## Fitting of rating curve

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.

Full reports of the quality of fit can be obtained and linear and double-logarithmic scale plots of the stage-discharge data and rating curve.

In addition to the above options a procedure is included to quantify shift adjustments on water levels to account for river bottom variations in the stage-discharge conversion.

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.

A graph of the stage-discharge data can be shown by pressing the <Graph> button. The data points in this graph are linked with the spreadsheet of the 'Fitting of Rating Curves' form, therefore it is easy to select outliers and flag them to zero. All data with a flag set to '0' are not taken into the rating curve computation. By clicking a point in the graph a label with the number of the stage-discharge data will be shown in the graph next to the selected point. By clicking the point in the graph again the label will disappear. By holding the shift key pressed on the keyboard and selecting the point in the graph the flag of the selected point will be set to zero in the spreadsheet. When a flag of a stage-discharge data is set to zero the point will be removed from the graph. The same can be achieved by changing the flag number to '0' in the spreadsheet. When the flag of a stage-discharge data is set to 1 again the point will be shown again in the graph with stage-discharge data.

**Simple rating curve**

When unsteady flow and backwater effects are negligibly small the stage-discharge data can be fitted by a single channel relationship, valid for a given period of time and water level range.

Following rating equations can be applied:

- parabolic type:

for h_{i}< h £h_{i+1}: Q = a_{1}+ b_{1}h + c_{1}h^{2} - power type:

for h_{i}< h £h_{i+1}:

where:

Q = discharge (m^{3} /s)

a_{i} , b_{i} , c_{i} = parameters

h_{i} , h_{i+1} = lower and upper water level for which the rating equation applies.

The coefficients a_{1} , b_{1} and c_{1} of the parabolic equation are determined by the least squares method. The shift parameter a_{2} 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} and subsequently the coefficients b_{2} and c_{2} are determined by the least squares method applied to the logarithms of Q and (h + a_{2} ). Next the estimate for a_{2} is varied within 2 m around its first estimate to obtain a set of parameters for which the mean square error is minimum.

Each rating curve may consist of at maximum 5 equations (5sets of parameters valid for specific water-level ranges).

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

Figure 8.4: multiple sections

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.

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} = percentage difference

Q_{i} = measured discharge

Q_{c} = computed discharge

The standard error S_{e} follows from:

where:

N = number of observations

The standard error is given for each rating equation separately as well as for all equations over the full range of data.

The standard error of the mean relation S_{mr} is computed from:

where:

t = Student t-value at 95% probability

P = value dependent on type of equation

with: P = h_{i} for parabolic equation

P = ln(h_{i} + a_{2} ) for power equation

= variance of P

### Compound channel rating curve (for power equation only)

The compound channel rating curve option has been introduced to avoid large values for the b-parameter and extremely low values for the c-parameter in the power equation for compound cross-sections (river+flood plain).

For this new option the last interval of water levels given will be treated as flood plain water levels. River discharge Qr will be computed for this last interval using the parameters computed for the one but last interval. A temporary floodplain discharge Qf is then computed by subtracting Qr from the observed discharge Qobs for the last water level interval: Qf = Qobs - Qr. Discharge Qf will be used to compute the parameters for a rating curve for the flood plain separately. These parameters together with the one(s) of the rating curve(s) for the other interval(s) will be stored.

**Rating curve with unsteady flow correction**

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:

S_{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/Svw

The factor 1/S_{0} v_{w} [day/m] varies with the water level. This factor is fitted by a parabolic function of h:

*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/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

### Constant-fall method

In this method the reference fall F_{r} is taken as a constant. A special case of the constant-fall method is the unit-fall method, where F_{r} = 1 m is applied.

In the computational procedure a value for 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} to the rating curve is shown, whereas in the error analysis the measured discharge Q_{m} is compared with the computed discharge according to the above equation.

### Normal-fall method

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

F_{r} = a_{4} + b_{4} h + c_{4} h^{2}

valid for h > h_{min} , where 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

To isolate the backwater free data from the rest of the measurements a data flag = 2 has to be added to the data. This is done by editing the spreadsheet column <Flag> from the screen and replacement of flag = 1 by 2 where applicable. Then to all backwater free data a rating curve is fitted analogous to the single channel rating curve procedure described in the previous section. Note that sufficient backwater free data have to be available for a proper fit, else data have to be added (temporarily) to shape the rating curve.

### Reference fall

Next all remaining data above h_{min} with flag =1 are used to fit a parabola to the reference fall computed as:

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

### Application

To execute the <Fitting of Rating Curve> function the following selections must be made.

### Station Codes

Select a station by clicking the checkbox in the stations list. HYMOSwill extract all Stage-Discharge data for this station within the specified period from the database and place them in the spreadsheet on the form. The specified period is the processing period defined in the <Time> tab of the HYMOS main window.

### Graph

From the data in the spreadsheet a graph can be made by pressing the <Graph> button on the <Fitting of Rating Curve> form. All water levels and discharges on dates with a 'flag' not equal to zero will be presented in the graph. In the graph window the user can select to view the data on a linear or logarithmic scale. There is also the possibility to subtract a fixed value from all the water levels, for example the gauge zero value. This allows the user to better analyse the stage-discharge data on a logarithmic scale when the water levels have a reference which is much below the measured values. By clicking a point in the graph a label with the number of the stage-discharge data will be shown in the graph next to the selected point. By clicking the point in the graph again the label will disappear. By holding the shift key pressed on the keyboard and selecting the point in the graph the flag of the selected point will be set to zero in the spreadsheet.To exit this graph window press the <Close> button.

### Rating Curve option

Select a Rating curve function by selecting the function in the list box. The special options will be enabled when selecting the function.

### Function curve and Function intervals

Select what kind of function you want to fit and enter the number of interval levels with their minimum and maximum values.

After you made your selections press <Execute>. HYmoswill fit a rating curve through the data and present the results as a graph, table and report. To show the graph on screen select <Graph> from the functions tab of the HYmosmain window. To analyse the report press <Report> from the functions tab.

If you are satisfied with the result you can save the rating curve by pressing <Save> on the <Fitting of Rating Curve> form.

Note

Only one rating curve type can be saved to the database for a specified period. If a rating curve relation already exists for the same period, Hymoswill ask to overwrite the existing relation in the database.