Hymod model for catchments
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The document discusses the Hymod rainfall-runoff model, which uses a storage distribution function for simulating runoff from precipitation. It details the algorithms for updating storage and runoff components, and presents various R scripts for estimating model functions. Key concepts include the separation of surface and subsurface runoff and the mathematical representation of storage systems.
Hymod model for catchments
- 1. Riccardo Rigon Hymod as an example of a rainfall-runoff model R. Rigon, G. Formetta, M. Bancheri, W.Abera S.Bertoni,2006?
- 2. 2 What Hymod does The basin is assumed to be composed by a group of storages which follow a distribution F(C) where C is the value of the storage which can vary from 0 to Cmax. If Cmax is exceeded, that water in excedance goes directly into runoff. If we call precipitation P, this is: RH = P + C(t) Cmax if P + C(t) > Cmax Generically, it is: RH = max(0, P(t) + C(t) Cmax) Which is true even if P(t)+C(t) < Cmax Introduction Riccardo Rigon
- 3. 3 There is a residual runoff RS produced by using the curve, which is valid even if C(t)+P(t) < Cmax: F(C) = 1 ✓ 1 C Cmax ◆b The volume below the curve goes into this residual runoff Storage (probability) function Riccardo Rigon
- 4. 4 Van Delft et al. 2009 figure said it properly for runoff Generating runoff What Hymod does Riccardo Rigon
- 5. 5 TC Petri net representation of Hymod Petri net description Riccardo Rigon
- 6. 6 In figure C(t)=2 P=2 Cmax =10. Therefore for a correct interpretation of the figure in previous slides, the area below a curve is the runoff produced. A correct interpretation of the plot says that all the precipitation below the curve is produced as R, the rest remaining stored at time t+1 . Let’s represent the curves in the right direction Storage (probability) function Riccardo Rigon
- 7. 7 The area below each one of the curve is The integral result can be written as: Z C(t)+P (t) C(t) F(C)dC = Z C(t)+P (t) C(t) 1 ✓ 1 C Cmax ◆b dC F(P(t), C(t), Cmax, b) = P(t) 1 Cb max(b + 1) h (Cmax C(t))) b+1 (Cmax C(t) P(t))) b+1 i Storage (probability) function So: Riccardo Rigon
- 8. 8 1) Update C* Summarizing 2) Update the Rs 3) Update S The algorithm of separation Riccardo Rigon
- 9. 9 Introducing AET S(t) continuously increases unless ET acts. In this case there is a fourth step: Where the left arrow means assignment, and AET is the actual ET AET(t) = S(t) Smax ET (t) The algorithm of separation Riccardo Rigon
- 10. 10 Say ↵ is coefficient to be calibrated R = Rsub + Rsup Runoff volumes is then split into surface runoff and subsurface storm runoff Riccardo Rigon
- 11. 11 Therefore, we have three LINEAR systems of reservoirs. The quick system SQ(t) = S1(t) + S2(t) + S3(t) Runoff volumes Riccardo Rigon
- 12. 12 The subsurface system: The groundwater system: Runoff volumes and groundwater Riccardo Rigon
- 13. 13 It seems a quite complicate system, but every hydrologist knows it can be “exactly” solved. For the quick system Three little reservoirs Riccardo Rigon
- 14. 14 And: And: Other two linear reservoirs Riccardo Rigon
- 15. 15 has the structure for some function f and input I, and, therefore, the storage part injected at time is: These formulas and their companions for Qi(t) and AET(t) can be used to estimate the various residence times. Riccardo Rigon Separating water of different ages
- 16. 16 Find this presentation at http://abouthydrology.blogspot.com Ulrici,2000? Other material at Questions ? R.F.B.A
- 17. 17 Simple R script to estimate the Hymod functions Annex F <- function(C,Cmax,b){1-(1-C/Cmax)^b} Intf <-function(P,C,Cmax,b){P+1/((Cmax^b)*(b+1))*((Cmax-C-P)^(b+1)-(Cmax-C)^(b+1))} If(10,0,10,0.5) Intf(10,0,10,1.5) Intf(10,0,10,5) Intf(10,0,10,1000) Intf(2,2,10,0.5) Intf(2,2,10,1.5) x <-seq(from=0,to=10,by=0.1) plot(x,F(x,10,0.5),type="l",col="blue",ylab="F(C)",xlab="C") text(6,0.25,"b=0.5", col ="blue",cex=.8) lines(x,F(x,10,1.5),type="l",col="red",ylab="F(C)",xlab="C") text(4.5,0.42,"b=1.5", col ="red",cex=.8) lines(x,F(x,10,5),type="l",col="darkblue",ylab="F(C)",xlab="C") text(2.5,0.65,"b=5", col ="darkblue",cex=.8) lines(x,F(x,10,1000),type="l",col="darkblue",ylab="F(C)",xlab="C") text(1,0.9,"b=1000", col ="darkblue",cex=.8) abline(v=2,color="grey") abline(v=4,color="grey") y <-seq(from=0,to=100,by=1) plot(y,Intf(10,0,10,y),type="l",col="darkblue",ylab="Runoff Production",xlab="b") y <-seq(from=0,to=20,by=0.2) plot(y,Intf(10,0,10,y),type="l",col="darkblue",ylab="Runoff Production",xlab="b") R.F.B.A
- 18. 18 y <-seq(from=0,to=20,by=0.2) plot(y,Intf(2,2,10,y),type="l",col="darkblue",ylab="Runoff Production",xlab="b",main="P(t)=2 Cmax =10") lines(y,Intf(2,1,10,y),type="l",col="black",ylab="Runoff Production",xlab="b",main="P(t)=2 Cmax =10") lines(y,Intf(2,3,10,y),type="l",col="blue",ylab="Runoff Production",xlab="b",main="P(t)=2 Cmax =10") lines(y,Intf(2,4,10,y),type="l",col="darkgreen",ylab="Runoff Production",xlab="b",main="P(t)=2 Cmax =10") lines(y,Intf(2,5,10,y),type="l",col="green",ylab="Runoff Production",xlab="b",main="P(t)=2 Cmax =10") lines(y,Intf(2,6,10,y),type="l",col="orange",ylab="Runoff Production",xlab="b",main="P(t)=2 Cmax =10") lines(y,Intf(2,7,10,y),type="l",col="red",ylab="Runoff Production",xlab="b",main="P(t)=2 Cmax =10") lines(y,Intf(2,8,10,y),type="l",col="darkred",ylab="Runoff Production",xlab="b",main="P(t)=2 Cmax =10") lines(y,Intf(2,8,10,y),type="l",col="violet",ylab="Runoff Production",xlab="b",main="P(t)=2 Cmax =10") y <-seq(from=0,to=20,by=0.2) plot(y,Intf(2,2,10,y),type="l",col="darkblue",ylab="Runoff Production",xlab="b",main="P(t)=2 Cmax =10") lines(y,Intf(2,1,10,y),type="l",col="black",ylab="Runoff Production",xlab="b",main="P(t)=2 Cmax =10") lines(y,Intf(2,3,10,y),type="l",col="blue",ylab="Runoff Production",xlab="b",main="P(t)=2 Cmax =10") lines(y,Intf(2,4,10,y),type="l",col="darkgreen",ylab="Runoff Production",xlab="b",main="P(t)=2 Cmax =10") lines(y,Intf(2,5,10,y),type="l",col="green",ylab="Runoff Production",xlab="b",main="P(t)=2 Cmax =10") lines(y,Intf(2,6,10,y),type="l",col="orange",ylab="Runoff Production",xlab="b",main="P(t)=2 Cmax =10") lines(y,Intf(2,7,10,y),type="l",col="red",ylab="Runoff Production",xlab="b",main="P(t)=2 Cmax =10") lines(y,Intf(2,8,10,y),type="l",col="darkred",ylab="Runoff Production",xlab="b",main="P(t)=2 Cmax =10") lines(y,Intf(2,8,10,y),type="l",col="violet",ylab="Runoff Production",xlab="b",main="P(t)=2 Cmax =10") Simple R script to estimate the Hymod functions Annex R.F.B.A