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(6.1)
In characteristic form, this becomes:
(6.2)
which gives the possibility to forecast the discharge at a given point (x, t) using the
measurements at another point located upstream at a distance D:
Q(x,t) = Q ( x ! D,t ! D/c )
(6.3)
It is therefore sufficient to measure the discharge at a given distance upstream and to
propagate it downstream with a lag time D/c = T .
However, the equation is not used in the form above, because a previous wrong
forecast can never be updated with the current measurement of the discharge. What is
used instead is the time variation of the discharge. Equation (6.1) is differentiated with
respect to time. Assuming that c remains constant in time, this yields:
(6.4)
This allows the forecast to be corrected for a previous erroneous forecast by
incorporating the current value of the discharge. Indeed, Equation (6.4) can be
discretized as:
Q ( x,t +' t )= Q(x,t) + Q ( x ! D,t ! T +' t )! Q ( x ! D, t ! T )
(6.5)
A temporal interpolation is needed if the time instant t ! T +' t does not correspond to a
time where the discharge has been measured. The temporal interpolation of Equation
(6.5) can be carried out as follows:
Q ( x,t +' t )= Q 1 +(1! r ) Q 2 +(2 r !1) Q 3 ! rQ 4
(6.6)
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