z-transform of unit hydrograph help !

Hi

i have a free online article : root selection methods in flood
analysis
http://www.hydrol-earth-syst-sci.net/7/151/2003/hess-7-151-2003.html
and there is a z transform for the unit hydrograph for a single
reservoir
(Nash-cascade) i can't figure out how its made !

in the time domaine single reservoir is
y=1/K*exp(- t/K)
general form for n- reservoirs
y=[ (  1/( K*(n-1)! )  )* (x/K).^(n-1).* exp(-t/K) ]

where N=100
t=1,2,3 ... N
K=storage const
n=1 single reservoire

Y(z^-1)=H(z^-1)X(z^-1)

and in the article there two mysterious statments
meaby with mistakes ?

H(z^-1)=[ ( exp(t/K) -1 )/t ] * exp(-t/K)
and below (its multiplicated or its a general form??)
{ ( 1-[exp(-t/K)z^-1]^N )/ ( 1-exp(-t/K)z^-1  )  }

how???
because if you see the analytical form of y
it should be a simple transform for the exp
set c= 1/K
then
y=c *exp (- ct)
Z{y}=Z{c *exp (- ct)}=c* Z{exp (- ct)} = c/(1-exp (- ct)z^-1) = Y
(z^-1)
Z{x}=Z{t}= (z^-1)/( (1-z^-1)^2 ) =  X(z^-1)

and H(z^-1) is simple division of Y(z^-1)/X(z^-1)

someone can werify this for me ? ;)

with regards
Wojtek

On Sat, 02 May 2009 01:18:08 -0700, wooopik wrote:

> Hi
> 
> i have a free online article : root selection methods in flood analysis
> http://www.hydrol-earth-syst-sci.net/7/151/2003/hess-7-151-2003.html and
> there is a z transform for the unit hydrograph for a single reservoir
> (Nash-cascade) i can't figure out how its made !
> 
> in the time domaine single reservoir is y=1/K*exp(- t/K)
> general form for n- reservoirs
> y=[ (  1/( K*(n-1)! )  )* (x/K).^(n-1).* exp(-t/K) ]
> 
> where N=100
> t=1,2,3 ... N
> K=storage const
> n=1 single reservoire
> 
> Y(z^-1)=H(z^-1)X(z^-1)
> 
> and in the article there two mysterious statments meaby with mistakes ?
> 
> H(z^-1)=[ ( exp(t/K) -1 )/t ] * exp(-t/K) and below (its multiplicated
> or its a general form??) { ( 1-[exp(-t/K)z^-1]^N )/ ( 1-exp(-t/K)z^-1  )
>  }
> 
> how???
> because if you see the analytical form of y it should be a simple
> transform for the exp set c= 1/K
> then
> y=c *exp (- ct)
> Z{y}=Z{c *exp (- ct)}=c* Z{exp (- ct)} = c/(1-exp (- ct)z^-1) = Y (z^-1)
> Z{x}=Z{t}= (z^-1)/( (1-z^-1)^2 ) =  X(z^-1)
> 
> and H(z^-1) is simple division of Y(z^-1)/X(z^-1)
> 
> someone can werify this for me ? ;)
> 
> with regards
> Wojtek

I tried, but there's too many distractions in the article.  Post equation 
numbers and maybe I'll try again.

Keep in mind that they may be using non-traditional forms.

-- 
http://www.wescottdesign.com

they use z^-1 not z

its eq (7)

as i say before single reservoire Nash - cascade is
y(t)=1/K*exp(- t/K)  and x(t)=t
for discrete time values t=T*n  T=1 n=1 .. N

and there is one little mistake in eq (7)
in { } should be 1 + exp... in numerator
because i have obtain the same results as in Table 1 with +
so it seem to be workig

i should try the inverse z transform but its a little
to hard , i dont have any program for
symbolic operations :]

they use z^-1 not z

its about eq (7)

for single reservoir in time domain whe have
y(t)=1/K*exp(- t/K)  and x(t)=t
i have no ide how they arrived until eq (7)
but it seem to be working because i
have coherent results

i should try a inverse z transform but it a little
to hard , i dont have any program for
symbolic operations

On Sat, 02 May 2009 11:36:55 -0700, wooopik wrote:

> they use z^-1 not z
> 
> its about eq (7)
> 
> for single reservoir in time domain whe have y(t)=1/K*exp(- t/K)  and
> x(t)=t
> i have no ide how they arrived until eq (7) but it seem to be working
> because i
> have coherent results
> 
> i should try a inverse z transform but it a little to hard , i dont have
> any program for symbolic operations

I don't see where they define the single reservoir response -- I think 
that the time-domain model that you are using is simpler than what 
they're using -- they seem to imply that the reservoir response has a 
significant delay component which (a) kinda matches a reservoir that's 
physically large compared to it's outlet size, and (b) matches my memory 
of how the water rises in most storms (i.e. it isn't at it's maximum 
immediately after a whopping big shower; instead it peaks a bit later).

-- 
http://www.wescottdesign.com