Entropy-based method for extreme rainfall analysis in Texas

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH: ATMOSPHERES, VOL. 118, , doi: /2011jd017394, 2013 Entropy-sed method for extreme rinfll nlysis in Texs. Ho, 1 nd V. P. Singh 1 Received 31 Decemer 2011; revised 12 Septemer 2012; ccepted 2 Novemer 2012; pulished 17 Jnury [1] Annul rinfll mxim re commonly used for rinfll nlysis, which entils the use of distriution for modeling extreme vlues. Anlysis of rinfll dt from different regions of Texs, USA, shows tht the form of the frequency distriution of nnul rinfll mxim chnges with different time durtions, climte zones, nd distnces from the Gulf of Mexico. Employing the entropy theory, n entropy-sed distriution for modeling nnul rinfll mxim is derived, which is expected to pply cross different time durtions, climte zones, nd distnces from the Gulf. The performnce of the proposed distriution is ssessed with synthetic dt from known distriutions, nd results show tht the performnce of the proposed entropy-sed distriution is generlly comprle with the generlized extreme vlue (GEV) distriution nd is preferle for highly skewed dt. Comprison sed on oserved rinfll dt lso shows this ttrctive property of the proposed distriution. Thus, the entropy-sed distriution provides promising lterntive for frequency nlysis of extreme rinfll vlues. The proposed distriution is then pplied to the nnul rinfll mxim, nd results show tht the entropy-sed distriution fits the empiricl proility distriution well nd lso performs well in modeling extreme rinfll vlues for different time durtions, climte zones, nd distnces from the Gulf. Cittion: Ho,., nd V. P. Singh (2013), Entropy-sed method for extreme rinfll nlysis in Texs, J. Geophys. Res. Atmos., 118, , doi: /2011jd Introduction [2] Rinfll frequency nlysis is used for constructing intensity-durtion-frequency (IDF) curves, which re needed for rnge of hydrologic designs, including dringe systems, culverts, rodwys, prking lots, runwys, nd so on. Extreme rinfll vlues, such s nnul rinfll mxim, re of interest in modeling floods nd quntifying the effect of climte chnge. From the fitted distriution, sttisticl properties of extreme rinfll vlues cn e investigted nd extrpolted eyond the ville dt for engineering purposes. [3] The generlized extreme vlue (GEV) distriution is one of the frequently employed proility distriutions for modeling nd chrcterizing extreme vlues. Derived from the extreme vlue theory, it is three-prmeter distriution encompssing three clsses of distriutions, nmely, Gumel, Frechet, nd Weiull. This distriution hs een used for extreme rinfll frequency nlysis in different res of the world. Schefer [1990] used the GEV distriution for frequency nlysis of nnul rinfll mxim of durtions of 2, 6, nd 24 hours for the stte of Wshington. Huff nd Angel [1992] selected the GEV distriution to model the distriution of nnul rinfll mxim for 1 Deprtment of Biologicl nd Agriculturl Engineering, Texs A&M University, College Sttion, Texs, USA. Corresponding uthor:. Ho, Deprtment of Biologicl nd Agriculturl Engineering, Texs A&M University, College Sttion, Texs. (hzc07@tmu.edu) Americn Geophysicl Union. All Rights Reserved X/13/2011JD durtions from 5 minutes to 10 dys in the mid-western United Sttes. Prrett [1997] lso used the GEV distriution to construct dimensionless frequency curves of nnul rinfll mxim of durtions of 2, 6, nd 24 hours within ech region in Montn. Using the L-moment rtio digrm, Asquith [1998] determined tht the GEV distriution ws n pproprite distriution for modeling the distriution of nnul rinfll mxim for durtions from 1 to 7 dys. Alil [1999] showed tht the nnul rinfll mxim of durtions from 5 minutes to 24 hours in Cnd were etter descried y the GEV distriution thn other distriutions, such s the generlized logistic nd EV1 distriutions. [4] Extreme rinfll exhiits different properties for different durtions in different regions. Anlysis of rinfll chrcteristics is importnt for choosing suitle rinfll distriution nd consequently estimting rinfll quntiles. Therefore, the ojective of this study is to investigte the chnge in the form of the nnul rinfll mxim frequency distriution with chnges in the time durtion, climte zone, nd distnce from the Gulf of Mexico nd then derive n entropy-sed distriution tht is sufficiently flexile for chrcterizing rinfll distriutions for different durtions in different climtic zones or t different distnces from the Gulf of Mexico. The performnce of the proposed entropy-sed distriution is ssessed using synthetic dt through Monte Crlo simultion nd oserved rinfll dt nd is shown to e promising lterntive distriution to the commonly used GEV distriution for modeling extreme rinfll vlues, especilly oservtions with high skewness. [5] This rticle is orgnized s follows. In section 2, the chnge in the form of empiricl distriutions of nnul 263

2 rinfll mxim is investigted. Using the entropy theory, generlized distriution is derived in section 3 nd the performnce of this distriution is ssessed y compring the GEV distriution in section 4. After the ppliction of the proposed entropy-sed distriution in section 5, conclusions re given in section Empiricl Frequency Distriutions 2.1. Study Are [6] The re selected for this study is the stte of Texs (longitude: W to W, ltitude: N to N). The climte of Texs is strongly influenced y physicl fetures, including the Gulf of Mexico. The pssge of frontl systems from northwest nd the moist ir moving inlnd from the Gulf of Mexico re the two competing influences tht dominte the climte of Texs, while proximity to the cost is the most importnt fctor tht determines the regionl climtic differences in Texs [North et l., 1995]. [7] There re three mjor types of climte in Texs, which re clssified s continentl, mountin, nd modified mrine, with no clerly distinguishle oundries, while the modified mrine zone is further clssified into four sutropicl zones [Lrkin nd Bomr, 1983]. The Mountin climte is dominnt in severl mountins of the Trns-Pecos region nd is not included in this study. The different climte zones of the Continentl nd Modified Mrine climte re revited s continentl steppe (CS), sutropicl rid (SA), sutropicl humid (SH), sutropicl suhumid (SSH), nd sutropicl steppe (SST), the oundries of which re pproximted with circles in Figure 1. In ddition, the U.S. Ntionl Wether Service (NWS) hs divided Texs into 10 climte divisions (including Upper Cost, Est Texs, High Plin, Trns-Pecos, nd so on) (ville t the wesite: Chrts_&_Mps/cwmp.htm) nd re lso used ccordingly in this study Dt Description [8] Rinfll dt for 15-minute, hourly, nd dily durtion for 99 NWS sttions, which re lso shown in Figure 1, were otined from the Ntionl Climtic Dt Center ( Not ll sttions hve rinfll dt of ll three durtions. To otin reltively long record of rinfll dt for different durtions, the 15-minute, hourly, nd dily rinfll dt were used s originl dt sources nd then rinfll dt of other durtions were produced sed on these originl dt. At different sttions, there exist missing vlues for some periods for ech time durtion. Only dt with no less thn 9-month oservtions for ech yer were selected for this study. The 15-minute dt re ville for few sttions nd they re of reltively short period. The hourly nd dily rinfll dt re ville for reltively more sttions nd for longer periods. The 45-minute nnul mxim were compiled from the 15-minute dt. Likewise, the rinfll dt for 12-hour durtion nd 7- nd 30-dy durtions were compiled from hourly nd dily dt, respectively. Annul rinfll mxim dt were then otined from these rinfll dt for different durtions, climte zones, nd distnces from the Gulf. Figure 1. Regions of climte zones in Texs nd rinfll sttions used in this study Chnge With the Time Durtion [9] Histogrms of nnul rinfll mxim of different durtions were prepred for ll sttions used in this study, nd the numer of ins ws pproximtely equl to the squre root of the numer of oservtions [Montgomery nd Runger, 2010]. The histogrms of smple sttion (411956) re shown in Figure 2 nd the length of oservtions (n) is lso shown in the figure. It ws oserved tht frequency distriutions for short durtions were more skewed, while those for long durtions were less skewed. For exmple, nnul rinfll mxim dt for sttion hd skewness vlue of 2.7 for 15-minute dt, ut 1.1 for 30-dy dt. To further show this chrcteristic, the ox plot of skewness vlues for 40 dt sets from reltively long record ( 22 yers) of different durtions is given in Figure 3. For exmple, the 75th percentile of skewness of the 15-minute durtion ws round 3.2, while tht for the 30-dy durtion ws 1.1. Though generl trend of the skewness of different time durtions cnnot e otined, sed on the selected dt sets, slight tendency of higher skewness for the short durtion is reveled. This is prtly ecuse for short durtions such s 15-minute, lrge mount of rinfll my occur within short time in certin cses exhiiting lrge skewness, while for long durtions, such s 30 dys, the rinfll is verged nd thus exhiits less skewness Chnge with the Climte one [10] In this section, frequency distriutions were nlyzed for different climte zones. No cler pttern of frequency distriutions in the SSH nd SST zones ws found from the dt from severl sttions selected in this study. Therefore, only the frequency distriutions for the rest of the climte zones were nlyzed. Two sttions for ech climte zone were selected to illustrte the typicl frequency distriution for 12-hour nnul rinfll mxim, s shown in Figure 4. The length (n) of record nd the vrince of dt (s 2 ) re lso shown in the figure SH [11] The SH zone lies in the estern prt of Texs, which is mostly noted for wrm summers [Lrkin nd Bomr, 1983]. 264

3 Figure 2. Histogrms nd proility density functions of nnul rinfll mxim of different durtions for sttion in the SH zone. Figure 3. Box plot of skewness vlues for nnul rinfll mxim of different durtions (40 dt sets for ech durtion). Ten sttions were selected for the study. This zone includes most prts of the Upper Cost nd Est Texs division. There re four rinfll-generting mechnisms tht exist in the Upper Cost re, leding to vrying ptterns from yer to yer s one or more of these controls chnge: In My, the typicl thunderstorm pttern is expected slightly inlnd, while the elt of mximum ctivity is long the cost y July; in Septemer, tropicl disturnces cn cuse very hevy rins for some yers, while in Decemer frontl ctivity ffects the region [Ntionl Fiers Informtion Center, 1987]. The Est Texs division is chrcterized y firly uniform sesonl rinfll, with slight mxim occurring in My nd Decemer, nd there is little vrition in the wether in the summer seson ecuse the influence of the Gulf of Mexico is dominnt [Ntionl Fiers Informtion Center, 1987]. The most widespred nd lengthy precipittion periods in Est Texs during spring nd utumn occur when the cold ir forms rrier, forcing the overriding moist Gulf ir to e deflected upwrd where it cools nd condenses [Crr, 1967]. [12] Two sttions ( nd ) were used for illustrtion of the typicl frequency distriution nd the histogrms re shown in Figures 4 nd 4. It cn e seen tht frequency distriutions re reltively smooth for this durtion, with higher vrince thn tht for climte zones CS nd SA. This region is long the cost, nd the rinfll pttern is ffected y the Gulf of Mexico. Since the proximity to the cost is the most determining fctor for regionl climte differences [North et l., 1995], the reson for this frequency distriution pttern my e due to the moderting influence of moisture from the Gulf of Mexico CS [13] The CS zone lies in the northwestern prt of Texs nd includes the regions similr to the High Plin division. The rinfll mount increses stedily through spring nd reches mximum in My or June, while the thunderstorm ctivity is lso on the rise during the spring seson [Ntionl Fiers Informtion Center, 1987]. In this region, summer is the wet seson nd thunderstorms re numerous in June nd July, ut egin to decrese in August [Ntionl Fiers Informtion Center, 1987]. Two sttions ( nd ) were used for illustrtion of the typicl frequency distriution nd the histogrms for 12-hour nnul rinfll mxim re shown in Figures 4c nd 4d. The vrinces for the two sttions re not s high s those for the SH climte zone. The frequency distriutions in this prt re reltively shrp, compred with those from the SH climte zone. The reson my e tht the mximum rinfll minly comes from the thunderstorms during the summer seson. 265

4 Figure 4. Histogrms nd proility density functions of 12-hour nnul rinfll mxim from different climte zones SA [14] The SA zone lies in the extreme western prt of Texs nd includes the region similr to the Trns-Pecos division. The sin nd plteu region of the Trns-Pecos fetures sutropicl rid climte, which is mrked y summertime rinfll nomlies of the mountin relief [Lrkin nd Bomr, 1983]. Rinfll reches its mximum in July nd in summer, where the rin comes minly from thunderstorms, often ffected y locl topogrphy [Ntionl Fiers Informtion Center, 1987]. In the Trns-Pecos region, the iggest percentge of rinfll occurring in this re is due to convective showers nd thundershower ctivity, while the thundershower ctivity is the primry contriutor of rinfll during lte-summer nd erly-utumn months [Crr, 1967]. Two sttions ( nd ) were selected for illustrtion of the typicl frequency distriution nd the histogrms of the 12-hour nnul rinfll mxim re shown in Figures 4e nd 4f. The vrinces for the two sttions re reltively smll nd the frequency distriutions re reltively shrp, compred with those from the SH climte zone. The reson for the vrition of rinfll my e tht the hevy rinfll in SA is minly produced due to the convective shower nd thundershower ctivity. [15] In generl, frequency distriutions for regions in extremely northern nd western prts (or the CS nd SA climte zones) were shrp; however, those for the regions in the southest ner the Gulf of Mexico (or the SH climte zone) were rther smooth. Although only few of the possile mechnisms of rinfll in ech region were investigted, the nlysis provided n insight into the reson for the specific rinfll frequency distriution pttern in ech climte zone Influence of the Distnce From the Gulf [16] The Gulf of Mexico is prticulrly importnt for the climte of Texs, s it provides the source of moisture nd modultes the verge sesonl nd diurnl cycles, prticulrly in the costl regions [North et l., 1995]. In generl, the verge nnul rinfll decreses with incresing distnce from the Gulf of Mexico. To ssess the effect of the Gulf of Mexico on the distriution of nnul rinfll mxim, 20 sttions were selected nd divided into two groups, ech with 10 sttions ccording to the distnce from the Gulf of Mexico. The histogrms of 12-hour mximum rinfll for four smple sttions re shown in Figure 5. It cn e seen tht the frequency distriutions in group II (more thn 250 miles wy from the Gulf) re not s smooth s those in group I (within 60 miles from the Gulf), nd the vrinces for the sttions in group II re not s high s those in group I. The smoothness of frequency distriutions in group I is prtly due to the closeness of rinfll sttions to the Gulf of Mexico. The effect of the Gulf of Mexico is reduced with distnce, nd the topogrphy my lso ply n importnt role in the rinfll-generting mechnism. The frequency distriution pttern for the two sttions in group II my e due to the mixed effect of the Gulf of Mexico nd topogrphy. [17] It is cler tht the frequency distriution vries with the durtion, climte zone, nd distnce from the Gulf. The question rises if proility distriution cn 266

5 Figure 5. Histogrms nd proility density functions of 12-hour nnul rinfll mxim of different distnces from the Gulf of Mexico (414309, 60 miles; , 20 miles; , 480 miles; , 450 miles). ccommodte the effect of these fctors. This is ddressed in wht follows. 3. Annul Rinfll Mxim Distriution Using Entropy Theory 3.1. Derivtion of Distriution [18] Let the nnul rinfll mxim for given durtion e represented s continuous rndom vrile, X є [, ], with proility density function, (pdf), f(x). For f(x), the Shnnon entropy, E, cn e defined s shown in eqution (1) [Shnnon, 1948; Shnnon nd Wever, 1949]: E ¼ fðþln x fðþdx x (1) where x is vlue of rndom vrile X with lower limit nd upper limit. Jynes [1957] developed the principle of mximum entropy, which sttes tht the proility density function should e selected mong ll the distriutions with the mximum entropy suject to the given constrints. The constrints cn e expressed in generl form s shown in eqution (2): g r ðþf x ðþdx x ¼ Eg ð r Þ r ¼ 0; 1; 2...; m (2) where function g r (x) is the known function with g 0 (x)=1, E(g r ) is the rth expected vlue otined from oservtions with E(g 0 ) = 1 (e.g., if g(x)=x, then E(g x ) is the men of x), nd m isthenumerofconstrints. [19] The mximum entropy-sed proility density function cn then e otined y mximizing the entropy in eqution (1), suject to eqution (2), using the method of Lgrnge multipliers, s shown in eqution (3) [Kesvn nd Kpur, 1992]: fðþ¼exp x ½ l 0 l 1 g 1 ðþ l x 2 g 2 ðþ... x l m g m ðþ x Š (3) where l r (r = 0,1,..., m) re the Lgrnge multipliers Mximum Entropy Distriution With Moments s Constrints [20] With the first four moments s constrints, the mximum entropy-sed proility density function (denoted s ENT4) defined on the intervl, [, ], with the function g(x) in eqution (2) expressed s g i (x)=x i (I =1,2,3,nd 4), cn e expressed s shown y eqution (4): fðþ¼exp x l 0 l 1 x l 2 x 2 l 3 x 3 l 4 x 4 [21] In this study, the lower limit of the intervl,, ws set to e zero, while the upper limit,, ws set to e 20 times the oserved mximum vlue. Since higher moments re involved in this distriution, reltively lrge dt set would e needed for the ccurcy of moment estimtion. [22] With the first four moments s constrints, the skewness, kurtosis, nd multiple modes cn e included in the resulting mximum entropy-sed distriution [ellner nd Highfield, 1988]. Ech mximum of the polynomil inside the exponentil corresponds to one mode, nd thus the multiple modes my exist in the mximum distriution [Smith, 1993]. This distriution hs een pplied for fitting imodl distriutions [Eisenerger, 1964]. Mtz [1978] exmined this distriution in detil nd ppliction of this distriution (4) 267

6 showed tht it fitted the oserved frequencies well. Compring this distriution with the Person distriution type III, ellner nd Highfield [1988] showed tht it provided etter fit, especilly t the tils. Smith [1993] used the mximum entropy-sed distriution with moments s constrints for decision nlysis to construct the distriution of vlue lottery nd showed tht the distriution with the first four moments s constrints performed well. [23] In this study, the entropy-sed distriution in eqution (4) is proposed s n lterntive for modeling extreme rinfll vlues. In ddition, the entropy-sed distriution with the first three moments s constrints ws lso selected s cndidte for modeling extreme rinfll vlues. From eqution (3), this distriution cn e expressed s shown y eqution (5): fðþ¼exp x l 0 l 1 x l 2 x 2 l 3 x 3 (5) [24] There re three prmeters ssocited with this entropy-sed distriution nd is denoted s ENT3 in this study Estimtion of Prmeters [25] The Lgrnge multipliers of eqution (4) cn e determined using eqution (2), where g r (r = 1,..,4) re the expecttion of the first four noncentrl moments. Generlly, the nlyticl solution does not exist nd the numericl estimtion of the Lgrnge multipliers is needed. To tht end, one cn mximize the function shown in eqution (6) [Med nd Ppnicolou, 1984; Wu, 2003]: Γ ¼ l 0 þ X4 l r g r r¼1 " # ¼ ln exp X4 l r g r ðþ x dx þ X4 l r g r (6) r¼1 [26] The mximiztion cn e chieved y employing Newton s method. Strting from some initil vlue, l (0), one cn solve for Lgrnge prmeters y updting l (1) through eqution (7) given elow: r¼1 l ð1þ ¼ l ðþ 0 H ; i ¼ 1; 2; 3; 4 i where the grdient, Г, is expressed s shown y eqution (8): " ¼ g i exp X4 l r g r ðþ x g i ðþdx; x i ¼ 1; 2; 3; 4 i r¼0 nd H is the Hessin mtrix whose elements re expressed s shown y eqution (9): H i;j ¼ exp exp exp X4 r¼0 X4 X4 r¼0! l r g r ðþ x g i ðþg x j ðþdx x! r¼0 l r g r ðþ x l r g r ðþ x! g i ðþdx x g j ðþdx; x i; j ¼ 1; 2; 3; 4 (9) 4. Model Evlution 4.1. Performnce Mesure [27] To quntify the performnce of the proposed distriution in modeling the extreme rinfll quntiles, the root men squre error (RMSE) ws used, which cn e defined s shown in eqution (10): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X n RMSE ¼ ðx i o i Þ 2 (10) n where n is the length of the oserved dt, x i re the quntiles estimted from the proposed distriution, nd o i re the oserved quntiles corresponding to the empiricl nonexceednce proilities estimted y the plotting position formul. In this study, the Gringorten plotting position formul ws used s shown in eqution (11) [Gringorten, 1963]: i¼1 i 0:44 P ¼ n þ 0:12 (11) where i is the rnk of the oserved vlues nd n is the length of the oserved dt Synthetic Dt From Known Distriution [28] Monte Crlo experiments were first crried out to compre the quntiles estimted from the GEV, ENT4, nd ENT3 distriutions. Two Monte Crlo simultions were conducted with rndom numers generted from the known GEV nd lognorml distriutions. Rndom numers of three different lengths (nmely, 40, 70, nd 100) were generted, which were used to pproximte the record length of the 15-minute, hourly, nd dily rinfll dt in this study. For the first simultion (S 1 ), the quntiles corresponding to different return periods (T = 5, 10, 25, 50, 100, nd 200 yers) were first ssessed with the rndom numers generted from the GEV distriution. For the second simultion (S 2 ), the quntiles corresponding to reltively long return period (T = 100 nd 200 yers) from the three distriutions were ssessed with the synthetic dt generted from lognorml distriution with different skewness vlues Rndom Numer From GEV Distriution [29] The GEV distriution hs een pplied extensively in hydrology for extreme rinfll nlysis. Its proility density function is defined s shown y eqution (12): 8 1 h x u i 1=k 1exp h x u i 1=k >< 1þ k 1þk ;k 6¼ 0 fðx; m; s; k Þ¼ s >: 1 s s h x u exp s s exp x u i ; k ¼ 0 s (12) where k, s, nd u re the shpe, scle, nd loction prmeters. In this study, the MATLAB (The Mthworks, Inc., Ntick, Mss.) function, gevfit, ws used for the prmeter estimtion of the GEV distriution with mximum likelihood method. [30] One thousnd dt sets of rndom numers with different smple sizes (n = 40, 70, nd 100) were generted from this prent distriution. The GEV, ENT4, nd ENT3 distriutions were then fitted to these dt sets nd the quntiles corresponding to different return periods were otined. Prmeters (k, s, nd u) of the prent proility density 268

7 function were ssigned (0.3, 0.3, nd 1.2), nd the proility density function is shown in Figure 6. The medin nd the RMSE vlues of the estimted quntiles for simultion S 1 re shown in Tles 1 nd 2. From the medin vlues, it cn e seen tht for short return periods (T 50 yers), the medin vlues from the ENT4 nd GEV distriutions were close to ech other for ech smple size. For exmple, for smple size n = 100, the medin vlues from GEV nd ENT4 for return period of 50 yers were 3.42 nd 3.40, respectively, while the oserved vlue ws Generlly, the RMSE vlues of the ENT4 distriution were slightly lrger thn those of the GEV distriution; however, these results were cceptle. For the quntiles corresponding to the reltively long return periods (100 nd 200 yers), the Figure 6. Prent distriutions for Monte Crlo simultion. () GEV distriution nd () lognorml distriution with different skewness(s). medin quntile from the ENT4 distriution is modertely underestimted, while tht from the GEV distriution ws close to the oserved vlue. This is not unexpected, since the rndom numers were generted from the GEV distriution nd then the GEV distriution ws fitted. Generlly, ENT4 modeled the dt generted from the GEV distriution well, especilly when the smple size ws reltively lrge. The ENT3 distriution lso estimted the quntiles reltively well for short periods (T 25 yers), while it did not model the quntiles well corresponding to reltively long return periods (T 50 yers) Rndom Numer From Lognorml Distriution [31] The proility density function of the log-norml distriution cn e expressed s shown in eqution (13):! 1 fðþ¼ x p x ffiffiffiffiffiffiffiffiffiffi exp ð ln x u Þ2 2ps 2 2s 2 (13) where u is the men in the log scle nd s 2 is the vrince in the rel scle. The skewness coefficient, pffiffiffiffiffiffiffiffiffiffiffiffiffiffis, is relted with the vrince, s 2,ss¼ e s2 þ 2 e s2 1. [32] One thousnd dt sets of rndom numers with different smple sizes (n = 40, 70, nd 100) with different skewness vlues of 1, 2, 2.5, nd 3 were generted from the lognorml distriution nd were used for comprison. Prmeter u ws ssigned vlue of 0.3, while the stndrd devitions corresponding to different skewness vlues were ssigned vlues of 0.31, 0.55, 0.64, nd 0.72, respectively. The pdfs of the prent distriution with these prmeters re shown in Figure 6. The ojective of this simultion ws to show the performnce of these distriutions in modeling dt with different vlues of skewness. The medin nd RMSE vlues of the estimted quntiles, x 100 nd x 200, for return periods of 100 nd 200 yers corresponding to nonexceednce proilities of 0.99 nd re shown in Tles 3 nd 4. Tle 1. Medin of Estimted Quntiles With Rndom Numer Generted From the GEV Distriution Estimted Quntile n =40 n =70 n = 100 Return Period (Yers) Oserved Quntile GEV ENT ENT3 GEV ENT ENT3 GEV ENT ENT Tle 2. RMSE of Estimted Quntiles With Rndom Numer Generted From the GEV Distriution RMSE n =40 n =70 n = 100 Return Period (Yers) GEV ENT ENT3 GEV ENT ENT3 GEV ENT ENT

8 Tle 3. Medin of Estimted Quntiles (x 100 nd x 200 ) With Rndom Numer Generted From the Lognorml Distriution With Different Skewness (k) Sknewness Quntile Oservtion Estimted Quntile n =40 n =70 n = 100 GEV ENT ENT3 GEV ENT ENT3 GEV ENT ENT3 k =1 x x k =2 x x k = 2.5 x x k =3 x x Tle 4. RMSE of Estimted Quntiles (x 100 nd x 200 ) With Rndom Numer Generted From the Lognorml Distriution With Different Skewness (k) Sknewness Quntile RMSE n =40 n =70 n = 100 GEV ENT ENT3 GEV ENT ENT3 GEV ENT ENT3 k =1 x x k =2 x x k = 2.5 x x k =3 x x [33] For the cse with skewness coefficient k = 1, the medin quntile from the ENT4 distriution ws not s close to the oserved vlues s from the GEV distriution. However, the difference etween the medin quntile estimted from GEV nd tht from ENT4 ws reltively smll, especilly for reltively lrge smple sizes. For exmple, for n = 100, the medin vlues from GEV nd ENT4 were 2.80 nd 2.74, with the oserved vlue eing Generlly, the RMSE vlues of the two distriutions were close to ech other. For exmple, the RMSE vlues of GEV nd ENT4 for x 200 were 0.43 nd 0.45, respectively, for n = 70. The performnce of ENT4 improved with the increse in smple size. Generlly, the performnces of ENT4 nd GEV were comprle in this cse. [34] For skewness vlues of k = 2 nd 2.5, the medin vlues from GEV distriution were modertely overestimted, while those from ENT4 were modertely underestimted. When the smple size ws reltively smll (n = 40), the GEV distriution performed slightly etter thn did the ENT4 distriution for the medin vlues. However, the RMSE vlue from the GEV ws higher thn tht from the ENT4 distriution. When the smple size ws reltively lrge (n = 100), the ENT4 distriution performed reltively etter thn did the GEV distriution for the medin vlue, while their performnce ws comprle for the RMSE vlues. For exmple, for the cse with k = 2.5 nd smple size n =100, the medin vlues from GEV nd ENT4 corresponding to the 100-yer return period were 6.60 nd 5.92, while the oserved vlue ws The corresponding RMSE vlues for GEV nd ENT4 were, respectively, 1.63 nd 1.69, which re comprle. The performnce of the ENT4 distriution improved with the increse in smple size. [35] For the skewness k = 3, the medin vlue estimted from GEV distriution ws overestimted significntly, while ENT4 still performed reltively well for estimting quntiles, especilly when the smple size ws reltively lrge. For exmple, the true quntile corresponding to the 100-yer return period ws 7.13, while the quntiles from GEV nd ENT4 with smple size n = 70 were 8.25 nd 6.83, respectively. The corresponding RMSE vlues were 3.05 nd 2.75, indicting tht ENT4 performed reltively etter. [36] Though the RMSE vlues from the ENT3 distriution were comprle with those from the ENT4 distriution nd sometimes even smller thn those from the ENT4 distriution, generlly the medin vlue from ENT3 ws underestimted significntly for ech smple size nd for ech cse with different skewness vlues. These results showed tht generlly ENT3 did not perform s well s the GEV nd ENT4 distriutions nd did not model extreme vlues stisfctorily Summry [37] The Monte Crlo simultion, S 1, showed tht generlly the ENT4 distriution ws comprle to the GEV distriution in modeling extreme rinfll vlues. Since the GEV distriution hs een extensively pplied for modeling extreme vlues, the results from the first simultion, S 1,showedtht the ENT4 distriution would lso e cndidte for modeling extreme vlues. The Monte Crlo simultion, S 2, showed tht the performnce of ENT4 distriution ws comprle with GEV for low skewness, especilly when the smple sizes were reltively lrge (n 70). When the skewness ws reltively high ( 2), the ENT4 distriution performed comprle with or reltively etter thn the GEV distriution for estimting quntiles corresponding to reltively long return periods, especilly when the smple size ws lrge. Botero nd Frncés 270

9 [2010] lso found tht the GEV distriution led to lrge errors for quntile estimtion corresponding to long return periods for high skewness. Synthetic dt from other distriutions (e.g., gmm distriution) were lso used for comprison, nd generlly similr results were otined (not presented). Thus, it cn e concluded from the Monte Crlo simultion tht generlly ENT4 provided n lterntive to the commonly used GEV distriution nd should e preferle for oservtions with high skewness. The ENT3 distriution ws not suitle for modeling extreme vlues. [38] The GEV distriution cn e pplied to ccount for nonsttionrity when the prmeters vry with set of covrites [Ktz et l., 2002; Towler et l., 2010]. From the structure of the ENT4 distriution, the four Lgrnge multipliers, l 1, l 2, l 3, nd l 4, correspond to the first four moments of the dt, ut they do not hve similr connottions of loction, scle, nd shpe prmeters. Thus, the incorportion of covrites (e.g., time) to ccount for nonsttionrity in the nlysis would e difficult, which would e disdvntge of the ENT4 distriution, compred with the GEV distriution Oserved Rinfll Dt [39] To further compre the performnce of the GEV distriution nd ENT4 distriution, the oserved rinfll dt for different durtions (15-minute, 45-minute, 1-hour, 12-hour, 1-dy, 7-dy, nd 30-dy) from 40 sttions were used. Using the RMSE mesure, the three distriutions were compred sed on the oserved nd estimted quntiles corresponding to the sme empiricl cumultive proilities s otined from eqution (11). Note tht the results from the oserved dt my not e s ccurte s those from the Monte Crlo simultion due to dt limittion nd error in pproximting the cumultive proility, ut still would e meningful for rough comprison. The numer of sttions for ech distriution performing the est (with the lest RMSE) is shown in Tle 5. For ll durtions, the ENT4 distriution performed the est for the lrgest numer of sttions. For exmple, for the nnul rinfll mxim of the 12-hour durtion of the 40 dt sets, the ENT4 distriution performed the est for 36 sttions ccording to RMSE. From these results, it cn e seen tht the ENT4 distriution would e good cndidte for modeling nnul rinfll mxim. 5. Appliction [40] The entropy-sed distriution ws used to fit the rinfll dt in section 2, s shown in Figures 2, 4, nd 5. These figures show tht the entropy-sed distriution (ENT4) fitted the empiricl histogrms well for the rinfll Tle 5. Numer of Sttions With the Lest RMSE From Ech Distriution Durtion ENT4 GEV ENT3 15-minute minute hour hour dy dy dy dt of different durtions, climte zones, nd different distnces from the Gulf of Mexico. [41] The GEV distriution ws lso pplied here for further comprison with the ENT4 distriution. For ech durtion (15-minute, 45-minute, 1-hour, 12-hour, 1-dy, 7-dy, nd 30-dy), totl of 10 dt sets were used in ech climte zone (except tht for the SA climte zone, where six dt sets were used for 15-minute nd 45-minute durtion due to dt limittion). The numer of sttions (nd percentge) tht ENT4 performed etter thn GEV in different climte zones is shown in Tle 6. Tking the result in the CS climte zone s n exmple, the ENT4 distriution performed etter for ll durtions for t lest 8 out of 10 dtsets (or 80%). [42] The ENT4 distriution ws lso compred with the GEV distriution for different distnces from the Gulf (groups I nd II) with totl of 10 dt sets in ech group. There were not enough sttions with reltively long record of 15-minute dt in group I nd thus only the hourly (1- nd 12-hour) nd dily dt (1-, 7-, nd 30-dy) were used. The numer of sttions (nd percentge) tht ENT4 performed etter thn GEV for the two groups is shown in Tle 7. It cn e seen tht generlly the ENT4 distriution performed etter thn the GEV distriution. Tking the 1-hour dt s n exmple, the ENT4 distriution hd less RMSE for 10 cses (or 100%) for group I nd eight cses (or 80%) for group II, respectively. [43] An IDF curve is defined s reltionship of rinfll intensity occurring over certin durtion, d, with different return periods. The nnul rinfll mxim distriution cn then e employed for the construction of IDF curves [Singh, 1992], which cn e utilized for hydrulic design, such Tle 6. Numer of Sttions (nd Percentge) With Better Performnce From ENT4 Distriution for Different Climte ones nd Durtions Durtion CS SA SH Numer Percentge (%) Numer Percentge (%) Numer Percentge (%) 15-minute minute hour hour dy dy dy For 15- nd 45-minute dt of SA climte zone, only six sttions re selected due to dt limittion. Tle 7. Numer of Sttions (nd Percentge) With Better Performnce From ENT4 Distriution for Different Distnces nd Durtions Durtion Group I Group II Numer Percentge (%) Numer Percentge (%) 1-hour hour dy dy dy

10 is comprle with the GEV distriution nd is preferle for the dt sets with high skewness. Furthermore, the ENT4 distriution performs etter for most cses thn the GEV distriution in modeling the quntiles sed on the oserved rinfll dt. These results from the synthetic dt nd oserved rinfll dt show tht the ENT4 distriution is good cndidte to model the nnul rinfll mxim. The ENT4 distriution is then pplied to the frequency distriution of nnul rinfll mxim of different durtions, climte zones, nd distnces from the Gulf, nd further comprison etween the ENT4 nd GEV distriutions shows tht the ENT4 distriution performs well in modeling extreme rinfll. Anlysis of the chnging ptterns of rinfll distriution with the durtion, climte zone, nd distnce from the Gulf of Mexico sheds some light on the nlysis of rinfll of different durtions in Texs. Figure IDF curves of different durtions for sttion s storm sewers nd prking lots, etc. The hourly nnul rinfll dt for sttion were used to construct the IDF curves, s shown in Figure 7. The empiricl return period (T E ) ws otined from the Gringorten plotting position formul s T E = 1/(1 P), where P is the nonexceednce proility. The empiricl return periods were lso plotted on the IDF curves. (The theoreticl return period ws extrpolted round the highest empiricl return period to void lrge errors.) Note tht the ccurcy of the empiricl return period for the highest-rnked pek flowsislimited [Stedinger et l., 1993; Beckers nd Alil, 2004]. Generlly, the return period from the IDF curves fitted the empiricl return period well. For exmple, for the return period 12.2 yers of 1-hour durtion, the theoreticl rinfll quntile from the ENT4 distriution ws 2.6 inches, while the oserved quntile ws 2.4 inches. 6. Conclusions [44] Frequency chrcteristics of nnul rinfll mxim from different sttions in Texs re nlyzed. [45] Results show tht frequency distriutions of nnul rinfll mxim re highly skewed for short durtions, such s 15-minute, nd tend to e smoothed when the durtion is reltively long. The distriutions lso show different ptterns cross different climte zones. In northern nd western prts, such s the CS nd SA climte zones, distriutions re shrp; however, they re reltively smooth in the southest, such s the SH climte zone. The possile reson is tht in the CS nd SA climte zones, hevy rinfll is minly produced y thunderstorms, while in the SH climte zone the moisture from the Gulf of Mexico is the moderting fctor. For the other climte zones, no cler pttern is found, which my e due to the mixed effect of different rinfll mechnisms. The frequency distriution of rinfll ner the Gulf of Mexico is smoother thn tht fr wy from the Gulf. The reson my e tht the Gulf of Mexico serves s the moisture source. 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