A STUDY ON THE NONLINEALITY OF RUNOFF PHENOMENA AND ESTIMATION OF EFFECTIVE RAINFALL

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1 A STUDY ON THE NONLINEALITY OF RUNOFF PHENOMENA AND ESTIMATION OF EFFECTIVE RAINFALL SHUICHI KURE Graduat school, Chuo Univrsity, Kasuga, Bunkyo-ku, Tokyo, Japan TADASHI YAMADA Dpt. of Civil Enginring, Faculty of Scinc and Enginring, Chuo Univrsity, Kasuga, Bunkyo-ku, Tokyo, Japan Th purpos of this study is to clarify th nonlinarity of runoff phnomna and to undrstand hydraulic procsss in mountainous basins. A Univrsal lumpd analysis mthod for runoff in a mountainous slop is proposd. Th partial diffrntial uation of kinmatic wav thory applid to th subsurfac flow is transformd to an ordinary diffrntial uation with rspct to th discharg along th slop by sparation of variabls undr th assumption that th slop-lngth is short and th thicknss of surfac soil layr is small. Runoff paramtrs ar dtrmind in trms of th slop gradint, slop lngth, thicknss of surfac soil layr, unsaturatd hydraulic conductivity and ffctiv porosity in th modl. This modl can xprss th charactristics of nonlinarity of runoff and bas-flow rcssion on small mountainous basins. Two nw mthods to stimat ffctiv rainfall ar proposd. On is to stimat ffctiv rainfall from obsrvd data of discharg. Anothr mthod is to stimat ffctiv rainfall by thory of watr holding capacity of a basin. W compard two mthods of stimating ffctiv rainfall and applid th mthods to stimat ffctiv rainfall to th Kusaki dam basin in Japan. Th rsults of runoff analysis match wll with th obsrvd data in Kusaki dam basin. It can b concludd that th proposd lumpd analysis mthod can xprss th runoff in a mountainous slop ffctivly. INTRODUCTION Th mthod of th runoff analysis as that of rational function, storag function, tank modl and kinmatic wav mthod hav bn rsarchd for a flood forcasting. Th conditions of a practical runoff modl ar showd as follows. First of all, it is asy to simulat a runoff in th mountainous basins. Scondary, th paramtr of runoff modl can b dtrmind in th mountainous basins that don t hav th past hydrological data. Thos mthods arn t filld with th formr conditions. Th purpos of th prsnt papr is to construct th practical and physical flood forcasting systm for th mountainous basins. In addition, this systm xprssd by runoff paramtrs can asily apply to th runoff analysis in any topography and morphology as to rivr, mountainous and urban basins. First, Intrflow is th main componnt of dirct runoff in a mountainous slop. W assum it to b th ovrland flow. Th partial diffrntial uation of kinmatic wav thory applid to th subsurfac flow

2 is transformd to an ordinary diffrntial uation with rspct to th discharg along th slop by sparation of variabls th mthod of storag function by th mountainous slop lumpd. Nxt, runoff paramtrs ar dtrmind by morphological or gological uantitis with using th unsaturatd thory. Finally, w stimat ffctiv rainfall from obsrvd data of discharg. LUMPING MODEL FOR RUNOFF ANALYSIS IN A SLOPE Lumping of subsurfac flow W mak th runoff modl of th intrflow. This flow is th main componnt of dirct runoff in a mountainous slop. E.() is a motion rul gnralizd for ach slop. Th consrvation of mass uation in a mountainous slop can b obtain as E.(2), v = α h m, = vh = αh m + () h + = r(t) (2) whr h(mm) is th ponding dpth in th surfac soil layr, v(mm/h) is th surfac flow vlocity, r(mm/h) is th ffctiv rainfall and is th discharg pr unit lngth along bottom of th slop. From E.() and E.(2), th xprssion for th can b obtain as follows, m+ m+ + a = a r() t (3) m whr ( ) m = m + m+ a α Th dirct runoff is composd by th discharg from th partial sourc ara. This ara is channl and moistur soil layr nar that. Thn, w can suppos that th slop lngth is short sufficintly. In this cas, w can considr that th longr slop lngth, th mor discharg incrass. W can transform (x, t) to E.(4) as th approximat uation and th sparation of variabls, E.(3) is lumpd as follows that is th ordinary diffrntial uation, ( x, t) x (t ) (4) d β (5) = a ( r( t) ) dt β ), whr + m+ a = al = ( m + α m L m β = m + whr * is runoff rat (mm/h), x is optional coordinat (m) on this slop. E.(5) bcoms a fundamntal uation which shows runoff from a slop. 2

3 Th rlationship btwn unsaturatd thory and kinmatic wav uation Bcaus of th runoff paramtrs hav not bn dtrmind by morphological uantitis until th formr, w invstigat rlationship btwn unsaturatd flow uation and kinmatic wav uation in this sction. W xprss that rlationship btwn th lumpd modl xprssd in this papr and th unsaturatd flow modl, that is proposd by Suzuki and Kubota, in th surfac soil layr as follows. Th unsaturatd flow is xprssd by Richards uation as follows, c ψ ψ ψ = k sin ω + k + cos ω (6) t x x z z whr Ψ is a prssur had, ω is a slop gradint, c is th spcific moistur capacity, θ is th volumtric soil moistur and k is th unsaturatd hydraulic conductivity. Suzuki rportd that th rcssion of discharg is not almost influncd by th fist and scond trms of E.(6) nglctd on th right-hand whn th surfac soil layr thicknss is thin. In this cas, w hav E.(7) as follows, ψ θ ψ θ k c = = = sin ω (7) ψ W intgrat th E.(7) with rspct to z-dirctions. This intgrats mans that th volumtric soil moistur and th unsaturatd hydraulic conductivity is avragd with rspctd to z-dirctions. W hav E.(8) as follows, θ k D = D sinω + r( t) (8) whr D is th thicknss of surfac soil layr and r(t) is th ffctiv rainfall. E.(8) transformd from th function with rspct to th z-dirctions to th function with rspct to th x-dirctions (slop-dirctions). E.(8) is kinmatic wav uation. W us Kozny uation (9) as th rlationship btwn th ffctiv saturation S and th unsaturatd hydraulic conductivity k. W us E.() as th rlationship btwn th volumtric soil moistur θand th ffctiv porosity w. Ths uantitis is rspctivly avragd with rspct to z-dirctions as follows, k = k s S (9) θ = S w + θ r () k = k s S () θ = S w + θ (2) r Insrting E.() and E.(2) in E.(8) yilds 3

4 4 S Dw S = Dk s sin ω + r( t) (3) whr k s is th saturatd hydraulic conductivity, is xponnt in hydraulic conductivity function and w is th ffctiv porosity. W xpand th Darcy s law into unsaturatd zon and avrag it with rspct to z-dirctions as follows, ψ ψ v = k = k ss = k ss (4) z z whr v is th infiltration vlocity of vrtical dirction. E.(4) is transformd with rspct to th discharg pr unit lngth. It is givn by Kubota as E.(5). W transform E.(5) as E.(6). Insrting E.(6) in E.(3) and transformd with rspct to th discharg pr unit lngth yilds = k D sin ω (5) s S S (, ) = sinω x t k s D D wk s sin ω + = r( t) (7) On th othr hand, insrting E. () in E. (2) yilds (6) m+ m + α + = r( t) (8) E.(7) and E.(8) is th uivalnt uations, bcaus th two point of viw hav mrgd ovr th on phnomnon of th subsurfac flow in th mountainous slop. Comparing E.(7) with E.(8) yilds = m + (9) k ω α s sin = D w (2) It is important that runoff paramtrs that hav bn calibratd by th actual discharg can b uniuly xprssd by uantitis of a surfac soil layr, a saturatd hydraulic conductivity, an xponnt in hydraulic conductivity function, th surfac soil layr thicknss and th ffctiv porosity.

5 5 THE ANALYTICAL SOLUTION WHICH SHOWS RUNOFF Analytical solution rlatd to rcssion curv of hydrograph E. (5) which is fundamntal uation of runoff has analytical solution rlatd to rcssion curv of hydrograph. To xprss rcssion curv of hydrograph, th origin of tim:t is point of tim in which th rainfall is stoppd and r(t)=. E.(2) as a sparation of variabls is obtaind from E.(5). W can b obtain analytical solution rlatd to rcssion curv of hydrograph from E.(2) on ** ()= ** and β. It is xprssd by E.(22). β + = d dt = a β / ( + a β t) β (2) (22) whr ** :runoff rat[mm/h] in rcssion curv. E.(22) is a pculiar solution of nonlinar uation. If condition is on β=, which mans linar condition, w obtain analytical solution E.(23) as th xponntial function. W know rcssion curv of hydrograph is diminishd as th fraction function from obsrvd data of discharg. W could say rcssion curv of hydrograph diminishd nonlinarly. If valu of β which valu taks from to. is.5 bcaus of lumping of runoff procss, w can b obtain analytical solution E.(24). () t = ( + / 2a ) 2 t (23) = xp ( a t) (24) E.(23) and E.(24) ar compltly sam with analytical solution which obtaind Wrnr and Sunuist, Roch and Takagi. It is vry intrsting that sam analytical solution is obtaind from th uations applid in diffrnt flow fild. Analytical solution rlatd to hydrograph E.(5) as fundamntal uation which shows runoff phnomna is nonlinar uation. Gnrally, nonlinar uations can obtain analytical solution undr pculiar conditions. E.(5) can obtain analytical solution undr condition of β=. On β=, E.(5) is transformd E.(25) which is th uation of Brnoulli typ as follows, On * ()= *i, w can b obtain analytical solution E.(26) as follows, d * * dt a r ( t ) = a * i * t a * i t a t 2 + a () = r() t dt r ( τ ) d τ t dt + (25) (26)

6 6 In analytical solution E.(26), Intgratd valu of th rainfall gos into xponntial function and th ffct of th rainfall has bn xprssd nonlinarly. Initial condition is xprssd in fraction function form not simpl function form. Th nonlinar ffct of initial condition is apparntly xprssd in analytical solution. NONLINEAR CHARACTERISTICS IN THE RUNOFF PHENOMENA It is said that runoff phnomna from th mountainous basins shows th nonlinarity gnrally. Th xampl of xamining th factor which shows th nonlinarity and charactristics of th nonlinar phnomna is littl. In ordr to xamin th charactristics of nonlinarity in runoff phnomna w carrid out numrical calculation using E.(5) which is fundamntal uation of runoff in this papr. Runoff rat [mm/h] r(t)=25+25sin(2π t π /2) a =.6 [m 4/5 h /5 ] β =.8 ()=.8[mm/h] ()=.4[mm/h] ()=.[mm/h] ()=.6[mm/h] ()=.2[mm/h] 25 5 Runoff rat [mm/h] r(t)=r +r sin(2π t π /2) a =.2 [m 4/5 h /5 ] β =.8 ()=.2[mm/h] r =2[mm/h] r =8[mm/h] r =6[mm/h] r =4[mm/h] r =2[mm/h] r =[mm/h] Tim[h] 8 Figur. Rlationship btwn initial runoff rat and runoff rat Tim[h] 8 Figur 2. Rlationship btwn rainfall intnsity and runoff rat Th nonlinarity of runoff phnomna about initial condition of soil moistur contnt Th calculation is carrid out by th chang of initial runoff rat In ordr to xamin th ffct of initial condition of soil moistur contnt. Th calculation rsult is shown in figur-. It is provn that runoff rat incrass nonlinarly without incrasing in proportion to linar incras of initial runoff rat, vn if initial runoff rat simply incrasd. And th hydrograph has bn sttld to th fixd curv with incras of initial runoff rat and progrss in th tim. Ths facts will b abl to b just calld th nonlinarity of runoff phnomna about initial condition of soil moistur contnt. Th nonlinarity of runoff phnomna about rainfall intnsity Rainfall intnsity was changd in this calculation in ordr to xamin th ffct of rainfall intnsity about hydrograph. Th calculation rsult is shown in figur-2. Th pak valu of

7 runoff rat bcoms also th doubl, if rainfall intnsity is mad to b th doubl in linar thory. Howvr, in this calculation rsult, it dos not bcom th simplicity for th doubl, whn rainfall intnsity was mad to b th doubl, and pak valu of runoff rat consists 3 tims nar. Thr is no linar rlation for runoff rat and rainfall intnsity. Ths facts will b abl to b just calld th nonlinarity of runoff phnomna about rainfall intnsity. Th nonlinarity of runoff phnomna about th slop lngth Th chang of hydrograph as slop lngth changs is shown in figur-3. What has bn ruird by this calculation is runoff rat [mm/h]. In linar thory, th slop lngth is not influncd about runoff rat. Th pak valu of runoff rat dcrass with th incras of slop lngth and it convrgs on th fixd curv with th progrss in th tim. Ths facts will b abl to b just calld th nonlinarity of runoff phnomna about th slop lngth. Th nonlinarity of runoff phnomna about th ffctiv porosity Finally, this calculation was carry out by th chang of ffctiv porosity for xamining th ffct of th diffrnc of spatial soil proprty of mountainous basins for hydrograph. Th rsult of this is shown in figur-4. It is provn that larg diffrnc occurs by th chang of ffctiv porosity at hydrograph. Th nonlinarity of runoff phnomna occurs from th spatial distribution of soil proprty such as ffctiv porosity in th basin. It can b concludd that th nonlinarity of runoff phnomna rmarkably ariss from th singl slop. 7 Runoff rat [mm/h] 2 r(t)=2+2sin(2π t π /2) β =.8 ()=.[mm/h] : L=[m] : L=2[m] : L=3[m] : L=4[m] : L=5[m] : L=6[m] 2 4 Runoff rat [mm/h] 3 2 r(t)=2+2sin(2π t π /2) β =.8 ()=.[mm/h] :w=.22 :w=.32 :w=.42 :w= Tim[h] Figur 3. Rlationship btwn slop lngth and runoff rat Tim[h] Figur 4. Rlationship btwn ffctiv porosity and runoff rat ESTIMATION OF EFFECTIVE RAINFALL Rainfall: r(t) in E.(5) which is fundamntal uation for runoff proposd in this papr is th ffctiv rainfall. Though various stimation mthods of ffctiv rainfall ar

8 proposd, thr is no xampl of showing its validity. W propos two mthods to stimat ffctiv rainfall in this sction. On is to stimat ffctiv rainfall from obsrvd data of discharg. Anothr mthod is to stimat ffctiv rainfall by thory of watr holding capacity of a basin. In this sction, runoff paramtrs a and β is th avrag valu which is simply dcidd from obsrvd rcssion curv of hydrograph using E.(22). Th application basin is Kusaki dam basin in Japan. Kusaki dam is locatd in th Wataras rivr upstram 78km sit of th Ton rivr watr systm. Th drainag ara of Kusaki dam basin is 254km 2, and it can b gnrally calld a mountainous basin. 8 Loss of rainfall F(R) [mm] :Obsrvd :approximat curv F=a*Tanh(b*R) a=28. b= Total rainfall R [mm] Figur 5. Rlationship btwn total rainfall and loss of rainfall in th Kusaki dam basin Th watr holding capacity distribution S(h)[/mm].5 th watr holding capacity distribution: S(h)=( AB)δ (h)+2ab 2 sinh(bh)/cosh 3 (Bh).4 a=28. b= th proportion of th non prmation ara part in a basin: ab= Th watr holding capacity h[mm] Figur 6. Th watr-holding capacity distribution in a Kusaki dam basin Th invrs stimation of ffctiv rainfall from discharg Th ffctiv rainfall is stimatd from obsrvation discharg data. Whn E.(5) which is th fundamntal uation is transformd, ffctiv rainfall function is shown as follows, r = + β a d dt (27) It is possibl to ruir th ffctiv rainfall by using this E.(27) from obsrvation discharg data. Th stimation mthod of ffctiv rainfall by thory of watr holding capacity in a basin Yamada proposd th thory of watr holding capacity that is to obtain watr-holding capacity distribution from th proportion occupid in th basin of th watr-holding capacity of th soil as th stimation mthod of ffctiv rainfall. Rainfall dos not contribut in th dirct runoff, until it rachs th som valus in which accumulation amount of rainfall dpnds on th soil proprty in a basin. This tim

9 accumulation amount of rainfall is dfind as a watr holding capacity. Th watr holding capacity taks various valus in actual basin. Thn, it is dfind as th proportion occupid in th whol basin in th soil with any watr holding capacity is th watrholding capacity distribution. Th watr holding capacity of th part with th basin is mad to b h, and th watr holding capacity distribution is mad to b S(h). Within total rainfall R(t) by th tim t, only th part which xcdd watr holding capacity h flows. It bcoms th ffctiv rainfall th rsult of intgrating th product btwn th ara proportion S(h)dh and xcss part by watr holding capacity. Thn, total loss F(R) is xprssd by E.(28). E.(28) is th first-kind Voltrra typ intgral uation. It is possibl to obtain th solution by Laplac transformation as a E.(29). 9 R F( R) = R ( R h) S( h) dh S( R) 2 d F df 2 + dr dr = R= δ ( R) (28) (29) whr δ(r) is th Dirac's dlta function, it mans th proportion of th non-prmation ara part in th basin. E. (29) is usd for th rgrssion formula for th rrationship Rainfall R(t) and Total loss F(R). whr a and b ar pculiar paramtrs in a basin. E.(3) which shows Watr-holding capacity distribution profil is obtaind from E.(29) and E.(3). F ( R) = a tanh( br) 2 sinh( br) S ( R) = ( ab) δ ( R) + 2ab 3 cosh ( br) (3) (3) Th calculation rsult of total loss curv is shown in figur-5 and watr-holding capacity distribution is shown in figur-6. Effctiv rainfall obtaind by th thory of watr holding capacity of a basin and th invrs stimation mthod is shown figur-7. In comparison with th ffctiv rainfall ruird from two diffrnt tchnius, in th initial stag, th ffctiv rainfall ruird from th thory of watr-holding capacity is small, and it is provn to largly appar in th pak tim. This rason is th thory of watr-holding capacity is obtaind from th accumulation uantity of rainfall. Finally, w carrid out runoff analysis for Kusaki dam basin using two ffctiv rainfall that is obtaind two thory proposd in this papr. Th calculation rsult of runoff analysis is shown in figur-8. Th rsults of runoff analysis match wll with th obsrvd data in Kusaki dam basin. It can b concludd that th proposd lumpd analysis mthod can xprss th runoff in a mountainous basin ffctivly.

10 Runoff rat [mm/h] Runoff cofficint: :~ Total rainfall :266[mm] a =.4 β =.9, m=. 2 :Effctiv rainfall calculatd by invrs stimation :Effctiv rainfall calculatd by th thory of watr holding capacity :obsrvd Effctiv total rainfall Invrs stimation:4.4[mm] th thory of watr holding capacity:55.8[mm] Tim[h] Figur 7. Calculatd ffctiv rainfall by invrs stimation and th thory of watr-holding capacity in a Kusaki dam basin 4 Runoff rat [mm/h] Tim[h] 2 :Calculatd by invrs stimation 3 :Calculatd by th thory of watr holding capacity :Obsrvd Initial runoff rat :.7[mm/h] Figur 8. Calculatd runoff rat using ffctiv rainfall in a Kusaki dam basin CONCLUSION ) Th lumping runoff modl proposd hr can xprss th runoff in a mountainous basin ffctivly. 2) Runoff paramtrs can b xprssd by uantitis of a surfac soil layr, such as a saturatd hydraulic conductivity, an xponnt in hydraulic conductivity function, th thicknss of surfac soil layr and th ffctiv porosity. 3) Th nonlinarity in th runoff phnomna vry gratly ariss from th singl slop. REFERENCES [] Kur, S., Koshizuka Y. and Yamada, T., Extraction of runoff charactristics from flow rcssion charactristics of hydrograph, In Japans, Annual Journal of Hydraulic Enginring, JSCE, Vol.48, (24), pp 3-8. [2] Yamada, T., Studis on nonlinar runoff in mountainous basins, In Japans, Annual Journal of Hydraulic Enginring, JSCE, Vol.47, (23), pp [3] Shimura, K., Yamada, T. and Ohara, N. and Matsuki,H., Study on runoff charactristics of th larg-scal channl ntwork using a physically basd modl, In Japans, Journal of Japan socity of hydrology and watr rsourcs,jsce, Vol.4, (2), pp27-228

EXST Regression Techniques Page 1

EXST Regression Techniques Page 1 EXST704 - Rgrssion Tchniqus Pag 1 Masurmnt rrors in X W hav assumd that all variation is in Y. Masurmnt rror in this variabl will not ffct th rsults, as long as thy ar uncorrlatd and unbiasd, sinc thy