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--- en:hydro:nash [2016/12/13 18:52] – ckuells
+++ en:hydro:nash [2024/04/10 10:02] (aktuell) – Externe Bearbeitung 127.0.0.1
@@ Zeile 3: / Zeile 3: @@
 The Nash cascade is a sequence of $N$ linear stores in series, each having a residence time of $K$ time units. The resulting mathematical form of the Nash-cascade is:
-$$h(t) = (t/K)^{N-1}*{exp(-t/K)}/{K \gamma (N)}$$
+$$h(t) = (t/K)^{N-1}*exp\left(\frac{-t/K}{K*\Gamma(N)}\right)$$
-where $h(t)$ is the unit hydrograph in terms of a unit discharge and $\gamma$ is the gamma function $\gamma(N)=(N-1)!$ for integer values $N$. $N$ can also have rational values.
+where $h(t)$ is the unit hydrograph in terms of a unit discharge and $\Gamma$ is the gamma function $\gamma(N)=(N-1)!$ for integer values $N$. $N$ can also have rational values.
+<code R |Nash.R>
+n <- 3
+k <- 0.2
+ks <- 0.1
+kl <- 0.3
+x <- seq(0.1,3,by=0.05)
+y <- 1/(k*gamma(n))*(x/k)^(n-1)*exp(-x/k)
+plot(x,y,col=2, xlab= " ", ylab="Q", xlim=range(0:3), ylim=range(0:3))
+abline( h = seq( 5, 25, 5 ),lty = 2,col = colors()[440])
+abline( v = seq( 1, 4, 1 ),lty = 2,col = colors()[440])
+curve(1/(ks*gamma(n))*(x/ks)^(n-1)*exp(-x/ks), add = TRUE, col="blue")
+curve(1/(k *gamma(n))*(x/k )^(n-1)*exp(-x/k ), add = TRUE, col="red")
+curve(1/(kl*gamma(n))*(x/kl)^(n-1)*exp(-x/kl), add = TRUE, col="green")
+</code>
 <code py |Nash.py>
 from pylab import *
+from scipy import *
 from math import *
-def Nash(t):
+from numpy import *
-    K = 2.0
-    N =  4
+N = 5
-    Q = 1/(K*math.factorial(N-1))*(t/K)**(N-1)*exp(-t/K)
+k = 0.5
-    return Q
+t = arange(0.0, 100, 1)
-t = arange(0.0, 35.0, 0.1)
-plot(t, Nash(t))
+def Nash(t,N,k):
-ytext = ylabel('Q (l/s)')
+    h = (t/k)**(N-1)*exp((-t/k)/(k*gamma(N)))
-xtext = xlabel('Time (t)')
+    return h
+plot(t, Nash(t,N,k))
+xtext = xlabel('time')
+ytext = ylabel('response')
 setp(xtext, size='medium', name='courier', weight='bold', color='g')
 setp(ytext, size='medium', name='helvetica', weight='light', color='b')
 show()
 </code>
+Model in R: [[https://thl-pro.shinyapps.io/Nash-Kaskade/#3|Nash-Kaskade mit Shiny/R Studio in HTML]]