Page History

...

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.

...

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.

...

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 (m^3^ m³ /s)
a~i~ a_i , b~i~ b_i , c~i~ c_i = parameters
h~i~ h_i , h~ih_i+1~ ₁ = lower and upper water level for which the rating equation applies.

The coefficients a~1~ a₁ , b~1~ b₁ and c~1~ c₁ of the parabolic equation are determined by the least squares method. The shift parameter a~2~ a₂ in the power equation is either input or determined by an adapted Johnson method. A computerised Johnson method (see for a description of the method e.g. wmoOperational Hydrology Report no. 13, 1980) is used to get a first estimate of a~2~ a₂ and subsequently the coefficients b~2~ b₂ and c~2~ c₂ are determined by the least squares method applied to the logarithms of Q and (h + a~2~ a₂ ). Next the estimate for a~2~ a₂ is varied within 2 m around its first estimate to obtain a set of parameters for which the mean square error is minimum.

...

Now let the rating equations for two successive water-level ranges be given by Q = f~if_i-1~ ₁ (h) and Q = f~i~ f_i (h) respectively and let the upper boundary used for the estimation of f~if_i-1~ ₁ be denoted by hu~ihu_i-1~ ₁ and the lower boundary used for the estimation of f~i~ f_i by hl~i~ hl_i . To force the intersection between f~if_i-1~ ₁ and f~i~ f_i to fall within certain limits it is necessary to choose: hu~ihu_i-1~ ₁ > hl~i~ hl_i , hence to use a certain overlap, see the underneath figure.

...

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~if_i-1~ ₁ and f~i~ f_i .
For the cases 2 and 3 the boundary between f~if_i-1~ ₁ and f~i~ f_i is set equal to the value of h for which the difference between Q = f~if_i-1~ ₁ (h) and Q = f~i~ f_i (h) is minimum in the range hl~i~ hl_i £h £hu~i£hu_i-1~ 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.

In the error analysis the difference in measured and computed discharge as well as the relative difference are presented, where the latter is computed from:

where:

DQ~i~ DQ_i = percentage difference
Q~i~ Q_i = measured discharge
Q~c~ Q_c = computed discharge

The standard error S~e~ S_e follows from:

where:

N = number of observations

...

The standard error of the mean relation S~mr~ S_mr is computed from:

where:
t = Student t-value at 95% probability
P = value dependent on type of equation
with: P = h~i~ h_i for parabolic equation
P = ln(h~i~ h_i + a~2~ a₂ ) for power equation
= variance of P

...

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~ Q_m and steady discharge Q~c~ 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 \S₀ = 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/Svw

Wiki MarkupThe factor 1/S~0~ v~w~ \S₀ 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~ Q_m = backwater affected discharge
Q~r~ Q_r = reference discharge
F~m~ F_m = measured fall
F~r~ F_r = reference fall
p = power, with: 0.4 £p £0.6

...

In this method the reference fall F~r~ F_r is taken as a constant. A special case of the constant-fall method is the unit-fall method, where F~r~ F_r = 1 m is applied.
In the computational procedure a value for F~r~ F_r is assumed. Then a rating curve is fitted to the values:

according to the standard procedure outlined in the previous section. The value for p is optimised between the boundaries 0.4 and 0.6 based on the least squares principle.

In the plot the fit of Q~r~ Q_r to the rating curve is shown, whereas in the error analysis the measured discharge Q~m~ Q_m is compared with the computed discharge according to the above equation.

...

In this method the reference fall F~r~ F_r is modelled as a function of the water level: F~r~ F_r = f(h). This function is represented by a parabola:

F~r~ F_r = a~4~ a₄ + b~4~ b₄ h + c~4~ h^2^ c₄ h²

valid for h > h~min~ h_min , where h~min~ h_min is a lower threshold of h above which the backwater correction is applied.

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

...

Next all remaining data above h~min~ h_min with flag =1 are used to fit a parabola to the reference fall computed as:

where Q~r~ Q_r is the reference discharge Q~r~ Q_r = f(h) computed from the backwater free stage-discharge data. The parameter p is optimised between 0.4 and 0.6.

...