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HYMOS includes following options to fit stage-discharge data by a rating curve:

• single and compound channel rating curve
• rating curve with unsteady flow correction
• rating curve with backwater correction.

The rating curve parameters are stored in the database. A rating curve is valid for a certain period of time. Each curve can be described by at maximum 5 sets of parameters valid for a specific water-level range. The curves may be of the parabolic or of the power type equation.

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Before computation
A graph-out of the stage-discharge data together with linear and double-logarithmic plots are useful prior to the determination of the parameters of the rating curve:

• to check the availability of data within water-level ranges, and
• to investigate distinct breaks in the double logarithmic stage-discharge plot, which marks the range of applicability of sets of parameters in the rating curve.

It is advised to use approximately the same amount of data points per unit of depth in the determination of a set of parameters of a rating equation applicable to a certain water-level range.

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Following rating equations can be applied:

• parabolic type:
  for h_i < h £h_i+1 : Q = a₁ + b₁ h + c₁ h²
• power type:
  for h_i < h £h_i+1 : Image Modified
  where:

Q = discharge (m³ /s)
a_i , b_i , c_i = parameters
h_i , h_i+1 = lower and upper water level for which the rating equation applies.

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The intersection between the equations is determined by computing the roots of a quadratic equation (parabolic type) or by the Newton-Raphson method, which is continued until the absolute difference between successive estimates of the intersection is less than 0.0001 m.
With respect to the intersection, 3 possibilities exist:

• intersection is in between the pre-set boundaries hu_i-1 and hl_i
• intersection is outside the pre-set boundaries hu_i-1 and hl_i
• no intersection exists.

For case 1 the intersection is taken as the boundary between f_i-1 and f_i .
For the cases 2 and 3 the boundary between f_i-1 and f_i is set equal to the value of h for which the difference between Q = f_i-1 (h) and Q = f_i (h) is minimum in the range hl_i £h £hu_i-1 . In the latter case the output shows the difference in discharge for the selected boundary as well as the real level of the intersection; in case no intersection exists the intersection is denoted by h = -999.

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If the rate of change of the water level is high the stage-discharge relation will not be unique but it will show loopings for the rising and falling stages. Omitting the acceleration terms in de dynamic flow equation the relation between the unsteady discharge Q_m and steady discharge Q_c , see also Figure, is given by the Jones equation:

with:

Wiki MarkupS{~}₀{~} = energy slope for steady flow
v{~}_w{~} = wave velocity
dh/dt = rate of change of the water level in time \ [m/day\]

Figure 8.5: Estimation of the adjustment factor 1/Svwunmigrated-wiki-markup

The factor 1/S{~}₀{~} v{~}_w{~} \ [day/m\] varies with the water level. This factor is fitted by a parabolic function of h: !02 Rating curves^image020.gif! _for h > h_ _{~}min{~}_ with: _h_ _{~}min{~}_ {~}{~} = the lowest water level for which the Jones correction has to be applied.
Image Added for h > h _min
with:
h _{min {}} = the lowest water level for which the Jones correction has to be applied.

In addition to h_min a maximum value for 1/S₀ v_w has to be entered as well to avoid that unacceptably high values of 1/S₀ v_w take part in the fit of equation. Remember that 1/S₀ v_w is expressed in [day/m Wiki MarkupIn addition to h{~}min{~} a maximum value for 1/S{~}0{~} v{~}w{~} has to be entered as well to avoid that unacceptably high values of 1/S{~}0{~} v{~}w{~} take part in the fit of equation. Remember that 1/S{~}0{~} v{~}w{~} is expressed in \[day/m\]!!

Rating curve with backwater correction

Stage-fall-discharge or the twin gauge station fall-discharge methods are used to include backwater effects on stage-discharge ratings. hymosincludes:

• constant-fall method
• normal-fall method.

In these methods the fall F between the water level at the discharge measuring site and a downstream station is considered as an additional parameter, to account for the effect of water surface slope on discharge. Both methods are based on the following equation:

where:
Q_m = backwater affected discharge
Q_r = reference discharge
F_m = measured fall
F_r = reference fall
p = power, with: 0.4 £p £0.6

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The normal fall method goes in two steps:

• computation of the backwater free rating curve to represent reference discharge Q_r
• fitting of normal fall equation to the reference falls.

Backwater free rating curve

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