58 2.32 0.90 1.71 4.98 2.35 2.36

May 4.66 3.13 0.67 1.11 1.72 4.67 4.70

June 7.85 4.18 0.53 0.94 1.14 8.23 8.37

July 6.98 3.19 0.46 0.58 0.44 7.06 7.23

August 7.03 3.18 0.45 0.91 1.78 7.12 7.23

September 7.23 3.78 0.52 1.09 1.81 7.22 7.28

October 4.72 3.82 0.81 1.63 4.36 4.48 4.45November 2.30 2.36 1.03 2.41 9.27 2.13 2.11

December 1.90 1.80 0.95 1.86 5.12 1.69 1.69

Figure 2. Thiessen Weighted Average of Monthly Rainfall Within Central and South Florida.

dropped. The Keys' annual rainfall average is approx-imately 40 inches. More results of Thiessen analysiscan be found in Ali et. al. (1999).

FREQUENCY ANALYSIS OFMONTHLY RAINFALL

A frequency analysis of monthly rainfall for Cen-tral and South Florida requires the selection of thebest-fit probability distribution for each month's rain-fall. The selection approach of a probability distribu-tion (model) was to test the goodness of fit of themajor and commonly applied statistical distributionson all available stations within the study area (Cen-tral and South Florida). The parameters of the best-fit

probability distribution are then identified at everystation throughout the study area. The candidate dis-tributions were Normal, Log Normal (2-parameter),Log Normal (3-parameter), Gamma (2-parameter),Gamma (3-parameter), Weibull and Log Pearson TypeIII.

For each month and for each station, each of theprobability distributions were fitted, and both tabularand computed Chi-square (x2) were generated usingthe Frequency analysis program by Ahn (1990) (pro-gram developed to select parameters for various prob-ability distributions suited for frequency analysis,SFWMD, 1990). A distribution is rejected when the

Chi-square (x2) ratio Xomputed is greater than 1.Xtabular

Table 2 presents the number of rejections for each



TABLE 2. Number of Rejections and Rejection Average Rate for Each DistributionBased on Chi-Square Ratio> = 1, and 5 Percent Significant Level.

Month NormalLog-Normal

(2-Parameter)Log-Normal

(3-Parameter)Gamma

(2-Parameter)Gamma

(3-Parameter WeibullLog Pearson

Type Ill

January 120 34 30 11 40 9 68February 54 59 15 11 33 9 112March 101 38 21 4 38 6 74April 105 85 27 10 33 10 116

May 33 46 16 9 18 5 113June 35 22 13 7 16 16 59July 13 19 14 8 24 13 80August 18 12 17 6 13 11 39September 30 18 16 13 15 13 63October 78 33 28 14 29 14 83November 125 32 33 9 57 13 65December 120 31 45 7 48 12 62Rejection Rate* 0.48 0.25 0.16 0.06 0.21 0.08 0.54

*Rejection rate is the total number of rejections for a distribution divided by 145 sites and 12 months.

(1)

(4)x

Gamma distribution fittings for each month for onesample station (Belle Glade) with 69 years of record,are shown in Figure 3. Monthly statistics for eachmonth, x2 test of fitness, a and 13 parameters forgamma distribution for this station are shown inTable 3.

Frequency Analysis Application

For each data set, Equations (2), (3), and (4) wereused to identify the two parameters cx and 13. GivenEquation (1), the following integral equations aresolved for the value "p" for a given return period "n:"

= f(P)dP, for dry return period (5)

(2) - = $ f(P)dP, for wet return period (6)

This analysis produces sets of rainfall frequency( estimates of each month and several wet and dry

return periods. The reader is referred to Au et. al.(1999) for a full presentation of these results. In thispaper, January and July rainfall estimates for twowet and two dry return periods are used for subse-quent spatial analysis. Sets of historical averagemonthly rainfall for January, May, July, and October


month and each distribution. Table 2 shows that therejection rates for Gamma (2-parameter) and Weibulldistributions are significantly lower than the otherfive distributions. The average Chi-square (x2) ratiofor the rejected cases of Gamma (2-parameter) is"1.2." For simplicity and consistency with statisticalhomogeneity across the entire site, Gamma (2-param-eter) distribution was selected to fit all sites and allmonths. The 2-parameter Gamma distribution (Haan,1977) is expressed through the following equation:

RIXPef(P)

F(a)

where f(P) is the probability density function of P(rainfall amount), a is a shape parameter, 13 is a scaleparameter, and F(a) is the gamma function of a.These parameters vary from month to month. Estima-tions of a and 13 were based on the method of Maxi-mum Likelihood (Greenwood and Durand, 1960).

0.5000876 + 0. 1648852Y — 0.0544274Y2Y

Y=lnI

where X is the arithmetic mean and G is the geomet-ric mean for non-zero monthly rainfall historical datafor each month. Zero monthly rainfall values weresubstituted with 0.01 inches.

'I)CU)0

Temporal and Spatial Characterization of Ramfall Over Central and South Florida

Monthly Rainfall (inches)

Figure 3. Gamma Distribution Fittings for January Through December at One Sample Station (Belle Glade).

TABLE 3. Monthly Rainfall Statistics, x2 Thstof Fitness, and a and (3(parameters for Belle Glade Station).

Month Mean Standardx2

(table)x2

(computed) a (3

January 2.22 2.18 14.1 3.0 1.02 0.463

February 1.90 1.47 14.1 10.3 1.85 0.877

March 2.97 2.73 14.1 7.4 1.17 0.394

April 2.86 2.12 14.1 8.5 1.78 0.625

May 4.78 2.55 14.1 4.2 3.47 0.730

June 8.61 4.59 14.1 7.7 3.46 0.401

July 7.88 3.17 14.1 5.1 6.09 0.775

August 7.98 3.17 14.1 5.3 6.24 0.781

September 8.05 3.68 14.1 1.9 4.71 0.585

October 4.47 3.25 14.1 9.1 1.85 0.417

November 2.24 2.37 14.1 4.5 0.87 0.394

December 1.74 1.76 14.1 2.2 0.96 0.556


0.8

0.7

Cp

0.1

0 2 4 6 8 10 12 14 16 18 20 22 24


are also used in the same analysis. This analysis pro-vides point rainfall maps for various return periods.These maps are used for rainfall frequency estimationat ungaged locations. The construction of these mapsis the focus of the next section.

RAINFALL SPATIAL CHARACTERIZATION

The characterization of rainfall spatial variabilityis of great interest to water resources planners, regu-lators, and decision makers. Such studies have directapplications in missing data estimation, water budgetanalyses, extreme events forecasting (e.g., flood anddrought), hydrometeorologic network design, andmany other hydrologic modeling applications.

In the preceding section, rainfall frequency analy-sis was performed on over 145 stations within Centraland South Florida. For a given return period and agiven month, historical rainfall data (monthly sum)were used to provide a set of rainfall estimates overthese stations for each month. Each set of these esti-mates, for the purpose of this study, is considered as arealization of a stochastic process. The goal in thissection is to use this realization to spatially character-ize the process on a regular grid using a traditionalgeostatistical tool (Kriging). To achieve this goal, theanalysis is divided into two stages: (1) variogramanalysis and (2) mapping.

Variogram Analysis

Variogram analysis is an essential tool in theclassic geostatistics. The purpose of this analysis is to(1) identify the spatial structure of a stochastic pro-cess by computing an "experimental" variogram, and(2) fit the "best" variogram model for subsequentKriging mapping. Variogram analysis assumes a sec-ond order stationarity of the underlying stochasticprocess (random function). That is, the first momentof this process (mean) is constant, and the secondmoment (two-point covariance or variogram) is depen-dent on the relative locations, rather than the loca-tions of the data points. While there is no advantageof using variogram over covariance, the latter cannotbe defined if the underlying function has unboundedvariance. Furthermore, the covariance is difficult todefine if the domain over which the random functionis defined is not sufficiently larger than the correla-tion length (called "range" in the geostatisticalliterature). The reader is referred to Journel and Hui-jbregts (1978) and Isaaks and Sirvastava (1989) formore details on variogram analysis.

Experimental Variogram

The experimental variogram is an inverse measureof the two-point covariance function for a stationarystochastic process. For a given separation distance"h," it is defined as the half of the expected value ofthe squared difference between a pair of measure-ments with separation distance "h." Given a set ofobservations from a stochastic process, a two-pointexperimental variogram at separation distance h,y(h), is expressed as:

y(h) =E[c(h)2] = E[(z(xi)

—

z(x2))2] (7)

where x1, x2 are the pair locations, and h = x1 - x2 =separation distance (lag).

The variogram, y(h), in Equation (7) represents theensemble (population) value. In reality, only one real-ization of this ensemble is available through a fewobservations (measurements). Assuming ergodicity(i.e., the properties of this realization are the same asthe ensemble), an estimate for y(h) can be computedby taking the expectation over this realization only.This is accomplished by dividing the domain into Nintervals. Each interval 'j" is defined by a separationdistance value "hi" and a tolerance I6h . The expecta-tion is taken over all pairs of observations with sepa-ration distance h falling within h - h � h} � h + h.Given N pairs of observations within this interval,Equation (7) can be expressed as follows:

i=N.1

(z(xl)_z(x2))2,J i=1

h +h (8)

Central and South Florida monthly rainfall is nottemporally (seasonally) stationary (i.e., its spatialstructure may vary from month to another) and,hence, variograms must be computed for each month.Assuming isotropy (Van Lent and Tracy, 1994), Equa-tion (8) was applied to "estimated" rainfall average forfour months, and rainfall estimates for four returnperiods (10 and 100 years for both dry and wet condi-tions) for January and July. The smallest separationdistance is 2,500 feet and the tolerance was 2,500feet. The relationship between the separation dis-tance and the associated number of pair-measure-ments is shown in Figure 4. It is noticed that thereare at least 40 pairs of stations with separation dis-tances ranging between 20 miles and 140 miles.

JAWRA 840 JOURNAL OF THE AMERICAN WATER RESOUF1CES ASSOCIATION


Fewer pairs are observed outside this range. Fewerpairs indicate a less reliable estimate for the van-ogram, particularly at larger separation distances.

Figure 4. Number of Station Pairs WithSeparation Distance (lag).

Figure 5 depicts the results of the experimentalvariograms for the average monthly rainfall for Jan-uary, May, July, and October. The reader is referred toAli et. al. (1.999) for more variogram analysis results.It is clear that July semi-vanogram exhibits highervalues for the same separation distance compared to

that of January, indicating a higher magnitude of spa-tial variability with higher rainfall. This is consistentwith the general characteristics of dry season and wetseason rainfall types (Rosenthal, 1994). Some van-ograms exhibit weak stationarity and unboundedvariance beyond a 150-mile separation distance. Thiscould be due to lack of pairs at high separation dis-tance values (see Figure 4) or that this distance maybe too large to sustain the same characteristics of thehydrologic process (i.e., spatial trend). In general, var-iograms exhibit stationarity and a pronounced spatialstructure for all months.

Variogram Fitting

While the experimental variogram is important forcharacterizing the spatial structure, it cannot bedirectly used for Kriging mapping because it mayexhibit nonstationarity and unbounded variance. Toovercome this problem, a "best" theoretical model isfitted to the experimental variogram manually or sta-tistically. As discussed earlier, many rainfall analysisstudies select exponential models for variogram fit-ting. In this study, exponential variograms appear tobe appropriate for most months. Although some van-ograms exhibit some departure from the exponentialmodels beyond a distance of 150 miles, the exponen-tial model was selected for all vaniograms in thisstudy. The theoretical form is expressed in Equation(9).

May2.5 F—Fitted

2 Experimental1.5

;.

.

0 50 jOO . 150 201Lag in miles

October8 •r:_Fjd

________________ 6 Experimental

C)4 .*

0 50 100 . 150 20'Lag in miles

Figure 5. Semi-Variograms for Monthly Average Rainfall in Central andSouth Florida for January, May, July, and October.


0.0

0 50 100 150 200

Lag (miles)

0.50.40.3

!.0.20.l0

January

o 50 lOQ 150 201Lag in miles

July

—Fitted4 . Experimental :.;

0 50 100 . 150 20Lag in miles


-Ihiy(h)=+w* 1—e

where ö = nugget (error due to lack of data at a reso-lution smaller than the network density); w = thevalue of the sill (model variance) after subtracting thenugget; and 0 = one-third of range (the practicalrange is the separation distance "h" that correspondsto 95 percent of the sill value and it is the distancebeyond which data pairs are not correlated).

By estimating the three parameters, the model inEquation (9) is fitted to the experimental variogram.In this study, the least squared error criterion wasused to select these parameters of the variogram foraverage rainfall, and four return periods for Januaryand July. Table 4 presents the variogram parametersfor the monthly average rainfall for all the 12 months.Monthly variogram models for January, May, July,and October are depicted in Figure 5.

MonthSill

(inches2)Nugget

(inches2) Nugget/SillRange(miles)

January 0.225 0.027 0.12 39.77February 0.177 0.035 0.20 99.43March 0.343 0.022 0.06 113.6April 0.266 0.027 0.10 59.66May 0.794 0 0.00 96.59June 1.375 0.134 0.10 90.91July 1.358 0.15 0.11 105.1August 0.842 0.171 0.20 79.55September 1.116 0.223 0.20 53.98October 2.734 0 0.00 88.07November 0.665 0 0.00 56.82December 0.172 0 0.00 53.98

The variogram analysis indicates a specific spatialstructure of rainfall average for the four months (Fig-ure 5). The nugget effect is relatively small comparedto the sill of most variograms. It ranges from 0 to 12percent of the sill value, indicating a reasonable datanetwork in resolving monthly rainfall spatial variabil-ity. All variograms exhibit some correlation length,with strong stationarity where the sill is relativelyconstant or weak stationarity where the sill fluctuatesor approaches asymptotic value relatively slowly. Thewet season (July) exhibits weak stationarity withlarge correlation length, while the dry season (Jan-uary) exhibits strong stationarity with shorter corre-lation length. May and October variograms exhibit a

level of stationarity and correlation length between(9) those of January and July.

Spatial Mapping and Estimation Variance

As mentioned earlier, variogram computation is anessential step to perform spatial interpolation usingKriging (Best Linear Unbiased Estimator with Mini-mum Estimation Variance). Given a group of observa-tions Z(x) of a stochastic process, a Kriging estimateof Z at an unsampled location, x0, is expressed as:

Z(x0) =oj * Z(x) (10)

where w10 is a set of weights.Given the estimation error E = Z(x0) -

an unbiased estimate for Z(x0) is obtained whenE(E) = 0. This is achieved by constraining the sum of

the weight w to 1, (i.e., cio = 1). An optimal set of

these weights is obtained by minimizing the estima-tion error variance, & = E[c21, (henceforth called esti-mation variance), as described in the followingKriging system of equations:

* y(h) = y(h0), i = 1.. .n (11)

where y(h) is the variogram as a function of the sepa-ration distance between location i and location j.

Since the estimation variance is minimized toobtain the set of Kriging weights, it is natural to useits "minimum" score as a measure of the estimation"uncertainty" (the lower this score the better the esti-mation). The availability of such a metric is one of theunique features in Kriging methodology To obtain anexpression for the minimum estimation variance,Equations (10) and (11) are incorporated into thescore, = E[E2}, leading to:

= * y(h) (12)

Theoretical review and mathematical formulationof Kriging can be found in Journel and Huijbregts(1978), and Isaaks and Sirvastava (1989). KrigingModule in the commercial package ARC/VIEW wasapplied to the data described earlier. Kriging generat-ed contour (isopluvial) maps reveal interesting char-acteristics of rainfall in Central and South Florida.

i= 1

TABLE 4. Exponential Variograin Parameters for 12 Months.



Results

Figure 6 shows the spatially interpolated monthlyaverage rainfall using Ordinary Kriging for four keymonths throughout the year. These months are Jan-uary, May, July, and October representing dry, dry towet, wet, and wet to dry seasons, respectively. Rain-fall contours extend generally from south to northwith rainfall values changing in the east-west direc-tion. January rainfall peaks up to 3.0 inches at someareas along the east coast, decreases steeply west-ward to 2.0 inches, and remains at 2.0 inches withoutsignificant variation throughout the rest of the area.May contours extend generally from southwest tonortheast with rainfall values changing northwest-ward from 6.0 inches at the southeastern corner to 4.0inches in Central Florida and the west coast. Julyrainfall pattern changes significantly from that ofMay. July rainfall is higher in the west coast than it isin the east coast. It changes from 8.5 inches at thewest coast, and goes eastward to 6.0 inches along theeast coast. October rainfall contours extend fromsouthwest to northeast with rainfall values changingfrom the east coast (8.0 inches) northwestward to 3inches in Central Florida and the west coast. Rainfallspatial mapping and interpretation for the rest of themonths can be found in Ali et. al. (1999).

The estimation variance is computed as presentedin Equation (12) for rainfall maps presented in Figure6. The corresponding estimation variance maps aredepicted in Figure 7. The estimation variance valuesare higher in the wet season (July) than those in thedry season (January). The estimation variance is sig-nificantly higher in the wet-to-dry month (October)than that in the dry-to-wet month (May). Given theexisting gage network, this implies that rainfall isless predictable at the end of the wet season than it atthe beginning of that season. High values of estima-tion variances are generally observed in two areas forany given map — the southwestern zone where dataare sparse and the boundary areas due to the bound-ary effect.

Figure 8 depicts January rainfall for the dry returnperiods of 100 years and 10 years, and the wet returnperiods of 10 years and 100 years. Rainfall estimatesfor the dry return periods range between 0 and 0.5inch with values in the western part lower than theeastern part of Central and South Florida. The esti-mates for the wet return periods have similar spatialpatterns and they range from 4.5 to 6.5 inches for the10 year return period and from 9.0 to 12.0 inches forthe 100 year return periods. Rainfall in this group ishigher in the eastern area than in the western area.

Figure 9 depicts July rainfall for the same returnperiods. Rainfall pattern reversed in this month

compared to January rainfall. For the dry and wetreturn periods, July rainfall estimates are higher inthe western area than in the eastern area of SFWMD.These values range from 1.5 to 3.00, and from 3.5 to5.50 inches for the 100 year and 10 year dry returnperiods, respectively. For the 10 year and 100 yearwet return periods, they range from 8.50 to 11.00 andfrom 10.00 to 13.00 inches, respectively. The reader isreferred to Ali et. al. (1999) for more detailed resultspertaining to this study.

CONCLUSION

Comprehensive frequency analysis and spatialcharacterization for rainfall in Central and SouthFlorida have been presented. The objective of thisstudy was to estimate average monthly rainfall andmonthly rainfall with different frequencies for dryand wet conditions at gaged and ungaged locations.The frequencies of interest were 10 years and 100years. A prescribed procedure for selecting the "best"representative data, temporally and spatially, wasadopted. Continuous uninterrupted time series ofmonthly rainfall were selected at 145 sites withrecords of 25 years or more. Frequency analysis wasperformed at the 145 sites to identify the best-fitmodel for monthly rainfall frequency over Central andSouth Florida. The model's parameters were thenestimated over the study area and, subsequently,rainfall estimates for various return periods werecomputed. These estimates were used for subsequentgeostatistical analysis. In this analysis, experimentalvariograms were constructed and exponential van-ogram models were fitted for each month and eachreturn period. Using the variogram models and esti-mated rainfall data, maps were constructed usingOrdinary Kriging for monthly rainfall averages forfour months, and rainfall estimates for various returnperiods for January and July. Estimation error vari-ance maps for average rainfall were also generated.

Global weighted monthly average rainfall is higherthan 4 inches for wet conditions and below 3 inchesfor dry conditions. Two-thirds of the annual rainfall isaccumulated in the wet season. The point frequencyanalysis conducted at the 145 stations has shown thatthe 2-parameter Gamma probability density functionis the best model for monthly rainfall frequency overCentral and South Florida. Experimental and theoret-ical variognams computed for the rainfall estimatesshow spatial coherence with a strong or weak station-arity. Wet season variognams exhibited higher nuggeteffects than dry season ones. This indicates that theexisting rain gage network is less capable of resolving


January


May

October

Figure 6. Spatial Distribution of Monthly Average Rainfall (inches)for Selected Months Within Central and South Florida.


+— I

0-

0:...L +

0

50

I

• 0 8 I8U

July

.:_

,'. 7.

70

70

I k' I

• 0 8 I8 '.7.——,'oS


January

July

May

October

Figure 7. Estimation Variance (inch2) for Monthly AverageRainfall Maps Depicted in Figure 6.


- 1L

:0.050

0 0H.+ !II :••• :'i' \\

I,4 .. ....

...., ..

0

d

1

L•::L.

L0'I+ I.H0..

o60 0100

...

I I I 10111, I 0 I 11111,

L

+ +

I 0 I III1 I 0 I II


100-year Dry Return Period 10-year Dry Return Period.

Figure 8. January Rainfall (inches) for Two Dry and Two WetReturn Periods Within Central and South Florida.


L- - 0.3

L+ -J.

-1.0.06

0.02.

L.+ o:o2

I- 0.06.

o:o�. 0.04

,ç(

I 0 I IS-

$ 0 I I$I6,

Wet Return Period


10-year Wet Return Period 100-year Wet Return Period

Figure 9. July Rainfall (inches) for Two Dry and Two Wet ReturnPeriods Within Central and South Florida.


Return Period Return Period

+ +

I 0 I III, I 0 S 0$


rainfall variation in the wet season than the dry sea-son. Rainfall maps show a changing pattern betweenthe wet and the dry seasons. Average rainfall is high-er along the east coast area in the dry season, and itis higher along the west coast area in the wet season.This reflects the changing nature of the rainfall stormtype between seasons. October rainfall exhibits aunique pattern reflecting the transition from the wetseason to the dry season. May exhibits a significantincrease of rainfall compared to January. Spatialmaps for various return periods at a given monthhave similar patterns with some offsets.

Point rainfall frequency analysis conducted in thisstudy was useful in providing rainfall data estimatesthat were used for monthly rainfall spatial mappingover Central and South Florida for various returnperiods. These maps can be used to infer rainfall esti-mates at ungaged locations for a given return periodor to infer the return period corresponding to a givenrainfall measurement or estimate. They also can beused to infer average rainfall depth (or volume) oversmall areas with same hydrometeorological character-istics. These maps, however, cannot be used to assessrainfall volume over larger areas, particularly forhigh return period events, since a simultaneous occur-rence of rainfall with the same frequency at all gageswithin such an area is not guaranteed. Regional fre-quency analysis can only provide an average estimatefor this volume provided that enough data are avail-able within that area. Therefore, rainfall volume maynot be easily estimated over an area without data, orwith insufficient data. Ideally, a frequency analysisthat considers the joint probability density function ofall gaged locations is to be implemented.

Rainfall statistics used in this study represent theaverage behavior of the process and may not charac-terize rainfall spikes (extreme events). Also, spatialinterpolation methods usually introduce somesmoothing to the estimation process and, hence, arenot capable of capturing extreme events at unsampledlocations. Extreme events can be captured, with somedegree of confidence, using a stochastic model. In astochastic model, rainfall data can be generated "ran-domly" such that the spatial and temporal structureof the rainfall process is honored. The generation ofdifferent realizations with the same structure pro-vides several rainfall estimates at any location for agiven return period. The highest and lowest valuescan then be identified.

In addition to measurement errors, rainfall esti-mates of this study are subject to such other errortypes as errors due to frequency analysis estimation,errors due to mapping to a regular grid, and errorsdue to isopluvial generation. While the estimationerror due to mapping to a regular grid was quantified,the other estimation errors are yet to be quantified.

LITERATURE CITED

Abtew, W., J. Obeysekera, and G. Shih, 1993. Spatial Analysis forMonthly Rainfall in South Florida. Water Resources Bulletin29(2): 179-188.

Mi, A., W. Abtew, S. Van Horn, and N. Khanal, 1999. Frequencyand Spatial Analyses for Monthly Rainfall in Central and SouthFlorida. Technical Publication WRE 371, Scuth Florida WaterManagement District, West Palm Beach, Florida.

Greenwood, J. A. and D. Durand, 1960. Aids for Fitting the GammaDistribution by Maximum Likelihood. Technometrics 2(1):55-65.

Haan, C., 1977. Statistical Methods in Hydrology. The Iowa StateUniversity Press, Ames, Iowa, 378 pp.

Isaaks, E. H. and R. M. Sirvastava, 1989. Applied Geostatistics.Oxford University Press, New York, New York.

Journel, A. G and J. Huijbregts, 1978. Minmg Geostatistics. Aca-demic Press, London, New York, San Francisco, 600 pp.

MacVicar, T. K., 1981. Frequency Analysis of Rainfall Maximumsfor Central and South Florida. Technical Publication No. 81-3,South Florida Water Management District, West Palm Beach,Florida.

Markovic, R. D., 1965. Probability Function of Best Fit to Distribu-tion of Annual Precipitation and Run Off. Hydrology Papers,Colorado State University, Fort Collins, Colorado, No. 8.

Miller, J. F., 1964. Two- to Ten-Day Precipitation for Return Periodsof 2 to 100 Years in the Contiguous United States. TechnicalPaper No. 49, U. S. Department of Commerce, Weather Bureau.

Moss, M., 1996. Testing the Adequacy of a Kriging Model of theSFWMD Rain Gage Network. Bayeswater, Inc., Tucson, Ari-zona.

Rosenthal, S. L., 1994. Statistical Aspects of the PrecipitationRegimes at Miami International Airport (MIA) and Palm BeachInternational Airport (PBI): 1961-1990. NOA.A Technical Memo-randum ERL AOML-80.

Sculley, S. P., 1986. Frequency Analysis of SFWfD Rainfall. Tech-nical Publication 86-6, South Florida Water Management Dis-trict, West Palm Beach, Florida.

Trimble, P., 1990. Frequency Analysis of One and Three-Day Rain-fall Maxima for Central and Southern Florida. Technical Memo-randum, South Florida Water Management I)istrict, West PalmBeach, Florida.

U. S. Army Corps of Engineers, 1953. Central and Southern FloridaProject for Flood Control and Other Purpose3. Part VI. GeneralStudies and Reports. Section 6 — Design Memorandum, RainfallFrequency Estimates. U.S. Army Corps of Engineers, Office ofthe District Engineer, Jacksonville, Florida.

Van Lent, T. J. and J. C. Tracy, 1994. Assessment of the Rain GageNetworks in the South Florida Water Management District. Areport submitted to the National Park Service under ContractCA4250-4-9015.

Wanielista, M., R. Eaglin, and L. Eaglm, 1996. Isopluval ContourCurves for Long Duration Storms in Florida. Florida Depart-ment of Transportation, WPI No. 0510680.

Zhao, S. and D. A. Chin, 1995. Assessment of the Evaporation andRain Gage Networks in the South Florida Water ManagementDistrict. Technical Report No. CEN-95-4.


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TEMPORAL AND SPATIAL CHARACTERIZATION OF RAINFALL OVER CENTRAL AND SOUTH FLORIDA