May 28, 2002 - forecast rainfall over the Upper Parramatta River Catchment in Sydney is based on the application of a pattern recognition technique using an ...
C HAP TER 10
A neural network approach to rainfall forecasting in urban environments J AMES E. BALL & KIN C HO I LUK Water Research Laboratory, School of Civil and Environmental Engineering, The University of New South Wales, Sydney, New South Wales, Australia
ABS TR AC T: An e ffe ctive flood wa rning s ys te m in urba n a re a s m us t provide the wa rning with s ufficie nt le a d tim e for a n a ppropria te re s pons e by the re le va nt e m e rge ncy s e rvice s a nd the a ffe cte d com m unity. This re quire m e nt pos e s a critica l proble m a s m os t urba n ca tchm e nts a re cha ra cte ris e d by a fa s t hydrologic re s pons e to s torm e ve nts . The a pproa ch us e d he re to fore ca s t ra infa ll ove r the Uppe r P a rra m a tta R ive r C a tchm e nt in S ydne y is ba s e d on the a pplica tion of a pa tte rn re cognition te chnique us ing a n Artificia l Ne ura l Ne twork. It a s s um e s tha t the future ra infa ll is a function of a dis cre te num be r of pa s t s pa tia l a nd te m pora l ra infa ll re cords ; a n im porta nt ta s k, the re fore , is the de te rm ina tion of the num be r of s pa tia l a nd te m pora l ra infa ll re cords ne ce s s a ry for a ccura te pre diction of future ra infa ll. The ra infa ll pre diction m ode l pe rform e d be s t whe n a n optim a l a m ount of s pa tia l a nd te m pora l ra infa ll inform a tion wa s provide d to the ne twork.
1 INTR O DUC TIO N Fla s h floods a re a life -thre a te ning phe nom e non, which a ls o re s ults in e conom ic los s e s a nd s ocia l dis ruption. Ha ndm e r et al. (1988), for e xa m ple , e s tim a te d the dire ct e conom ic los s e s for re s ide ntia l prope rty in the Toonga bbie C re e k ca tchm e nt (a s ubca tchm e nt within the Uppe r P a rra m a tta R ive r C a tchm e nt in the we s te rn s uburbs of S ydne y, Aus tra lia ) a s be ing a pproxim a te ly $5 m illion (1986 Aus tra lia n dolla rs ) for the 1% Annua l Exce e da nce P roba bility (AEP ) e ve nt. C om m e rcia l a nd indus tria l a ctivitie s within the ca tchm e nt we re a ls o e s tim a te d to s uffe r e conom ics los s e s of a s im ila r m a gnitude . In the de ca de s ince the s e e s tim a te s we re m a de , s ignifica nt a dditiona l urba nis a tion, which de cre a s e s the pote ntia l re s pons e tim e of wa rning s ys te m s , toge the r with infla tion, m a ke the s e a n unde re s tim a te in pre s e nt da y te rm s . De ve lopm e nt of a n e ffe ctive flood wa rning s ys te m ca n be e xpe cte d to m itiga te the s e los s e s . For a n e ffe ctive flood wa rning s ys te m , the re ne e ds to be s ufficie nt tim e be twe e n the re cognition of a like ly flood proble m a nd its 1
2 James E. Ball and Kin Choi Luk occurre nce for dis s e m ina tion of flood wa rning m e s s a ge s a nd the a ctiva tion of a ppropria te e m e rge ncy s e rvice s . Accura te fore ca s ts of flood s hould he lp to incre a s e the tim e be twe e n re cognition of a like ly flood a nd its s ubs e que nt occurre nce . This ne e d ha s prom pte d the de ve lopm e nt of a n e ffe ctive ra infa ll fore ca s ting s ys te m . In s uch a s ys te m , it m us t be re cognis e d tha t ra infa ll is a dyna m ic proce s s , which va rie s in s pa ce a nd tim e . The re is a ne e d to cons ide r the s pa tia l va ria bility in a ny ra infa ll fore ca s t m ode l a nd to tra ns form the point da ta into a re a l dis tributions us ing s pa tia l tools within a G e ogra phic Inform a tion S ys te m (G IS ). The re a re two ba s ic a pproa che s s uita ble for the de ve lopm e nt of a ra infa ll fore ca s ting m ode l. The s e ca n be ca te goris e d a s (i) the proce s s m ode l a pproa ch, in which the phys ica l proce s s e s influe ncing ra infa ll a re a na lys e d a nd proce s s m ode ls a re de ve lope d, but this a pproa ch m a y not be fe a s ible be ca us e :
ra infa ll is a com ple x dyna m ic s ys te m which va rie s both in s pa ce a nd tim e re s ulting in proble m s a s s ocia te d with the de finition of s olution s pa ce bounda rie s ; e ve n if the ra infa ll proce s s e s ca n be de s cribe d concis e ly a nd com ple te ly, the volum e of ca lcula tions involve d m a y be prohibitive ; a nd the da ta tha t a re a va ila ble to a s s is t in the de finition of control va ria ble s for the proce s s m ode ls , s uch a s pre s s ure , e va pora tion, wind s pe e d a nd dire ction a re lim ite d in both the s pa tia l a nd te m pora l dim e ns ions .
a nd (ii) the “bla ck box” m ode l a pproa ch, in which pa tte rn re cognition te chnology is us e d to pre dict the m os t like ly future pa tte rn of ra infa ll in tim e a nd s pa ce . The a im is to e xtra ct from the his torica l ra infa ll re cords the e s s e ntia l pa tte rns ne ce s s a ry for the pre diction of future ra infa ll e ve nts . The re a re m a ny a lte rna tive te chnique s for the e xtra ction of the e s s e ntia l fe a ture s from his torica l re cords ; the te chnique us e d he re is ba s e d on a n Artificia l Ne ura l Ne twork (NN). R a infa ll fore ca s ts a t ra in ga uge loca tions provide only s ca tte re d da ta a nd fore ca s ts of ra infa ll ove r the ca tchm e nt. Although the ra infa ll fore ca s ts m a y ha ve a fine te m pora l re s olution, the true a re a l ra infa ll which produce s runoff is not known. This highlights a critica l proble m in conve ntiona l ra infa ll-runoff m ode lling whe re s im plifie d a pproa che s , s uch a s Thie s s e n P olygons , a re us e d without ta king the s pa tia l dis tribution nor the dyna m ic prope rtie s of the ra infa ll into a ccount. The s e s im plifie d a pproa che s ca n re s ult in la rge e rrors in runoff e s tim a tion (Fonta ine 1991; Urbona s et al. 1992). In re s pons e to this ne e d, Ba ll a nd Luk (1998) de ve lope d a m e thod to m ode l the s pa tia l va ria bility of ra infa ll us ing point m e a s ure m e nts of ra infa ll a s the input inform a tion. The ra infa ll fore ca s t produce d from a NN ca n the re fore be pa s s e d to this m ode l for the de te rm ina tion of future ra infa ll pa tte rns ove r a ca tchm e nt. The inte gra tion of the s e tools provide s a powe rful fore ca s ting s olution.
Rainfall forecasting in urban environments 3 2 THE UP P ER P AR R AMATTA R IVER C ATC HMENT
2.1 Catchment Details The Uppe r P a rra m a tta R ive r C a tchm e nt is us e d a s the s tudy a re a for the de ve lopm e nt of the ra infa ll fore ca s ting s ys te m . This ca tchm e nt in the we s te rn s uburbs of S ydne y, Aus tra lia (Figure 1), dra ins into S ydne y Ha rbour. The tida l lim it of the rive r is the C ha rle s S tre e t W e ir. Im m e dia te ly ups tre a m of the W e ir, is the the P a rra m a tta ce ntra l bus ine s s dis trict which ha s s uffe re d cons ide ra ble flood da m a ge ove r a num be r of ye a rs . The re a re two m a in tributa rie s of the P a rra m a tta rive r within the Uppe r P a rra m a tta R ive r C a tchm e nt: the Toonga bbie C re e k a nd Da rling Mills C re e k. The ca tchm e nt is s te e p with a n a ve ra ge s lope of 1.2% .
Figure 1 - Location of Sydney, Australia
The dom ina nt la nd us e is a m ix of re s ide ntia l, com m e rcia l, indus tria l a nd ope n s pa ce (pa rkla nd) a re a s . C ons ide ra ble de ve lopm e nt a s a re s ult of the ra pid incre a s e in popula tion a nd dwe llings ha s occurre d within the ca tchm e nt ove r the pa s t two de ca de s , a s s hown in Ta ble 1 (Aus tra lia n Bure a u of S ta tis tics , C e ns us of P opula tion a nd Hous ing, pe rs ona l com m unica tion). The incre a s e in the num be r of dwe llings from 1986 to 1991 is 18% , which is s ignifica nt from a hydrologica l a nd flood m a na ge m e nt vie wpoint. A rough e s tim a tion of the incre a s e in im pe rvious a re a ca n be obta ine d by a s s um ing a n a ve ra ge dwe lling s ize of 200 s qua re m e tre s a nd a llowing 50% of this a re a for a s s ocia te d im pe rvious a re a s uch a s roa ds a nd footpa ths . Us ing this a pproa ch, a n e s tim a te for the incre a s e in im pe rvious a re a is 6.5 km 2 or 6% . Table 1 - Population and Dwellings in Parramatta Total Population Year
% Change to Previous 5 Year
Total Occupied Dwellings
% Change to Previous 5 Year
4 James E. Ball and Kin Choi Luk 1976 1981 1986 1991
348398 374190 384601 435478
-7.4 2.8 13.2
100246 111064 119229 140900
-10.8 7.4 18.2
The e ffe ct of this urba nis a tion ha s be e n continuing incre a s e s in e s tim a te s of the pe a k le ve l for a ll e ve nts . To m itiga te the s ocia l a nd e conom ic los s e s a s s ocia te d with the s e floods , the Uppe r P a rra m a tta R ive r C a tchm e nt Trus t (UP R C T) wa s ins titute d in 1989 with the ta s k of m a na ging flood m itiga tion m e a s ure s within the ca tchm e nt, a m ong othe r dutie s .
2.2 Rainfall Records The re a re s ixte e n continuous ly re cording ra in ga uge s within the ca tchm e nt (Figure 2). The m a jority of the s e ga uge s ha ve be e n ins ta lle d by the UP R C T s ince its form a tion. C ons e que ntly, long-te rm re cords a re not a va ila ble . This give s a n a ve ra ge point ra infa ll s a m ple for e ve ry 7 km 2 of ca tchm e nt. W hile this is a high de ns ity of inform a tion from ra in ga uge s for m os t ca tchm e nts , Urbona s et al. (1992) s ugge s t tha t a n e ve n highe r de ns ity of s pa tia l inform a tion is re quire d if a ccura te pre dictions of ca tchm e nt re s pons e a re to be obta ine d for conve ctive s torm e ve nts (Ta ble 2).
Figure 2 - Rain Gauges within the Upper Parramatta River Catchment
Table 2 - Accuracy of Rainfall-Runoff models (after Urbonas et al. 1992) Gauge Density (km2/gauge) 8.0 4.0 2.7 2.0 1.6
Range (%) -100.0 to 150.0 -75.3 to 94.5 -32.2 to 63.66 -32.2 to 18.8 0.0 to 0.0
Mean Deviation (%) -24.2 0.5 15.8 -0.9 0.0
Rainfall forecasting in urban environments 5
R e cords from the 16 ra in ga uge s within the Uppe r P a rra m a tta R ive r ca tchm e nt we re obta ine d from J a nua ry 1991 to S e pte m be r 1996. During this pe riod, 34 s torm e ve nts occurre d whe re the da ily ra infa ll tota l e xce e de d 20 m m . Am ong the s e s torm s , m ore tha n 70% of the s torm s we re conve ctive , the re s t we re fronta l s torm s . The conve ctive s torm s occurre d pre dom ina ntly during the s um m e r a nd a utum n s e a s ons , while the fronta l s torm s we re m ore e ve nly dis tribute d through the ye a r. The da ta s e rie s we re e xtra cte d in 15-m inute inte rva ls . Any m is s ing ra infa ll va lue s due to m a lfunctioning of ga uge s or e rrors during tra ns m is s ion of the da ta we re e s tim a te d from ne ighbouring ra in ga uge s us ing the s pa tia l ra infa ll m ode l of Ba ll a nd Luk (1996; 1998).
3 R AINFALL FO R EC AS TING - P O INT P ATTER NS
3.1 Concepts To de ve lop the propos e d ra infa ll fore ca s t m ode l, the continuous proce s s of ra infa ll wa s re pre s e nte d by a dis cre te Ma rkovia n proce s s , in which the ra infa ll va lue a t a give n loca tion in s pa ce a nd tim e is a function of a finite s e t of pre vious re a lis a tions . W ith this a s s um ption, a s im ple m ode l s tructure ca n be e xpre s s e d a s :
X ( t 1) f [X( t ), X( t 1), X( t 2),..., X ( t k 1)] e( t )
(1)
whe re X(t) = [x 1t, x 2t, …, x Nt]T re pre s e nts a ve ctor of ra infa ll va lue s x 1t, x 2t,.., x Nt a t N diffe re nt ga uge s ite s a t tim e t, whe re T de note s the tra ns pos e ope ra tor, f[ ] is a non-line a r m a pping function, which s ha ll be a pproxim a te d us ing a n NN, e (t) is a m a pping e rror (to be m inim is e d) a nd k is the (unknown) num be r of pa s t re a lis a tions contributing to ra infa ll a t the ne xt tim e s te p, re fe rre d to a s the m ode l la g. If k is e qua l to 1, ra infa ll a t the ne xt tim e s te p is re la te d only to the pre s e nt ra infa ll, re pre s e nting a la g-1 m ode l. The m ulti-la ye r fe e d-forwa rd ne ura l ne twork (MLFN) offe rs a s tra ightforwa rd a pproa ch to re pre s e nt the a bove ra infa ll m ode l. Furthe r pa rticula rs a bout the workings of this ne twork a re provide d in cha pte r 2. The MLFN is pre s e nte d with the curre nt a nd pa s t ra infa ll va lue s a s inputs , e .g. X(t), …, X(t-k+1), a nd the ne xt ra infa ll va lue X(t+1) is us e d a s the ne twork output. The re a re , howe ve r, s e ve ra l dra wba cks a s s ocia te d with this a pproa ch. Firs tly, s ince the m ode l la g k is unknown, a le ngthy tria l proce s s is re quire d to de te rm ine the optim a l va lue of k. S e condly, for a ne twork with a high orde r of la g, la rge num be rs of input node s a re re quire d. C ons e que ntly, the num be r of pa ra m e te rs will incre a s e , m a king the ne twork unne ce s s a rily com ple x. Fina lly, the MLFN is a s ta tic m ode l which m a y not be a ble to m ode l the dyna m ic na ture of ra infa ll proce s s e s . The tim e de la y ne ura l ne twork (TDNN) offe rs a n a lte rna tive s olution, which ca n e ffe ctive ly m ode l the ra infa ll proce s s while ke e ping a m inim um num be r of pa ra m e te rs . More de ta ils a bout this type of ne twork a re provide d in cha pte r 5. A dis tinctive fe a ture of a TDNN is the us e of pa rtia l conne ctions ; this dra m a tica lly re duce s the num be r of we ights pre s e nte d in the ne twork com pa re d with a fully conne cte d MLFN a rchite cture . In a ddition,
6 James E. Ball and Kin Choi Luk the TDNN ha s be e n de ve lope d for de te cting loca l fe a ture s within a la rge r pa tte rn; this fe a ture de te ction a bility is ve ry us e ful for the ta s k of ra infa ll fore ca s ting. A TDNN, howe ve r, is s till a s ta tic m ode l. This s ta tic re pre s e nta tion ha s s e ve ra l dra wba cks . Firs tly, if the ra infa ll proce s s ha s a long te rm m e m ory, a la rge num be r of inputs node s a re re quire d, re s ulting in a ne twork conta ining a la rge num be r of fre e pa ra m e te rs . S e condly, the num be r of pa s t ra infa ll inputs ha s to be de te rm ine d e xpe rim e nta lly. A le ngthy tria l a nd e rror proce s s is re quire d. A dyna m ic m ode l m a y ove rcom e this proble m . A dyna m ic m ode l ca n be re pre s e nte d by a NN with fe e dba ck conne ctions to fe e d pa s t s ta te s of the s ys te m ba ck to the ne twork. S uch a ne twork with fe e dba ck conne ctions is ca lle d a re curre nt ne twork. A re curre nt ne twork pos s e s s e s the cha ra cte ris tic of dyna m ic m e m ory. In a ddition, a re curre nt ne twork re duce s the num be r of inputs a nd cons e que ntly the num be r of pa ra m e te rs , s pe e ding up the ca lcula tions . R e curre nt ne tworks ca n be cla s s ifie d a s fully or pa rtia lly re curre nt. Fully re curre nt ne tworks ca n ha ve a rbitra ry fe e dforwa rd a nd fe e dba ck conne ctions , a ll of which a re tra ina ble . Howe ve r, the tra ining of the s e ne tworks is ve ry com plica te d. For pra ctica l a pplica tions , the pa rtia l re curre nt ne ura l ne twork (P R NN) is m ore a ppropria te be ca us e the tra ining of s uch ne tworks is s im ila r to tha t of the MLFN a nd the re fore m uch e a s ie r tha n a fully re curre nt ne twork. In pa rtia lly re curre nt ne tworks , the m a in ne twork s tructure is fe e dforwa rd. The fe e dforwa rd conne ctions a re tra ina ble . The fe e dba ck conne ctions a re form e d through a s e t of "conte xt" units tha t a re not tra ina ble , which s im plifie s the tra ining proce s s . The conte xt units s tore s om e pa s t s ta te s of the s ys te m , a nd s o the outputs of the ne twork de pe nds on a n a ggre ga te of the pre vious s ta te s a s we ll a s the curre nt input. In a ddition, a P R NN ca n be de ve lope d for re a l tim e a pplica tions . Am ong the a va ila ble pa rtia lly re curre nt ne tworks , the Elm a n ne twork (Elm a n 1990) is one of the s im ple s t type s tha t ca n be tra ine d us ing s ta nda rd ba ck propa ga tion le a rning a lgorithm s (R um e lha rt et al. 1986). This type of
ne twork wa s a dopte d a nd cus tom is e d for the pre s e nt s tudy. Figure 3 is the s tructure of a n Elm a n ne twork a s ca n be us e d for ra infa ll fore ca s ting. Figure 3 - An Elman Network for Rainfall Forecasting
As s ta te d pre vious ly, a n im porta nt fe a ture of the Elm a n ne twork is the inclus ion of a s pe cia l s e t of conte xt units to re ce ive fe e dba ck s igna ls from
Rainfall forecasting in urban environments 7 the hidde n node s . The function of the conte xt units is to s tore inform a tion from the pre vious tim e s te ps . To a chie ve this , the conte xt units m a ke a copy of the a ctiva tion of hidde n node s in the pre vious tim e s te p. The re fore , a t tim e t the conte xt units ha ve s om e s igna ls re la te d to the s ta te of the ne twork a t tim e t-1. As a re s ult, the ra infa ll a t tim e t+1 is a function of the ra infa ll a t tim e t a nd the pre vious s ta te s of the s ys te m re pre s e nte d by the a ctiva tion of the hidde n node s a t tim e t-1, e xpre s s e d by X(t+1) = g(X(t), O (t-1)) + e (t)
(2)
whe re X(t+1) a re ra infa ll a t tim e t+1, which a re outputs of the ne twork, X(t) a re ra infa ll a t tim e t, which a re inputs of the ne twork, O (t-1) a re the a ctiva tions of the hidde n node s a t tim e t-1 a nd copie d ba ck to the conte xt units for input a t tim e t, g() is a re curre nt m a pping function a nd e (t) is the m a pping e rror. As s hown in Figure 3, the input node s conta in N e le m e nts re pre s e nting the s pa tia l dim e ns ion of the ra infa ll. For e xa m ple , if ra infa ll fore ca s ts a t 16 ga uge pos itions a re re quire d, the n the num be r N s hould be s e t to 16. The tota l num be r of hidde n node s is H, which is the s a m e a s the num be r of conte xt units . H is the ke y va ria ble to be s pe cifie d a nd controls the com ple xity of the ne twork. All ne tworks ha ve the ir s tre ngths a nd we a kne s s e s , a nd the y a re powe rful tools whe n us e d a ppropria te ly. The P R NN a nd TDNN we re s pe cifica lly de ve lope d to m ode l the s tructure s in tim e s e rie s , s o the y a re cons ide re d to be the m os t s uita ble ca ndida te s for the curre nt s tudy. The MLFN, howe ve r, is the m os t popula r m ode l a nd ha s a re la tive ly s im ple s tructure . The MLFN wa s include d in this s tudy to provide a ba s e line for com pa ris on. 3.2 Methodology MLFN, TDNN a nd P R NN m ode ls we re de ve lope d for fore ca s ting the s torm e ve nts occurring ove r the Uppe r P a rra m a tta R ive r C a tchm e nt. The ne tworks we re de ve lope d through: (i) da ta pre pa ra tion, including da ta pre -proce s s ing, (ii) s e le ction of tra ining a lgorithm a nd pe rform a nce indica tors , a nd (iii) de te rm ina tion of the num be r of inputs a nd output s ta ge s . In the da ta pre pa ra tion s te p, two da ta s e ts we re e s ta blis he d for tra ining a nd te s ting the ne twork; the third da ta s e t wa s us e d for m onitoring the tra ining to e ns ure tha t the ne twork le a rns the pe rtine nt inform a tion a nd not the nois e a s s ocia te d with the da ta us e d for tra ining the ne twork. To obta in unbia s e d s a m ple s for e a ch of the da ta s e ts , the 34 s torm e ve nts we re divide d ra ndom ly into:
a tra ining s e t - 16 s torm e ve nts with a tota l of 748 ra infa ll pe riods . This da ta s e t wa s us e d to ca libra te the conne ction we ights of the va rious ne tworks te s te d. a m onitoring s e t - 8 s torm e ve nts with a tota l of 376 ra infa ll pe riods . This da ta s e t wa s us e d to m onitor the pe rform a nce of the tra ining a nd to provide a n indica tion of whe n to ce a s e tra ining. a va lida tion s e t - 10 s torm e ve nts with a tota l of 625 ra infa ll pe riods . This fina l da ta s e t wa s us e d to e va lua te the pe rform a nce of the ne tworks on da ta pre vious ly uns e e n by the ne twork.
8 James E. Ball and Kin Choi Luk As the s e le ction of individua l e ve nts for a pa rticula r da ta s e t wa s ra ndom , a ll e ve nts ha ve s im ila r cha ra cte ris tics (Ta ble 3). Table 3 - Summary of Storm Characteristics Characteristics Storm Type Storm Duration (hour) Time to Max. Rainfall
Training 10 convective, 6 frontal 3 to 22 30 min to 10 hr
Monitoring 4 convective, 4 frontal 2 to 21 1 hr to 18 hr
Validation 8 convective, 2 frontal 6 to 24 45 min to 21 hr
P rior to tra ining, the da ta we re s ca le d to a s m a lle r ra nge [0,1], which is ne ce s s ita te d by the choice of a s igm oid a ctiva tion function. The re a re m a ny a lte rna tive da ta tra ns form a tion a pproa che s which ca n be us e d, with e a ch a pproa ch ha ving its own a dva nta ge s a nd dis a dva nta ge s . In this ca s e , a loga rithm ic a lgorithm wa s us e d to e ns ure tha t the re corde d ra infa ll va lue s we re tra ns form e d into the de s ire d ra nge . The ne xt s te p in the im ple m e nta tion of the ne twork is the s e le ction of the tra ining a lgorithm a nd pe rform a nce indica tor. The norm a lis e d m e a n s qua re d e rror (NMS E) wa s chos e n a s the pe rform a nce indica tor for the com pa ris on of the a lte rna tive type s of ne twork. O ne proble m with the us e of s um s qua re d e rror for the ne twork com pa ris on is tha t the ra infa ll s e rie s ha d diffe re nt le ngths which introduce s proble m s . This proble m wa s ove rcom e by us ing a norm a lis e d ve rs ion of s um s qua re d e rror, na m e ly the norm a lis e d m e a n s qua re d e rror (NMS E). W e ige nd et al. (1992) de fine d the NMS E a s
d NMSE d N
N
np
y np
d np
P
np
2
2
1 NP 2
d N
np
y np
2
(3)
P
P
whe re N is the tota l num be r of output node s , P is the tota l num be r of da ta s a m ple s , d np a re the ta rge t outputs , ynp a re the ne twork’s outputs a nd 2 is the va ria nce of the ta rge t outputs . In e s s e nce , the NMS E is the s um of s qua re d e rrors norm a lis e d by the num be r of da ta s a m ple s ove r a ll output node s a nd the e s tim a te d va ria nce of the da ta . The fina l s te p in the da ta pre pa ra tion is the de te rm ina tion of the input a nd output da ta re pre s e nta tion. For the Uppe r P a rra m a tta R ive r C a tchm e nt, the re a re a num be r of pos s ible configura tions of input a nd output inform a tion. The thre e m os t fe a s ible a pproa che s a re to:
divide the s tudy ca tchm e nt into grids (439 pixe ls of 500m x 500m ) a nd us e ra infa ll a t the 439 pixe ls a s inputs to fore ca s t ra infa ll a t a ll 439 pixe ls s im ulta ne ous ly. The re s ulting outputs will be ra infa ll a t e a ch pixe l of the ca tchm e nt; us e ra infa ll a t the 16 ga uge s a s inputs to fore ca s t ra infa ll for the 16 ga uge s s im ulta ne ous ly. In this ca s e , one ne twork re pre s e nts ra infa ll for a ll 16 ga uge s . Afte r the ra infa ll fore ca s ts for the 16 ga uge s a re obta ine d, a s pa tia l ra infa ll m ode l is us e d to ge ne ra te the ra infa ll a t e ve ry point of the ca tchm e nt; a nd us e ra infa ll a t the 16 ga uge s a s inputs to fore ca s t ra infa ll for a s ingle ga uge . This will e nd up with 16 ne tworks for 16 ga uge s of
Rainfall forecasting in urban environments 9 the s tudy ca tchm e nt. Aga in, a fte r the ra infa ll fore ca s ts for the 16 ga uge s a re obta ine d, a s pa tia l ra infa ll m ode l is us e d to ge ne ra te the ra infa ll a t e ve ry point of the ca tchm e nt. P re lim ina ry a s s e s s m e nt of the s e thre e configura tions s ugge s ts tha t the option of us ing inform a tion from the 16 ra in ga uge s a s input inform a tion a nd us ing the s a m e loca tions a s the output inform a tion (i.e . ra infa ll fore ca s t) wa s the m os t de s ira ble option. R e a s ons for this a re :
inform a tion from a ll m e a s ure m e nt points a re us e d s im ulta ne ous ly to produce fore ca s ts for e a ch of the m e a s ure m e nt points ; a nd the fore ca s t re s ults ca n be us e d re a dily in the s pa tia l ra infa ll m ode l pre s e nte d e a rlie r.
The option of fore ca s ting a s ingle ga uge wa s re je cte d be ca us e the s a m e proce s s wa s re quire d once for e a ch pre diction loca tion a nd, the re fore , 16 ne tworks we re ne e de d. In a s im ila r m a nne r, the option of us ing 439 pixe ls a s both input a nd output wa s re je cte d be ca us e it involve d the us e of a la rge num be r of input a nd output node s a nd the cons e que nt ne twork would conta in a la rge num be r of pa ra m e te rs (we ights ). For e xa m ple , a one -hidde n la ye r MLFN with 439 input node s , 439 output node s a nd 2 hidde n node s com pris e s 1756 conne ctions (439x2 + 439x2, e xcluding the bia s e s ). The a va ila ble ra infa ll da ta (m a x. 1749 da ta points ) we re not s ufficie nt to tra in a ne twork of this s ize . 3.3 Development of Alternative Networks The thre e a lte rna tive type s of ne tworks cons ide re d in this s tudy we re tra ine d a nd va lida te d with ra infa ll da ta colle cte d from the s tudy ca tchm e nt, i.e . a tota l of 34 s torm e ve nts with de pth re corde d e ve ry 15-m inute s . Va rious ne twork configura tions we re e xplore d to de te rm ine the e ffe ct of the ke y va ria ble s : la g of the ne twork a nd the num be r of hidde n node s . For the MLFN, ne tworks with la gs of 1, 2, 3 a nd 4 we re us e d, with hidde n node s of 2, 4, 8, 16, 24, 32, 64, a nd 128. W e a ls o e xplore d ne tworks with two la ye rs of hidde n node s . For the TDNN the s ize of input windows us e d wa s 2, 3 a nd 4. Fina lly, for the P R NN, the la g wa s fixe d a t 1, while the num be rs of conte xt units trie d we re 2, 4, 8, 16, 24, 32 a nd 64. The com ple te lis ting of a ll ne twork re s ults us ing the s e pa ra m e te rs is pre s e nte d in Luk (1998). P e rus a l of ne twork pe rform a nce with va rious num be rs of hidde n node s indica te d tha t the ne tworks with a gre a te r num be r of hidde n node s re s ulte d in a lowe r tra ining e rror a t the m a xim um tra ining e poch, which wa s 1000 e pochs . In this conte xt, a n e poch is a com ple te s we e p through the tra ining pa tte rns . S ince the conne ction we ights of the ne twork we re upda te d only a t the e nd of e a ch e poch, a m a xim um of 1000 e pochs m e a ns tha t the we ights we re upda te d a t m os t 1000 tim e s . This re s ult is to be e xpe cte d a s the ne tworks with m ore hidde n node s ha ve m ore fre e pa ra m e te rs , thus re s ulting in lowe r tra ining e rrors . During the va lida tion te s ts , howe ve r, the s e ne tworks ha d poore r pe rform a nce s ince the y ha d ove r-le a rne d the tra ining da ta . This e ffe ct is illus tra te d in Ta ble 4. Irre s pe ctive of the orde r of la g in the ne twork,
10
James E. Ball and Kin Choi Luk
a n MLFN with 128 hidde n node s re s ulte d in a s m a lle r NMS E during tra ining but a m uch highe r NMS E during va lida tion. For e xa m ple , the la g-4 MLFN with 128 hidde n node s ha d the s m a lle s t tra ining e rror of 0.27 while the va lida tion e rror for the s a m e ne twork wa s the highe s t a t a va lue of 2.33. Table 4 - Effect of Hidden Nodes and Time Lag of MLFN Normalised Mean Squared Error (NMSE) Network Training Validation Lag-1 MLFN with 2 hidden nodes 0.53 0.71 Lag-1 MLFN with 128 hidden nodes 0.40 1.20 Lag-2 MLFN with 2 hidden nodes 0.51 0.73 Lag-2 MLFN with128 hidden nodes 0.36 0.96 Lag-3 MLFN with 2 hidden nodes 0.49 0.72 Lag-3 MLFN with 128 hidden nodes 0.32 1.26 Lag-4 MLFN with 2 hidden nodes 0.49 0.78 Lag-4 MLFN with 128 hidden nodes 0.27 2.33 Note: the networks were trained to 1000 epochs
The influe nce of the orde r of la g ca n a ls o be a s s e s s e d from Ta ble 4. P e rus a l of the da ta s hown in this ta ble indica te s tha t the MLFN with a highe r orde r of la g te nde d to le a rn the tra ining da ta be tte r. For va lida tion, howe ve r, the re ve rs e is the ca s e . The s e re s ults do not s ugge s t tha t a ne twork with highe r orde r la g will give poore r re s ults . R a the r the re s ults indica te tha t a ne twork with highe r orde r of la g conta ine d m ore conne ction we ights , a nd he nce m ore fre e pa ra m e te rs . The re fore , like a ne twork with m ore hidde n node s , the ne twork with highe r la g te nde d to ove r-le a rn the tra ining da ta . From this a na lys is it wa s conclude d tha t the pe rform a nce of a ne twork de pe nde d m ore on the ne twork c om ple xity tha n the inclus ion of m ore inform a tion through the us e of da ta from pre vious tim e pe riods . The e ffe ct of us ing highe r orde rs of la g wa s to incre a s e the ne twork com ple xity without provis ion of pe rtine nt a dditiona l inform a tion. C ons e que ntly, it is s ugge s te d tha t the pe rform a nce of a ne twork de pe nds not on the num be r of hidde n node s or the orde r of la g but ra the r on the com bina tion of the s e two a s pe cts . 3.4 Comparison of Alternative Networks The ne twork re s ults of Luk (1998), in a ddition to be ing us e d for de te rm ina tion of the ne twork com ple xity, ca n a ls o be us e d a s the ba s is of a com pa ris on be twe e n a lte rna tive ne tworks . Ba s ing the s e le ction of the be s t ne twork configura tion on the NMS E for the va lida tion da ta , the e ight ne tworks with the lowe s t va lida tion NMS E a re s hown in Ta ble 5. Ea ch row of Ta ble 5 re pre s e nts a ne twork with a s pe cific la g. For e xa m ple , the firs t row s hows the re s ults for a la g-1 MLFN with 24 hidde n node s , while the s e cond row s hows the re s ults for a la g-2 MLFN with 8 hidde n node s . In ge ne ra l, a ll thre e type s of ne tworks ha ve com pa ra ble pe rform a nce . The NMS E of the va lida tion s a m ple s for a ll ne tworks we re in the ra nge of 0.63 to 0.67 with only ve ry s m a ll diffe re nce s be twe e n the ne tworks . This com pa ra ble pe rform a nce is due to the ne tworks be ing de ve lope d to the ir optim a l com ple xity a s de fine d by the inte ra ction be twe e n the la g a nd the num be r of hidde n node s . For e xa m ple , the la g-1 MLFN re quire s m ore hidde n node s to a chie ve a n optim a l com ple xity while the m ore com plica te d la g-4 MLFN only re quire s two hidde n node s due to the la rge num be r of pa ra m e te rs introduce d by the high orde r of la g. This re s ult is cons is te nt with
Rainfall forecasting in urban environments 11 the conce pt of the e xis te nce of a n optim a l com ple xity for a ne twork which wa s dis cus s e d e a rlie r. Table 5 - Comparison of Alternative Networks Network MLFN Lag 1 (16-24-16) MLFN Lag 2 (32-8-16) MLFN Lag 3 (48-4-16) MLFN Lag 4 (64-2-16) PRNN (16-4-16) TDNN Lag 2 (32-16-16) TDNN Lag 3 (48-32-16) TDNN Lag 4 (64-32-16)
Training (NMSE)
Monitoring (NMSE)
Validation (NMSE)
Stopping epoch
Training Error at 1000 epoch (NMSE)
0.50
0.68
0.64
200
0.49
0.51
0.69
0.66
100
0.47
0.48
0.69
0.67
700
0.47
0.52
0.71
0.65
200
0.49
0.49
0.67
0.64
300
0.48
0.50
0.67
0.63
100
0.41
0.50
0.69
0.64
100
0.41
0.51
0.69
0.65
100
0.40
Notation: The network configuration is denoted by three figures (x-y-z), where x= no. of input nodes, y = no. of hidden nodes and z = no. of output nodes.
Figure s 4 a nd 5 com pa re thre e one -s te p a he a d fore ca s ts of ra infa ll de pth a t a s ingle ga uge . The hye togra ph in e a ch figure s hows the a ctua l ra infa ll re corde d a t tha t ga uge s ite a nd the fore ca s t ra infa ll, a nd s im ila r plots for othe r ga uge s a nd s torm e ve nts pre s e nte d in Luk et al. (1998), Luk (1998) a nd Luk et al. (2000).
Figure 4 Forecasting Rainfall at Gauge No. 7253 for the Storm Event on 2 January 1996
12
James E. Ball and Kin Choi Luk
Figure 5 - Forecasting Rainfall at Gauge No. 7253 for the Storm Event on 6 January 1996
Ana lys ing the fore ca s t e rrors for one s torm with e a ch of the thre e s olutions , we conclude tha t: (i) the fore ca s t e rror incre a s e s a s the ra te of cha nge of ra infa ll inte ns ity incre a s e s ; (ii) the ne tworks m a de be tte r pre dictions a fte r the pe a k of the s torm e ve nt; a nd (iii) a ll the ne tworks te nde d to unde r-pre dict the ra infa ll whe n the ra te of cha nge in the ra infa ll inte ns ity wa s pos itive , a nd to ove r-pre dict the ra infa ll whe n the ra te of cha nge in the ra infa ll inte ns ity wa s ne ga tive . S um m a ris ing a ll the com pa ris on te s ts , it wa s found tha t
a ll the thre e a lte rna tive type s of ne tworks ha ve com pa ra ble pe rform a nce ; a MLFN with lowe r orde r of la g ha s a m a rgina lly be tte r pe rform a nce tha n a ne twork with a highe r orde r of la g; MLFN ne tworks with highe r la gs te nde d to ove r-le a rn the tra ining da ta , re s ulting in s m a lle r tra ining e rrors but la rge r va lida tion e rrors ; MLFN ne tworks with lowe r la gs re quire d m ore hidde n node s , a nd vice ve rs a s ugge s ting a n optim a l le ve l of ne twork com ple xity; the P R NN ne twork s howe d com pa ra ble pe rform a nce with the la g-1 MLFN a nd outpe rform e d the MLFN with a highe r orde r of la g; a m ong a ll the ne tworks , the la g-2 TDNN yie lde d the m inim um va lida tion e rror; a nd for a ll thre e a lte rna tive ne tworks , fore ca s t e rrors incre a s e a s the ra te of cha nge of ra infa ll inte ns ity incre a s e s in e ithe r a pos itive or ne ga tive dire ction.
Rainfall forecasting in urban environments 13 4 R AINFALL FO R EC AS TING – AR EAL DIS TR IBUTIO NS
4.1 Integrating a GIS and an NN The inte gra tion of the G IS a nd a NN provide s a powe rful ra infa ll fore ca s ting m ode l by m e rging the m e rits of the two toge the r. For the purpos e s of de ve loping a n a re a l ra infa ll fore ca s ting s ys te m , it wa s a s s um e d tha t the only a va ila ble ra infa ll da ta we re tha t m e a s ure d a t the ra infa ll ga uge s within the ca tchm e nt. The re a re two a lte rna tive a pproa che s , howe ve r, to the ge ne ra tion of future ra infa ll fore ca s ts ; the s e two a pproa che s a re
Us e of the G IS to e s tim a te ra infa ll a t e a ch pixe l within the ca tchm e nt ba s e d on m e a s ure d ra infa ll a t the ga uge s . This will ge ne ra te a s ignifica nt num be r of s pa tia lly dis tribute d ra infa ll e s tim a te s . The NN could the n be us e d to m a p the s e e s tim a te s to dire ctly produce a ra infa ll fore ca s t a t e a ch pixe l within the ca tchm e nt. Us e of the NN to fore ca s t ra infa ll a t the e a ch of the ra infa ll ga uge s a nd the n us e the G IS to ge ne ra te the s pa tia lly dis tribute d ra infa ll fore ca s t for a ll the pixe ls within the ca tchm e nt.
The la tte r a pproa ch wa s a dopte d be ca us e the num be r of pa ra m e te rs for the NN would be dra m a tica lly re duce d com pa re d to the firs t a pproa ch. More ove r, the tim e re quire d for tra ining the NN a nd the da ta re quire d for the tra ining proce s s would both be s ignifica ntly re duce d. The firs t a pproa ch of us ing the ca tchm e nt pixe ls a s the loca tion of both the input a nd output inform a tion wa s not cons ide re d fe a s ible due to the la rge num be r of input a nd output node s tha t would re s ult. The tota l num be r of 0.25 km 2 pixe ls in the Uppe r P a rra m a tta R ive r ca tchm e nt is 439 which would re s ult in 1756 conne ctions for a 3-la ye r MLFN. Tra ining s uch a la rge ne twork would be e xtre m e ly tim e cons um ing. More ove r, a ne twork with s o m a ny fre e pa ra m e te rs re quire s a la rge num be r of da ta s e ts , which ge ne ra lly a re not a va ila ble . 4.2 Methodology To te s t the a ccura cy of the ra infa ll fore ca s ting s ys te m , we ge ne ra te d a tota l of 150 a rtificia l s torm e ve nts due to a n ina de qua te num be r of re corde d s torm e ve nts . The fore ca s t ra infa ll a t e a ch pixe l wa s the n com pa re d with the a ctua l va lue of ra infa ll (from the a rtificia l e ve nt) to a s ce rta in the fore ca s t a ccura cy. O f the s e e ve nts , 100 s torm e ve nts we re us e d for tra ining of the ANN, 25 s torm e ve nts for m onitoring the tra ining proce s s a nd 25 s torm e ve nts for va lida tion of the ra infa ll fore ca s ting s ys te m . The s e a rtificia l s torm e ve nts we re a s s um e d to be a ra ndom proce s s with s om e de gre e of m e m ory. Accordingly, the y we re ge ne ra te d by a m ixture of a utore gre s s ive a nd ra ndom te chnique s . The firs t s te p involve d ra ndom ly s ta rting a s torm ce ntre a t a point clos e to or within the s tudy ca tchm e nt. The loca tion of this s ta rting point wa s bia s e d a ccording to his torica l re cords of s torm s ove r the Uppe r P a rra m a tta R ive r C a tchm e nt. The s torm the n m ove d towa rds the ce ntre of the ca tchm e nt s ubje ct to de via tions from the initia l dire ction de fine d by the following a utore gre s s ive e qua tion:
14
James E. Ball and Kin Choi Luk
dire ction(t) = 0.8*dire ction(0) + 0.2*dire ction(t-1) + e (t)
(4)
whe re dire ction(t) is the dire ction of the s torm ce ntre a t tim e t, dire ction(0) is the initia l dire ction of s torm m ove m e nt a nd e (t) is ra ndom de via tion of s torm m ove m e nt, which ha d a m e a n of 0 o a nd a s ta nda rd de via tion of 15 o. The inte ns ity of the s torm a t its ce ntre wa s a n a utore gre s s ive proce s s with this proce s s gove rne d by P max = 0.2*P max(t-1) + 0.8*e (t)
(5)
whe re P max(t) is the ra infa ll inte ns ity a t the s torm ce ntre (m m /hr) a t tim e t, a nd e (t) is the ra ndom fluctua tion. The m ove m e nt of the s torm ce ntre wa s a ra ndom proce s s with a m e a n s pe e d of 12 km /hr, a nd a s ta nda rd de via tion of 2 km /hr, a s illus tra te d in Figure 6. This s torm m ove d a cros s the ca tchm e nt from the North Ea s t to S outh W e s t during 2.5 hours (15 m in x 10 tim e s te ps ).
Figure 6: Track of Storm Centres
4.3 Test Results and Discussions In a s ce rta ining the a ccura cy of the s pa tia l ra infa ll fore ca s ts , both vis ua l a nd a rithm e tic com pa ris ons we re e s ta blis he d. Va lida tion of the s pa tia l ra infa ll fore ca s ting s ys te m wa s on the ba s is of
re plica ting the re a l ra infa ll pa tte rns (vis ua l ins pe ction); tra cking the m ove m e nt of s torm ce ntre s (vis ua l ins pe ction); pre dicte d ra infa ll a t individua l pixe ls ; a nd pre dicte d ra infa ll for s ubca tchm e nts .
Figure 7 s hows the dis tribution of ra infa ll a t one ins ta nt during one of the va lida tion e ve nts . A com ple te his tory of this s torm e ve nt toge the r with othe r e ve nts is pre s e nte d in Luk (1998).
Rainfall forecasting in urban environments 15
Figure 7 – Predicted and Actual Rainfall during a Validation Storm Event
Figure 8 m a ps the a ctua l s torm ce ntre tra ck for the s a m e s torm e ve nt with the pre dicte d s torm ce ntre tra ck. The a ctua l tra ck of s torm ce ntre s is re pre s e nte d by num be rs , whe re a s the fore ca s t tra ck is s hown by a lpha be tic cha ra cte rs . Figure 8 s hows both tra cks ha d s im ila r cha ra cte ris tics ; the fore ca s t s ys te m did ve ry we ll in tra cking the s torm ce ntre s from tim e s te ps 2
16
James E. Ball and Kin Choi Luk
to 8. The re we re , howe ve r, e rrors in fore ca s ting the s torm ce ntre s a t tim e s te ps 6 a nd 7. The s e e rrors re s ulte d in a n ove re s tim a tion of the s pa tia l ra infa ll dis tribution.
Figure 8 - Forecasting Movement of Storm Centres
Table 6: Forecast Errors for Storm No. 138 Time Step Intensity (mm/15-min) 1 -4.0 2 -4.6 3 -5.8 4 -1.5 5 -3.5 6 1.8 7 -3.1 8 1.8 9 -2.0 10 -3.6
Distance (km) 0.11 0.64 -0.61 -0.66 -0.56 -0.22 -1.05 -1.00 -0.34 -0.17
Angle (degree) -47.2 12.3 28.3 16.4 -15.8 25.4 -10.7 -1.5 109.4 74.8
Table 7: Error Statistics for Storm No. 138 Statistic Intensity (mm/15-min) High 1.8 Low -5.8 Mean -1.7 Median -2.3 Standard Deviation 3.1 Skewness 0.1
Distance (km) -0.22 -1.05 -0.68 -0.63 0.31 0.18
Angle (degree) 28.3 -15.8 7.0 7.4 18.9 -0.1
Ta ble s 6 a nd 7 s how la rge e rrors in the firs t a nd la s t fe w tim e s te ps . It is s us pe cte d tha t the s e e rrors a re due to initia tion a nd bounda ry conditions for the s ys te m . Due to the unce rta inty in the s e tim e s te ps , the e rror cha ra cte ris tics pre s e nte d in Ta ble 7 do not include va lue s a t the s e tim e s te ps . The a ve ra ge inte ns ity of this s torm e ve nt wa s 28.9m m pe r 15 m inute
Rainfall forecasting in urban environments 17 pe riod (or a pproxim a te ly 76m m /hr), a nd the e rror in the fore ca s t ra infa ll inte ns ity is s m a ll. In a ddition, the pre diction of loca tion wa s e xce lle nt. The ra nge of e rror in loca tion wa s only -1.05 to -0.22 km with a m e a n e rror of 0.68 km , which m e a ns tha t the s torm ce ntre wa s pre dicte d in a n a dja ce nt pixe l to tha t whe re it a ctua lly occurre d. S im ila r cha ra cte ris tics we re re plica te d for a ll 25 va lida tion s torm e ve nts a nd s um m a ris e d in Ta ble 8. O ve ra ll, the NN ba s e d fore ca s t s ys te m produce d re lia ble pre dictions for the ra infa ll inte ns ity a nd loca tion of the s torm ce ntre , but this wa s not the ca s e for pre diction of the s torm m ove m e nt. Table 8: Error Statistics for All 25 Validation Storm Events Statistic Intensity (mm/15-min) High 8.2 Low -16.8 Mean -2.8 Median -2.9 Standard Deviation 4.3 Skewness -0.1
Distance (km) 0.73 -1.68 -0.62 -0.62 0.49 0.27
Angle (degree) 178.3 -172.5 -0.9 0.6 53.0 -0.03
The norm a lis e d m e a n s qua re d e rror for e a ch tim e s te p during a ll 25 va lida tion s torm e ve nts is 0.263, which is cons ide re d re a s ona ble va lida tion of the s ys te m re lia bility (Ta ble 9). The re we re , howe ve r, s e ve ra l a bnorm a l figure s , but a ll of the s e occurre d a t the firs t tim e s te p. Afte r this , the NN a ccura te ly fore ca s ts the re m a inde r of the s torm m ove m e nt a nd a s s ocia te d ra infa ll inte ns itie s . The s e e rrors a re a s cribe d to the ina bility of the NN to re cognis e the initia l pos ition of the s torm ce ntre s . Table 9: Normalised Mean Squared Errors For All 25 Validation Storm Events NMSE at Each Time Step Event 1 2 3 4 5 6 7 8 126 0.444 0.245 0.227 0.429 0.015 0.028 0.072 0.162 127 0.063 0.875 0.506 0.482 0.042 0.083 0.101 0.027 128 0.246 0.227 0.156 0.232 0.327 0.436 0.312 0.046 129 0.362 0.046 0.177 0.161 0.341 0.156 0.254 0.617 130 0.328 0.336 0.173 0.056 0.150 0.177 0.554 0.116
9 0.055 0.097 0.587 0.196
131 132
0.160 0.469
0.512 0.264
0.305 0.321
0.010 0.078
0.015 0.070
0.031 0.239
0.185 0.454
0.066 0.348
0.098 0.215
133 134 135 136 137 138 139 140
0.703 0.175 0.145 0.884 0.070 0.130 0.150 1.561
0.123 0.328 0.092 0.350 0.443 0.086 0.245 0.165
0.267 0.328 0.266 0.120 0.213 0.286 0.301 0.388
0.199 0.074 1.643 0.503 0.224 0.139 0.427 0.244
0.206 0.175 0.167 0.202 0.615 0.127 0.062 0.419
0.074 0.168 0.307 0.236 0.375 0.081 0.054 0.210
0.161 0.040 0.241 0.140 0.063 0.026 0.082 0.237
0.299 0.055 0.132 0.144 0.221 0.176 0.080 0.410
0.124 0.194 0.176 0.137 0.199 0.101
141
4.422
0.026
0.826
0.147
0.217
0.199
0.278
0.053
0.048
142 143 144 145
0.039 0.416 0.756 0.703
0.178 0.028 0.181 0.112
0.056 0.383 0.305 0.132
0.134 0.070 0.222 0.651
0.083 0.809 0.191 0.103
0.167 0.441 0.204 2.333
0.286 0.224 0.347 0.211
0.024 0.117 0.207 0.226
0.299 0.165 0.261 0.115
146 147 148
0.434 0.093 0.149
0.196 0.655 0.107
0.13 0.104 0.073
0.128 0.402 0.093
0.257 0.355 0.207
0.056 0.069 0.073
0.045 0.071 0.150
0.009 0.170
0.059 0.185
10 0.193 0.163 0.193 0.326 0.143 0.645 0.140 0.396 0.059 0.150 0.189 0.113 0.133 1.755 0.147 0.094 0.045
Mean 0.203 0.248 0.227 0.286 0.237 0.154 0.295 0.240 0.171 0.352 0.286 0.278 0.165 0.175 0.344
0.587 0.302 0.280 0.297 0.430 0.178 0.202 0.134
18 149 150
James E. Ball and Kin Choi Luk 0.063 1.075
0.186 0.432
0.09 0.391
0.102 0.097
0.476 0.566
0.120 0.296
0.285 0.251
0.041 0.051 0.080 0.073 0.156 Overall Mean NMSE
0.157 0.342 0.263
The inte gra te d G IS a nd NN produce d ra infa ll fore ca s ts of the s pa tia l dis tribution a nd de pth of ra infa ll ove r the ca tchm e nt for 15 m inute tim e inte rva ls . The fore ca s ting s ys te m a ccura te ly pre s e rve d the s pa tia l ra infa ll pa tte rns a nd produce d fore ca s ts with good a gre e m e nt to a ctua l ra infa ll va lue s . O n fore ca s ting ra infa ll a t e a ch pixe l of the s tudy ca tchm e nt, the ove ra ll a ve ra ge d norm a lis e d m e a n s qua re d e rrors ove r 25 va lida tion e ve nts wa s 0.263 a nd s o the s ys te m wa s cons ide re d to produce re lia ble pre dictions .
5
C O NC LUS IO NS
De ve lopm e nt of a ra infa ll fore ca s ting s ys te m us ing a G IS a nd NN te chnologie s ha s be e n the focus of this cha pte r. This de ve lopm e nt involve d inve s tiga tions into m ode ls of the s pa tia l dis tribution of ra infa ll a nd the a ppropria te form of a n a rtificia l NN for ra infa ll fore ca s ting a s we ll a s a n a s s e s s m e nt of the fore ca s t a ccura cy of the propos e d s ys te m . During de ve lopm e nt of the a rtificia l NN, thre e a lte rna tive type s we re ide ntifie d, de ve lope d a nd com pa re d: a m ulti-la ye r fe e d-forwa rd ne ura l ne twork; a pa rtia l re curre nt ne ura l ne twork; a nd a tim e de la y ne ura l ne twork. It wa s found tha t a ll thre e type s of NN we re a ble to m a ke re a s ona ble 15 m inute fore ca s ts of ra infa ll for m ultiple loca tions within c a tchm e nt. As a re s ult of the te s ts , the following points a re note d: For e a ch ne twork, the re wa s a n optim a l com ple xity de te rm ine d by the com bina tion of the num be r of hidde n node s a nd the la g of the ne twork; All thre e ne tworks ha d com pa ra ble pe rform a nce whe n the y we re de ve lope d a nd tra ine d to re a ch the ir optim a l com ple xity; a nd The ne tworks with lowe r la g outpe rform e d one s with a highe r la g due to the 15-m in ra infa ll tim e s e rie s not pos s e s s ing a long te rm m e m ory. The inte gra tion of G IS a nd NN te chnologie s e na ble d re a s ona ble 15 m inute fore ca s ts of the s pa tia l dis tribution of ra infa ll ove r the Uppe r P a rra m a tta R ive r C a tchm e nt. The s ys te m a ccura te ly pre s e rve d the s pa tia l ra infa ll pa tte rns a nd produce d fore ca s ts with good a gre e m e nt to the a ctua l ra infa ll va lue s . In conclus ion, the ra infa ll fore ca s ting s ys te m de ve lope d for the Uppe r P a rra m a tta R ive r C a tchm e nt ha d the following cha ra cte ris tics : A s pa tia lly a nd te m pora lly dis tribute d a rchite cture . It re ce ive s pre s e nt ra infa ll va lue s a t m ultiple ga uge pos itions a nd produce s ra infa ll fore ca s ts a t e ve ry pixe l of the s tudy ca tchm e nt; The s ys te m ha s be e n de ve lope d for re a l-tim e ope ra tion. The input inform a tion to the s ys te m is the ra infa ll da ta a t m ultiple ga uge loca tions while the output inform a tion is the ra infa ll va lue s for e a ch pixe l within the ca tchm e nt; a nd W ith colle ction of ne w ra infa ll da ta , the a rtificia l ne ura l ne twork com pone nt of the s ys te m ca n be re -tra ine d to im prove its a ccura cy.
Rainfall forecasting in urban environments 19 6 R EFER ENC ES Ball, J.E. & Luk, K.C. (1996) Determination of the Rainfall Distribution over a Catchment using Hydroinformatics Tools, Proc. 2nd International Conference on Hydroinformatics, Zurich, Switzerland, pp 369 - 376. Ball, J.E. & Luk, K.C. (1998) Modelling the spatial variability of rainfall over a catchment, ASCE, Journal of Hydrologic Engineering, 3(2):122-130. Luk, K.C., Ball, J.E. & Sharma, A. 2001. An application of artificial neural networks for rainfall forecasting. Mathematical and Computer Modelling, 33 (6-7): 683-693. Elman, J. L. (1990) Finding structure in time. Cognitive Science, 14:179-211. Fontaine, T.A. (1991) Predicting Measurement Error of Areal Mean Precipitation During Extreme Events, Water Resources Bulletin, 27(3), 509-520. Handmer, J.W., Smith, D.I. & Lustig, T.L. (1988) The Sydney Floods of 1986: Warnings, Damages, Policy and Future, Proc. Hydrology and Water Resources Symposium, 1988, Australian National University, Canberra, Australia, IEAust Nat Conf Pub 88/1, pp 206-210. Luk, K.C. and Ball, J.E. (1996) Application of GIS for Modelling of the Spatial Distribution of Rainfall, Research Report 191, Water Research Laboratory, School of Civil and Environmental Engineering, University of New South Wales. Luk, K.C. (1998) An application of hydroinformatic tools for rainfall forecasting, PhD Dissertation, School of Civil and Environmental Engineering, The University of New South Wales, Sydney, Australia Luk et al. 1998. – missing this reference? – can’t find on WOS Luk, K.C., Ball, J.E. & Sharma, A. 2000. A study of optimal lag and spatial inputs to artificial neural network for rainfall forecasting. Journal of Hydrology, v.227, 56-65. Rumelhart, McClelland, and the PDP Group (1986). Parallel Distributed Processing, Vols 1&2. Cambridge, MA: MIT Press. Urbonas, B.R., Guo, J.C.Y. & Janesekok, M.P. (1992) Hyetograph density effects on urban runoff modelling, Proc. Int. Conf. On Comp. Applications in Water Resources, Tamkang University, Tamsui, Taiwan, pp 3237. Weigend, A.S., Huberman, B.A. and Rumelhart, D.E. (1992), Predicting Sunspots and Exchange Rates with Connectionist Networks, In Casdagli, M. and S. Eubank (eds) Non-linear Modeling and Forecasting, 395431. Addison Wesley.
Filename: Preprint Ball and Luk.doc Directory: E:\ANN Book Template: E:\WINWORD\SJABLOON\BO2_47A4.DOT Title: CHAPTER 1 Subject: Author: A.A. Balkema Uitgevers B.V. Keywords: Comments: Creation Date: 28/05/2002 4:30:00 PM Change Number: 6 Last Saved On: 13/12/2002 9:43:00 AM Last Saved By: James E Ball Total Editing Time: 80 Minutes Last Printed On: 23/08/2015 12:26:00 PM As of Last Complete Printing Number of Pages: 19 Number of Words: 6,360 (approx.) Number of Characters: 36,257 (approx.)
