superPos: superposition of discharge, unit hydrograph

Description

superposition of precipitation along unit hydrograph (to simulate Q from P)

Usage

superPos(P, UH)

Arguments

Vector with precipitation values

Vector with discrete values of the Unit Hydrograph. This can be any UH summing to one, not just the storage cascade model.

Value

list with optimized n and k, Nash-Sutcliffe Index, and simulated discharge

References

Skript "Abflusskonzentration" zur Vorlesungsreihe Abwasserentsorgung I von Prof. Krebs an der TU Dresden http://tu-dresden.de/die_tu_dresden/fakultaeten/fakultaet_forst_geo_und_hydrowissenschaften/fachrichtung_wasserwesen/isiw/sww/lehre/dateien/abwasserbehandlung/uebung_ws09_10/uebung_awe_1_abflusskonzentration.pdf http://www.uni-potsdam.de/fs-g3/data/archiv/klausuren/Master_of_Science/HydroII_Lernzettel.pdf

Examples

Run this code

N <- c(9,5,2,14,1,3) # [mm/hour]
UH <- c(0.1, 0.4, 0.3, 0.1, 0.1) # [1/h]
sum(UH) # sum must be 1

superPos(N, UH)
# If catchment area = 34 km^2 and precipitation is homogenous:
superPos(N/10^3, UH) * 34*10^6 / 3600  # m^3/s  # Add baseflow and you're done...

SP <- data.frame(Prec=c(N, 0,0,0,0),
          P1=c( UH*N[1], 0,0,0,0,0),
          P2=c(0, UH*N[2], 0,0,0,0),
          P3=c(0,0, UH*N[3], 0,0,0),
          P4=c(0,0,0, UH*N[4], 0,0),
          P5=c(0,0,0,0, UH*N[5], 0),
          P6=c(0,0,0,0,0, UH*N[6] ),
          runoff=superPos(N, UH))
SP # SuperPosition

SPcum <- t( apply(SP[2:7], 1, cumsum) )

plot(N, type="h", col=2:7, lwd=3, xlim=c(1, 10), ylim=c(30,0), lend=1)
par(new=TRUE)
plot(1, type="n", ylim=c(0, 15), xlim=c(1, 10), axes=FALSE, ann=FALSE)
axis(4, las=1)
polygon(x=c(1:10, 10:1), y=c(SPcum[,1], rep(0, 10)), col=2)
for(i in 2:6) polygon(x=c(1:10, 10:1), y=c(SPcum[,i], rev(SPcum[,i-1])), col=i+1)
text(2.5, 1, "Shape of UH")

lines( superPos(N, UH), lwd=3)

plot(UH, type="o", ylim=c(0, 0.4), las=1)
lines(UH, type="h" )


# Effect of distribution of Prec:
P_a <- c(1,2,3,4,5,6,7,8)
P_b <- c(4,4,4,4,4,4,4,4,4)
P_c <- c(8,7,6,5,4,3,2,1)
sum(P_a) ; sum(P_b) ; sum(P_c)

UH_1 <- unitHydrograph(n=2, k=2.3, t=1:25)
UH_2 <- unitHydrograph(n=5.5, k=1.8, t=1:25)

par(mfrow=c(2,3), mar=c(2,3,2,1), las=1)
plot(P_a, type="h", col=3, lwd=3, ylim=c(0,8), main="Precipitation a")
plot(P_b, type="h", col=4, lwd=3, ylim=c(0,8), main="Precipitation b")
plot(P_c, type="h", col=5, lwd=3, ylim=c(0,8), main="Precipitation c")
#
plot(UH_1, type="l", main="unit hydrograph", ylab="",xlab="Zeit")
lines(UH_2, col=2)
text(c(7,14), c(0.12, 0.07), c("UH_1","UH_2"), col=1:2)
abline(h=0)
#
plot( superPos(P=P_a, UH=UH_1), col=3, ylim=c(0,5), type="l",
      main="Discharge", ylab="Q [m^3/s]")
lines(superPos(P=P_b, UH=UH_1), col=4)
lines(superPos(P=P_c, UH=UH_1), col=5)
legend("topright", c("P a","P b", "P c"), title="with UH_1", col=3:5, lty=1)
#
plot( superPos(P=P_a, UH=UH_2), col=3, ylim=c(0,5), type="l", 
      main="Discharge", ylab="Q [m^3/s]")
lines(superPos(P=P_b, UH=UH_2), col=4)
lines(superPos(P=P_c, UH=UH_2), col=5)
legend("topright", c("P a","P b", "P c"), title="with UH_2", col=3:5, lty=1)