A known limitation of rating curves is that their performance is generally poor, which can be partly attributed to the fact that rating curve methods neglect the hysteresis effects

1. Introduction

Concentration rating curves are useful for the analysis of the response of sediment or solute concentrations to changes

in stream discharge or for the interpolation of infrequent concentration measurements in time with discharge as

auxiliary variable, for example to estimate annual sediment or solute loads.

A known limitation of rating curves is that their performance is generally poor, which can be partly attributed to the fact that rating curve methods neglect the hysteresis effects

in the concentration response to changes in discharge. To

enhance the performance of rating curve models, they should account for these hysteresis effects.

2. Supply-based rating curve

Here, we present a supply-based concentration rating curve for total phosphorus (P) concentrations in the Rhine River,

the Netherlands, which does account for the above hysteresis effects. The supply-based concentration rating curve has four components:

1) The traditional power law rating curve of the form C = a Q ^b where C is the total P concentration [M L ^-3 ],

Q is the river discharge [L T ^-1 ], and a and b are constants [-] (Fig 1b);

2) A long-term linear trend (Fig. 2);

3) A seasonal trend of the form C(t) = A cos [2π(t – T _k )/T]

where A is the concentration amplitude [M L ^-3 ], t is the time (T), T _k is the phase shift (T), and T is the period [T]

(365.25 d) (Fig. 2).

4) A discharge dependent supply or loss term of the form C = -ΔS/(Q Δt), where S is the P stock [M].

The P stock was assumed to increase linearly during periods of deposition, i.e. when the discharge is below a critical dis- charge Q _crit . If Q > Q _crit for more than 16 days, the P stock was assumed to decrease proportionally to the excess dis- charge above the critical discharge.

A supply-based concentration rating curve

to predict total phosphorus concentrations in the Rhine River

MARCEL VAN DER PERK & MARJOLEIN F.A. VOGELS

Department of Physical Geography, Utrecht University, P.O. Box 80115, 3508 TC Utrecht, The Netherlands; e-mail: m.vanderperk@uu.nl

3. Model parameterization and calibration

The model parameters were calibrated by a step-wise

procedure which involved in some steps visual calibration (e.g. concentration amplitude, critical discharge for erosion/

deposition) and in other steps regression analysis (e.g. long- term linear trend, power law rating curve).

5. Conclusions

The results imply that inclusion of the long-term and

seasonal trends and a discharge dependent supply and loss term substantially enhances the performance and predictive power of the concentration rating curve model. As the

response to changes in discharge is different for dissolved and particulate total phosphorus, a further improvement of model performance can likely be achieved by deriving separate concentration rating curves for dissolved total phosphorus and sediment-associated phosphorus.

Source: https://beeldbank.rws.nl, Rijkswaterstaat / Joop van Houdt

Fig. 2. Observed total P concentrations, long-term seasonal trend and seasonal fluctuations.

-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5

0 1000 2000 3000 4000 5000 6000 7000

01/04/2004 01/04/2005 01/04/2006 01/04/2007 01/04/2008 01/04/2009 01/04/2010

To ta l P c onc ent ra tio n (m g/ l)

Di sc har ge (m

/s )

Discharge Total P

Fig. 1a. Discharge and total P concentrations at the Lobith- Bimmen monitoring station (Rhine River, the Netherlands)

(Q: wwww.waterbase.nl; total P: www.aqualarm.nl); b. Pow- er law rating curve for the total P concentrations.

4. Results

The total P concentrations show a long-term linear decrease of 1.0 × 10-5 mg l ^-1 d ^-1 . The amplitude of the seasonal

fluctuation in P concentrations was estimated to be 0.03

mg l ^-1 . The critical discharge for erosion and deposition was estimated to be 1900 m s ^-1 , the increase in phosphorus stock during deposition periods 9300 kg d ^-1 , and the supply from the phosphorus stock to the river water 32 kg d ^-1 per m ³ s ^-1 excess discharge.

The R ² between the observed and predicted total P increased from 0.19 for the traditional rating curve (Nash’s efficiency

coefficient = 0.18) to 0.35 for the supply-based rating curve (Nash’s e.c. = 0.34) (Fig. 3).

R ² = 0,3521 Nash e.c. = 0.34

R ² = 0,1927 Nash e.c. = 0.18

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

0 0.1 0.2 0.3 0.4 0.5 0.6

P re dic te d T ot al P (m g/l)

Observed Total P (mg/l)

traditional rating curve supply-based rating curve

Linear (traditional rating curve) Linear (supply-based rating curve)

Fig. 3. Predicted versus observed total P concentrations for traditional power law rating curve and supply-based rating curve.

R² = 0.1318

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2

2.8 3 3.2 3.4 3.6 3.8

Log( Tot al P )

Log(Q)

a.

b.

0 0.1 0.2 0.3 0.4 0.5 0.6

01/04/2004 14/08/2005 27/12/2006 10/05/2008 22/09/2009

O bs er ved tot al P c onc ent rat ion [m g/ l]

Observed Total P [mg/L]

Cosine function Moving average

Long-term linear trend