SPATIAL RAINFALL FIELD SIMULATION WITH RANDOM CASCADE INTRODUCING OROGRAPHIC EFFECTS ON RAINFAL
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1 Proc. of the d Asa Pacfc Assocato of Hydrology ad Water Resources (APHW) Coferece, July 5-8, 4, Sutec Sgapore Iteratoal Coveto Exhbto Cetre, Sgapore, vol., pp , 4 SPATIAL RAINFALL FIELD SIMULATION WITH RANDOM CASCADE INTRODUCING OROGRAPHIC EFFECTS ON RAINFAL Yasuto TACHIKAWA Dsaster Preveto Research Isttute, Kyoto Uversty Gokasho, Uj, 6-, Japa Masahro HIWASA Graduate School of Cvl Egeerg, Kyoto Uversty Yoshda Homach, Sakyo-ku, Kyoto, 66-85, Japa Kaoru TAKARA Dsaster Preveto Research Isttute, Kyoto Uversty Gokasho, Uj, 6-, Japa A space rafall model s costructed to geerate spatal rafall felds. The model s based o a radom cascade process whch dsaggregates a regoally averaged rafall amout to geerate spatal dstrbutos of rafall. To clude a structure of rafall feld heterogeety rafall smulatos, a determstc factor s troduced a radom cascade geerator. The factor represets a orographc effect o spatal dstrbutos of rafall, whch s determed by usg the depedece le o topographc elevatos (DLTE) developed by Nakakta et al. [] ad Suzuk et al. []. The model s appled to a 9 km 9 km area covered by the Myama radar ra gauge the Kk rego of Japa. It s foud that smulato results well preserve the characterstcs of orographc effects represeted by the DLTE. Ths suggests that the proposed method has the ablty to capture statstcal characterstcs of topography o rafall felds foud radar rafall data. INTRODUCTION A dowscalg techque to geerate local scale rafall dstrbutos from global scale formato s ewly devsed for local scale watershed maagemets ugauged bass. I the method, a statstcal dowscale techque usg a radom cascade theory cosderg orographc effects o rafall dstrbutos s proposed. The WGR model based o the pot process theory developed by Waymre et al. [3] s a well-kow stochastc model to geerate a space-tme rafall dstrbuto; however t s qute dffcult to detfy the model parameters. O the other had, radom cascade models, whch assume self-smlarty spatal rafall dstrbutos, have recetly used as a rafall feld geerator, for example, see Over ad Gupta [4]. These models have few model parameters whch are easly obtaed usg observed spatally dstrbuted rafall formato. To represet o-homogeeous spatal features of rafall patters,
2 Proc. of the d Asa Pacfc Assocato of Hydrology ad Water Resources (APHW) Coferece, July 5-8, 4, Sutec Sgapore Iteratoal Coveto Exhbto Cetre, Sgapore, vol., pp , 4 Jothtyagkoo et al. [5] troduced the regoal characterstcs of rafall to the cascade geerator wth the beta-logormal model developed by Over ad Gupta [4]. I ths study, a topographc effect o rafall spatal dstrbutos s corporated to a radom cascade geerator to realze more realstc rafall feld smulatos. It s kow that a spatal dstrbuto of accumulated rafall s well related to a topographc elevato patter. To clude the topographc effects to smulated rafall dstrbutos, the depedece le o topographc elevatos (DLTE), whch represets the depedece propertes of spatal rafall dstrbutos o topographc elevatos developed by Nakakta et al. [] ad Suzuk et al. [], s corporated to a radom cascade geerator. DLTE s a regresso le betwee the spatal averages of accumulated rafall ad topographc elevatos. Nakakta et al. [] ad Suzuk et al. [] foud that the correlato coeffcet of the relatoshp obtaed usg radar rafall data was more tha.9. Usg the DLTE, determstc topographc multpler matrces are troduced to a radom cascade geerator usg the beta-logormal model. The method s appled to a 9 km 9 km rego the Kk area of Japa. The parameters of the radom cascade geerator are obtaed usg radar rafall data. The values of the determstc topographc multpler matrces for each dowscale level are determed wth the DLTE ad average elevato of each sub-area. By usg the cascade geerator, spatal rafall dstrbutos wth 3km resoluto are smulated over the rego. The statstcal characterstcs of the geerated data are examed. It s foud that the accumulated values of geerated rafall felds wth 3 km grd resoluto shows a lear relatoshp wth the elevatos, whch s good agreemet wth the relatoshp troduced by the DLTE. Ths suggests that the proposed method has the ablty to capture statstcal characterstcs of topography o rafall felds foud radar rafall data. RANDOM CASCADE MODEL A radom cascade process starts spatal dsaggregato of rafall wth a gve volume of rafall a tal area (level ). The area s subsequetly subdvded to sub-areas (four areas ths case) at each dsaggregato step ad the volume gve at the prevous step s assged to the four sub-areas. The rafall volume the sub-area at the th level of dsaggregato step s gve by µ = R L b W () j j = where R s a average rafall testy (mm/h) of the tal area, L s the legth of the study area ( ths case 9 km), b s a brachg umber (b = d ), ad d s a dmeso costat (d = ). The multplers W j are radom cascade geerators whch are weghts to assg average rafall testy to four sub-areas. As the cascade geerator W j, we adopt the beta-logormal model proposed by Over ad Gupta [4]. I the model, W s a product of depedet radom varables B ad Y, that s W=BY. B ad Y are represeted as
3 Proc. of the d Asa Pacfc Assocato of Hydrology ad Water Resources (APHW) Coferece, July 5-8, 4, Sutec Sgapore Iteratoal Coveto Exhbto Cetre, Sgapore, vol., pp , 4 β β β β B = wth probablty P ( B = ) = b, B = b wth probablty P ( B = b ) = b () Y = b ( σ l b ) / +σx where X s a stadard ormal radom varate ad β ad σ are parameters of the betalogormal model. The expectato ad varace of B ad Y are probed to be β E [ B ] =, Var [ B ] = b (4) ( σ l b ) E[ Y ] =, Var [ Y ] = e (5) From the Eqs. (4) ad (5), E[W] = s satsfed. The radom varate B expresses the codto whether t ras the sub-area or ot, ad the radom varate Y represets the weght for assgg a rafall volume to the four sub-areas. Fgure shows a example of geerated rafall felds at each dsaggregato step (from step to step 6) by usg the beta-logormal model. The study area s a 9 km 9 km area covered by the Myama radar ra gauge the Kk rego of Japa. 3 (3) rafall testy(mm/h) rafall testy(mm/h) rafall testy(mm/h) y (km) y (km) y (km) rafall testy(mm/h) rafall testy(mm/h) rafall testy(mm/h) y (km) y (km) y (km) Fgure. A example of dsaggregated rafall felds from level to level 6 usg the beta-logormal model appled at a 9 km 9 km area covered by the Myama radar ra gauge the Kk rego of Japa. PARAMETER ESTIMATION OF BETA-LOGNORMAL MODEL The values of model parameters β ad σ are detfed usg the data observed at the Myama radar ra gauge. The observato rage s the sde of the radar radus km.
4 Proc. of the d Asa Pacfc Assocato of Hydrology ad Water Resources (APHW) Coferece, July 5-8, 4, Sutec Sgapore Iteratoal Coveto Exhbto Cetre, Sgapore, vol., pp , 4 The study area s a 9 km 9 km square area wth the radar observato rage. The space resoluto of the radar rafall data s 3 km. Thus, the study area s composed of a grd system. The tme resoluto s 5 mute. Radar data of observatos durg August 989 to September 989 were used to detfy the values of model parameters β =.389 R σ = exp [-.875( l R ) -.963( l R)-.938] β σ Average rafall testy, R (mm/h) Average rafall testy, R (mm/h) (a) β (b) σ Fgure. Estmated model parameters of β (a) ad σ (b) as fuctos of average rafall testy R. The method to detfy the parameters s the same as show Jothtyagkoo et al. [5]. I Fgure, values of obtaed parameters are plotted as a fucto of spatal average rafall testy R (mm/h). The sold les the fgures are regresso curves derved by a least square method. The emprcal fuctos of the regresso curves adopted here are the same fucto type used by Jothtyagkoo et al. [5]. It s foud that whe the value of average rafall testy R creasg, the value of β decreases. Ths mples whe the regoal average of rafall testy s hgher, the possblty that rafall occurs at somewhere the area becomes large. For the relatoshp betwee σ ad R, t shows postve correlato whe the value of R s less tha several mm/hr. Ths shows whe the regoal average of rafall testy s hgher the extet, the varace of Y becomes large, whch leads to large varatos of spatal rafall testy. These relatoshps are smlar to the results obtaed by Over ad Gupta [4] ad Jothtyagkoo et al. [5]. INTRODUCING DEPENDENCE PROPERTIES OF SPATIAL RAINFALL DISTRIBUTION ON TOPOGRAPHY INTO CASCADE GENERATOR To corporate a structure of rafall feld heterogeety rafall smulatos, a determstc factor s troduced a radom cascade geerator. The factor represets a orographc effect o spatal dstrbutos of rafall, whch s determed by usg the
5 Proc. of the d Asa Pacfc Assocato of Hydrology ad Water Resources (APHW) Coferece, July 5-8, 4, Sutec Sgapore Iteratoal Coveto Exhbto Cetre, Sgapore, vol., pp , 4 depedece le o topographc elevatos (DLTE) developed by Nakakta et al. [] ad Suzuk et al. []. Nakakta et al. [] ad Suzuk et al. [] preseted a obvous depedece relato betwee accumulated rafall dstrbutos ad topographc elevatos. They examed the relatoshp betwee the spatal mea values of accumulated rafall obtaed the classes dvded by m terval of topographc elevatos ad ther mea topographc elevatos. They foud a hgh value of the correlato coeffcet over.9 betwee the logarthmc values of average accumulated rafall amout each class ad ther mea topographc elevatos. The relato (the Depedece Le o Topographc Elevato, DLTE) s represeted as ( ( T ) / ( T )) = a z b log µ µ (6) k k + where µ (T ) (mm) s a accumulated value of spatally averaged rafall over the etre study rego durg the accumulato tme scale T; µ k (T ) (mm) s the spatal average value of accumulated rafall amout the elevato class k; z k (m) s the spatal average of topographc elevato the elevato class k; ad a, b are parameters. Suzuk et al. [5] obtaed the DLTEs for varous µ (T ) the study area by usg the radar rafall data. Whe µ (T ) = 3 (mm) ad settg y = µ k ( T ) / µ ( T ), the DLTE s obtaed as log y =.775 z + log.8399 (7) By usg ths relatoshp, determstc topographc multpler matrces G are troduced to a radom cascade geerator. For the subdvso area ( =,,, ) at the dsaggregato level (=,,, 6), the topographcal weght G s corporated accordg to Jothtyagkoo et al. [5] as W = BYG (8) wth the codto that the average value of G over the sub-areas s equal to at each dsaggregato step. G s gve by 5 G y y e µ ( T ) = µ / µ, µ = e µ ( T ), µ = (9) = where y s the value of y calculated usg Eq. (7) at the th level of dsaggregato the th sub-area. Thus the values of the topographc multpler matrces G are determed by the DLTE ad topographc elevato formato. Fgure 3 shows smulated rafall felds for the 9 km 9 km rego wth spatal resoluto of 3 km at the sxth level of dsaggregato usg the cascade geerator Eq. (8) for the rafall feld at :4 o September
6 Proc. of the d Asa Pacfc Assocato of Hydrology ad Water Resources (APHW) Coferece, July 5-8, 4, Sutec Sgapore Iteratoal Coveto Exhbto Cetre, Sgapore, vol., pp , 4 6 Fgure 3. Smulated rafall felds for the 9 km 9 km rego wth spatal resoluto of 3 km at the sxth level of dsaggregato usg the cascade geerator troducg topographc effects for the rafall feld at :4 o September To exame the effect of the topographc multpler matrces G, realzatos of rafall dstrbutos were smulated. I the smulatos, the tal rafall testy for the 9 km 9 km area at the level was set to (mm/h). Fgure 4 shows the relato betwee the accumulated rafall volume ad the elevato usg the smulated rafall at the sxth level (spatal resoluto 3km). The fgures (a) ad (b) are the cases of 5 ad tmes accumulatos respectvely. The sold le represets the DLTE set the smulatos. The umber of tmes of accumulato creases, the more clear correlato relatoshp betwee topographc elevatos ad the amouts of accumulated rafall appears. Ths shows the topographc effects o spatal rafall dstrbutos descrbed by usg the DLTE are well corporated to the radom cascade rafall smulatos through the determstc topographc multpler matrces G. However, most of results Fgure 4 are plotted over the DLTE ad the gradet calculated usg the plotted pots s steeper tha the oe of the DLTE. I the smulatos preseted Fgure 4, G was corporated at all dsaggregato levels. For the reaso, the topographc depedece o rafall dstrbutos appears strogly.
7 Proc. of the d Asa Pacfc Assocato of Hydrology ad Water Resources (APHW) Coferece, July 5-8, 4, Sutec Sgapore Iteratoal Coveto Exhbto Cetre, Sgapore, vol., pp , 4 To avod the overdepedece, the smulatos that G was corporated oly at the 6 th dsaggregato level were carred out. Fgure 5 shows the results of the smulatos. I both cases of 5 tmes ad tmes accumulatos, the results are well plotted aroud the DLTE, whch are better results tha show Fgure 4. 7 Σ R / 5 (mm/hr) Σ R / (mm/hr) y =.775x + log(.8399) alttude(m) y =.775x + log(.8399) alttude(m) (a) 5 tmes accumulatos (b) tmes accumulatos Fgure 4. The relato betwee the accumulated rafall volume ad the elevato usg the smulated rafall at the sxth level (spatal resoluto 3km). I the cases, G s corporated at all dsaggregato levels. The fgures (a) ad (b) are the cases of 5 ad tmes accumulatos respectvely. The sold le represets the DLTE set the smulatos. Σ R / 5 (mm/hr) Σ R / (mm/hr).775x + log(.8399) y =.775x alttude(m).775x + log(.8399) y =.775x alttude(m) (a) 5 tmes accumulatos (b) tmes accumulatos Fgure 5. The relato betwee the accumulated rafall volume ad the elevato usg the smulated rafall at the sxth level (spatal resoluto 3km). I the cases, G s corporated oly at the 6 th dsaggregato level. The fgures (a) ad (b) are the cases of 5 ad tmes accumulatos respectvely. The sold le represets the DLTE set the smulatos.
8 Proc. of the d Asa Pacfc Assocato of Hydrology ad Water Resources (APHW) Coferece, July 5-8, 4, Sutec Sgapore Iteratoal Coveto Exhbto Cetre, Sgapore, vol., pp , 4 8 CONCLUSION I ths study, a topographc effect o rafall spatal dstrbutos was corporated to a radom cascade geerator to realze more realstc rafall feld smulatos. To clude the topographc effects to geerated rafall dstrbutos, the depedece le o topographc elevatos (DLTE) of rafall dstrbutos developed by Nakakta et al. [] ad Suzuk et al. [] was corporated to the radom cascade geerator. Usg the DLTE, determstc topographc multpler matrces were troduced to a radom cascade geerator usg the beta-logormal model. The method was appled to a 9 km 9 km rego the Kk area of Japa. The parameters of the radom cascade geerator were obtaed usg radar rafall data. By usg the cascade geerator, spatal rafall dstrbutos wth 3 km resoluto were smulated over the rego. Two kds of radom cascade geerators were tested; oe s a geerator corporatg the determstc topographc multpler matrces G at every dsaggregato step, ad aother s the oe wth troducg G oly at the 6 th level of dsaggregato. The statstcal characterstcs of the smulated rafall data were examed. It was foud that the accumulated values of geerated rafall felds wth 3 km grd resoluto showed a lear relatoshp wth the elevatos, whch was good agreemet wth the relatoshp troduced by the DLTE, especally whe usg the latter radom cascade geerator. The results suggest that the proposed method has the ablty to capture statstcal characterstcs of topography o rafall felds foud rafall data. Ackowledgmets: The authors thak the Yodo Rver Dams Cotrol Offce, Mstry of Lad, Ifrastructure ad Trasport, Japa for provdg radar rafall data, ad Dr. Suzuk at Departmet of Cvl Egeerg, Utsuomya Uversty for makg commets about DLTE. REFERENCES [] Nakakta, E., Y. Suzuk, ad S. Ikebuch () Herarchcal tme-scale structure the depedece of rafall dstrbuto o topography, Joural of Hydroscece ad Hydraulc Egeerg, Japa Socety Cvl Egeers, 9 (), pp. -. [] Suzuk, Y., E. Nakakta, ad S. Ikebuch () A study of depedece propertes of rafall dstrbuto o topographc elevato, Joural of Hydroscece ad Hydraulc Egeerg, Japa Socety Cvl Egeers, (), pp. -. [3] Waymre, E., V. K. Gupta ad I. Rodrguez-Iturbe (984) A spectral theory of rafall Itesty at the meso-beta Scale, Water Resour. Res., (), pp [4] Over, T. M., ad V. K. Gupta (996) A space-tme theory of mesoscale rafall usg radom cascades, Jour. Geophys. Res., (D), pp. 6,39-6,33. [5] Jothtyagkoo, C., M. Svapala, ad N. R. Vey () Tests of a space-tme model of daly rafall Southwester Australa based o ohomogeeous radom cascades, Water Resour. Res., 36(), pp
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