Fitting and Interpretation of Sediment Rating Curves

Fitting and interpretation of sediment rating curves

N.E.M. Asselman1

Department of Physical Geography, Marine and Atmospheric Research, The Netherlands Centre for Geo-Ecological Research (ICG),Utrecht University, P.O. Box 80.115, 3508 TC Utrecht, The Netherlands

Received 28 September 1999; revised 28 April 2000; accepted 1 May 2000

Abstract

A large part of the sediment in lowland rivers is transported as wash load. As wash load is a non-capacity load it often ismodelled using empirical relations, such as the rating curve technique. Sediment rating curves in the form of a power functionare derived for several locations along the river Rhine and its main tributaries, using different fitting procedures. Inaccuracies inestimated sediment loads are analysed, and spatial differences in the shapes of the fitted rating curves are related to watershedcharacteristics.

Rating curves obtained by least squares regression on logarithmic transformed data underestimate long-term sedimenttransport rates by 10–50%. Better estimates are obtained when nonlinear least squares regression is applied. The steepnessof the fitted rating curves decreases along the main channel of the Rhine in a downstream direction. Contrary to what isgenerally believed this is not related to changes in the sediment transport regime. A better indication of the sediment transportregime is obtained when the slope/intercept pairs of the fitted rating curves are plotted in a graph. All locations that plot on thesame line have a common discharge-concentration value and appear to be characterised by a similar sediment transport regime.At locations of which the slope/intercept pairs plot on higher lines, a larger part of the annual sediment load is transportedduring high discharge.q 2000 Elsevier Science B.V. All rights reserved.

Keywords: Rhine River; Sediment transport; Suspension material; Modelling; Regression analysis

1. Introduction

In many lowland rivers a major part of the sedimentis transported in suspension. In the German lowlandrivers suspended sediment makes up about 85% of thetotal solid sediment load (Hinrich, 1974). In 1984,93% of the total sediment load of the river Rhinenear the German–Dutch border was transported insuspension (Bundesanstalt fu¨r Gewasserkunde,1987). About 85% of this load consisted of silt and

clay, i.e. wash load. It can thus be concluded that washload plays an important role in the sediment transportin the river Rhine.

As the finest fraction of the suspended sedimentload often is a non-capacity load it cannot be predictedusing stream power related sediment transportmodels. Instead, empirical relations such as sedimentrating curves often are applied. A sediment ratingcurve describes the average relation betweendischarge and suspended sediment concentration fora certain location. The most commonly used sedimentrating curve is a power function (e.g. Walling, 1974,1978):

C � aQb �1�

Journal of Hydrology 234 (2000) 228–248www.elsevier.com/locate/jhydrol

0022-1694/00/$ - see front matterq 2000 Elsevier Science B.V. All rights reserved.PII: S0022-1694(00)00253-5

E-mail address:[email protected](N.E.M. Asselman).

1 Present address: WL/DELFT HYDRAULICS, P.O. Box 177,2600 MH Delft, The Netherlands.

with C is suspended sediment concentration (mg/l),Qis water discharge (m3/s), anda andb are regressioncoefficients. Eq. (1) covers both the effect of increasedstream power at higher discharge and the extent towhich new sources of sediment become available inweather conditions that cause high discharge. Despiteits general use several problems are recognised thatregard the accuracy of the fitted curve as well as thephysical meaning of its regression coefficients.

Inaccuracies in predicted instantaneous suspendedsediment concentrations are related to the statisticalmethod used to fit the sediment-rating curve and to thescatter about the regression line. Statistical inaccura-cies related to the fitting procedure are discussed byFerguson (1986, 1987), Jansson (1985), Singh andDurgunoglu (1989), and Cohn et al. (1992). Theyconcluded that the sediment load of a river is likelyto be underestimated when concentrations are esti-mated from water discharge using least squaresregression of log-transformed variables. Scatterabout the regression line is, among other things,caused by variations in sediment supply due to, forinstance, seasonal effects, antecedent conditions in theriver basin, and differences in sediment availability atthe beginning or the ending of a flood. This is notaccounted for by the rating curve.

As a sediment rating curve can be considered a‘black box’ type of model, the coefficientsa and bin Eq. (1) have no physical meaning. Nevertheless,some physical interpretation is often ascribed tothem. Peters-Ku¨mmerly (1973) and Morgan (1995)state that thea-coefficient represents an index oferosion severity. Higha-values indicate intensivelyweathered materials, which can easily be transported.According to Peters-Ku¨mmerly (1973), theb-coeffi-cient represents the erosive power of the river, withlarge values being indicative for rivers where a smallincrease in discharge results in a strong increase inerosive power of the river. Others state that theb-coefficient indicates the extent to which new sedimentsources become available when discharge increases.Several authors compare the values of theb-coeffi-cient obtained for different rivers to discuss differ-ences in sediment transport characteristics in thedifferent basins (Peters-Ku¨mmerly, 1973; Walling,1974; Sarma, 1986; Morgan, 1995; Kern, 1997).

In this study sediment rating curves are derived fordifferent locations along the river Rhine and its main

tributaries. The study has three objectives. The firstobjective is to determine spatial differences in therelation between discharge and suspended sedimenttransport for different locations in the Rhine drainagebasin and to relate these differences to differences inthe sediment transport regime. Secondly, inaccuraciesin estimated sediment loads will be analysed andrelated to the type of rating curve and the statisticalmethodology applied to fit the rating curves. Finally,the study aims at determining the physical meaning ofthe regression coefficients. Questions that need to beanswered with regard to these objectives include‘Which rating curve produces the most accurate esti-mates of the suspended sediment load at differentlocations in the Rhine drainage basin?’, ‘How doesthe shape of the fitted curves differ between differentlocations?’, and ‘Do sediment rating curves provideinformation on the sediment transport regime of theriver and if so, what kind of information can bederived from them?’

2. The Rhine drainage basin

The Rhine is one of the largest rivers in northwest Europe. Its drainage basin is situated betweenthe Alps and the North Sea (Fig. 1). The drainagearea upstream of the German–Dutch border isabout 165 000 km2 (CHR/KHR, 1976). The totallength of the river is about 1320 km. The Rhinebasin is characterised by a temperate climate.Mean annual precipitation varies between 600 mmin the lower downstream parts and 2500 mm in theAlps (CHR/KHR, 1976). Average annual dischargeincreases from about 370 m3/s at the outlet of theBodensee, to about 2300 m3/s near Rees at theGerman–Dutch border. The average annualsuspended sediment load of the river Rhine atRees is 3.14× 106 ton.

3. Discharge and suspended sediment data

Daily discharge and suspended sediment concen-tration measurements have been carried out by theBundesanstalt fu¨r Gewasserkunde (BfG) in Germanyat several locations along the river Rhine and its maintributaries. Suspended sediment concentrations aredetermined from samples collected about 1 m

N.E.M. Asselman / Journal of Hydrology 234 (2000) 228–248 229

below the water surface. Details on the measurementmethods are provided by Deutscher Verband fu¨rWasserwirtschaft und Kulturbau (1986). Data wereprovided by the BfG for the measurement locationsindicated in Fig. 1. The data records vary in length,but at most locations measurements have beencarried out since the early or mid seventies (Table1). Average water discharge and average annual sedi-ment loads are also given in Table 1. Sediment loadsare computed by multiplication of daily riverdischarge values and daily suspended sedimentconcentrations.

4. Method

4.1. Fitting of sediment rating curves

In most studies, a power function is used, in whichthe regression coefficients a and b are obtained byordinary least squares regression on logarithms ofconcentration and discharge data (e.g. Walling,1974, 1977a,b, 1978; Church and Gilbert, 1975;Mossa, 1989):

log C � log a 1 b log Q 1 log 1 �2�

N.E.M. Asselman / Journal of Hydrology 234 (2000) 228–248230

Fig. 1. Measurement locations in the Rhine drainage basin.

which is then back-transformed to obtain:

C � aQb1 �3�in which 1 is a lognormally distributed error. Statis-tical considerations show that the sediment load of ariver is likely to be underestimated when concentra-tions are estimated from water discharge, using leastsquares regression of log-transformed variables(Ferguson, 1986, 1987; Jansson, 1985; Singh andDurgunoglu, 1989; Cohn et al., 1992). To correct forthis underestimation, or bias, several forms of biascorrection factors have been suggested (Ferguson,1986, 1987; Jansson, 1985). Since the degree ofunderestimation is proportional to the degree of scat-ter about the regression, Ferguson (1986) proposedthe following bias correction factor (CF):

CF � exp�2:651S2� �4�whereS2 is the mean square error of the log-trans-formed regression (in log-10 units). It is importantto notice that application of this correction factoronly results in unbiased estimates of suspendedsediment concentrations when the residuals ofC arelog-normally distributed andC is a power function ofQ. This assumption often is not met.

An alternative approach is to assume a power func-tion with additive error, which consists of random,normally distributed white noise, with zero meanand variances 2. In this case unbiased estimates ofaandb can be obtained using a nonlinear least squaresregression procedure. However, in statistical terms the

method is inappropriate since the assumption ofconstant variance or scatter of the dependent variable,homoscedasticity, often is not met. The degree ofscatter in a graph of sediment concentrations plottedagainst discharge usually increases with discharge.Also, the estimate of the variance of the normal errorswill often be biased because there can be no negativeconcentration values.

As both fitting procedures have their own statisticalshortcomings, they are both applied and evaluated onthe measurement locations along the main channel ofthe Rhine. First, a regression was carried out on thelogarithmic transformed data Eq. (2). Then, twopower functions were fitted through the data, usingnonlinear regression. The first power function isgiven in Eq. (5):

C � aQb 1 d �5�whered is a normally distributed error. The secondpower function consists of a power function with anadditive constant term (p):

C � p 1 aQb 1 d �6�Finally, four sets of rating relationships wereconstructed, using data sets subdivided after seasonand changes in discharge (i.e. summer and wintermonths, and rising or falling limbs of the hydrograph).

4.2. Selection of the calibration period

A sediment rating curve can only be regarded asrepresentative for a certain location under the present


Table 1Measurement periods, drainage area, annual average discharge and suspended sediment transport of various locations along the river Rhine andits tributaries

Location Period Area (km2) Discharge (m3/s) Sediment load (106 t/year)

Rheinfelden (Rhine) 1977–1990 34 550 1100 1.12Maxau (Rhine) 1975–1990 50 200 1300 1.16Worms/Nierstein (Rhine) 1984–1990 68 827 1450 1.40Kaub/Bacharach (Rhine) 1971–1990 103 730 1688 2.16Andernach/Weissenthurm (Rhine) 1975–1990 139 800 2219 3.48Rees/Emmerich (Rhine) 1975–1991 159 300 2386 3.14Rockenau (Neckar) 1972–1990 12 676 142 0.47Schweinfurt/Viereth (Main) 1973–1989 12 715 108 0.10Kleinheubach (Main) 1987–1990 21 505 191 0.34Kalkofen (Lahn) 1971–1990 5305 46 0.08Hauconcourt (Mosel) 1975–1980 9400 131 0.24Cochem/Brodenbach (Mosel) 1982–1990 27 088 384 0.88

range of environmental and climate conditions, whenthe relationship between discharge and suspendedsediment concentration is consistent over the entiremeasurement period. Homogeneity of the data waschecked using a double-mass curve approach (Nordinand Sabol, 1973). For most stations except forRheinfelden and Basel, the double-mass curve is analmost straight line, indicating that the relationbetween sediment and water discharge is consistentfor the period of record. The bend in the double-masscurve of Rheinfelden was hardly visible. Also, noinconsistency occurred during the first 8 years of themeasurement period. Therefore the data wereassumed applicable. The data of Basel were notused for further analyses.

As sediment rating curves should be based on awide range of discharges including a sufficientnumber of measurements carried out at highdischarge, it was decided to fit the rating curves ona relatively wet period. A relatively wet period withdischarge exceeding the long-term average dischargewas selected after examination of the cumulativemonthly mean discharge anomalies, as described byLozowski et al. (1989). At all gauging stations a simi-lar trend was observed, with low discharges in 1976,and during the period 1989–1991. Relatively highdischarges occurred during the periods 1979–1983and 1986–1988. The period 1979–1983 was the long-est wet period. Also, during this period no inconsis-tencies were found in the data. Therefore it wasdecided to develop a complete set of sediment ratingcurves for the period January 1979 until December1983. All fitting procedures and bias correctionfactors described previously Eqs. (1), (2), (4), and(5) were applied on the 9 sub samples of the dataset. The 9 sub samples are (1) all data, (2) summerand (3) winter months, (4) rising discharge and (5)falling discharge, (6) rising discharge during summer,(7) falling discharge during summer, (8) risingdischarge during winter, and (9) falling dischargeduring winter. As 4 rating curves were fitted on eachdata set, this resulted in a total of 36 curves perlocation. Because data records for some of the maintributaries start after 1979, sediment rating curveswere also fitted for a 5 year period including thesecond wet period (1986–1990), and for all availabledata, i.e. the period 1975–1990 for most gaugingstations. Comparison of the curves fitted on different

measurement periods provides an indication of therobustness of sediment rating curves under slightlydifferent discharge regimes.

5. Results

5.1. Fitted rating curves

Sediment rating curves were fitted for locationsalong the river Rhine and its tributaries. The regres-sion coefficients of the fitted curves for locationsalong the main channel are given in Table 2. Althoughcorrelation coefficients for some curves are low, theyare all significant at the 0.05 level, because thenumber of data points is large. Several of the regres-sions given in Table 2 have negativep-values Eq. (6).This may suggest that when discharges are extremelylow negative concentration estimates can be produced.However, during the selected periods no flows of lessthan the critical discharge levels occurred.

Large differences in regression coefficients arefound for different fitting procedures as well as forthe different calibration periods. To facilitate compar-ison, some fitted curves obtained for locations alongthe main channel of the Rhine are shown in Fig. 2. Therating curves based on least squares regression of thelog-transformed data seem to underestimate concen-tration values at high discharge (curve 1). The degreeof underestimation decreases when the bias correctionfactor is used (curve 2), or when the rating curve isobtained by fitting a power function based onnonlinear least squares regression (curve 3). However,at gauging stations where suspended sedimentconcentrations are relatively high at low dischargeand increase only slightly with increasing discharge,none of the first three methods fit the data well. This isbest shown by the data obtained from the gaugingstation near Rees. This shortcoming can be overcomeby fitting a rating curve in the form of a power func-tion, based on nonlinear least squares regression, withaddition of a constant term (curve 4). At all locations,the rating curves obtained by fitting a power function(curves 3 and 4) visually result in the best fit.

Fig. 3 shows sediment rating curves for locationsalong the river Rhine as well as its main tributaries,fitted by means of nonlinear least squares regression(curve 4). The rating curves are based on all data


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Table 2Regression coefficients of rating curves fitted for gauging stations along the river Rhine (callibration period: 1979-1983, CF� correction factor,N� number of data points, fall/rise� falling/rising discharge)

Location Data Log-transformed datalog C � log a 1 b × log Q

NonlinearC � a × Qb

NonlinearC � p 1 a × Qb

N

10a b CF r2 a b r2 p a b r2

Rheinfelden All 3.35× 10203 1.21 1.18 0.34 7.70× 10204 1.44 0.27 13.7 8.00× 10214 4.43 0.35 1764Winter 2.42× 10203 1.25 1.21 0.36 1.22× 10206 2.34 0.36 10.7 2.12× 10211 3.73 0.40 849Summer 6.44× 10203 1.12 1.16 0.24 1.31× 10204 1.67 0.24 17.1 2.10× 10217 5.47 0.30 915Fall 6.60× 10203 1.10 1.17 0.30 7.40× 10204 1.43 0.22 14.1 1.10× 10215 5.70 0.32 1473Rise 1.17× 10203 1.38 1.20 0.36 4.26× 10206 2.16 0.34 16.4 1.70× 10211 3.73 0.36 291Winter-rise 1.71× 10203 1.33 1.23 0.37 1.10× 10205 2.05 0.33 15.7 4.70× 10210 3.32 0.34 140Winter-fall 5.52× 10203 1.13 1.20 0.30 1.71× 10207 2.60 0.33 11.9 3.92× 10218 5.78 0.48 709Summer-rise 3.08× 10204 1.55 1.17 0.37 2.81× 10206 2.21 0.37 19.3 5.44× 10216 5.06 0.39 151Summer-fall 1.92× 10202 0.96 1.15 0.19 3.97× 10203 1.20 0.15 16.8 1.06× 10216 5.24 0.19 764All 75-90 8.84× 10203 1.10 1.28 0.28 1.41× 10203 1.40 0.18 14.8 3.38× 10208 2.75 0.19 5047

Maxau All 8.09× 10203 1.07 1.13 0.37 2.92× 10203 1.22 0.37 7.9 3.90× 10205 1.75 0.38 1826Winter 5.08× 10203 1.13 1.14 0.43 3.87× 10204 1.49 0.49 12.3 1.50× 10209 3.03 0.53 911Summer 3.33× 10202 0.88 1.11 0.47 2.40× 10202 0.94 0.23 27.4 1.38× 10201 0.74 0.23 915Fall 1.32× 10202 1.00 1.13 0.32 7.29× 10203 1.09 0.32 4.4 9.10× 10204 1.35 0.32 1455Rise 5.13× 10203 1.14 1.12 0.40 1.23× 10203 1.35 0.40 16.6 1.70× 10208 2.70 0.42 371Winter-rise 2.83× 10203 1.23 1.14 0.45 4.49× 10204 1.49 0.43 22.3 4.40× 10217 5.16 0.48 189Winter-fall 1.06× 10202 1.02 1.13 0.36 8.50× 10204 1.38 0.48 11.4 2.86× 10209 2.95 0.54 722Summer-rise 1.52× 10202 1.00 1.09 0.32 6.15× 10204 1.13 0.37 14.7 2.00× 10206 2.07 0.38 182Summer-fall 5.37× 10202 0.81 1.12 0.18 4.91× 10202 0.84 0.17 2161.7 7.97× 10101 0.12 0.18 733All 75-90 1.48× 10202 1.01 1.15 0.36 2.83× 10203 1.25 0.41 11.1 4.35× 10206 2.05 0.43 6148

Kaub/Bacharach All 8.30× 10203 1.06 1.11 0.44 3.70× 10203 1.19 0.36 3.8 1.25× 10203 1.31 0.36 1826Winter 2.05× 10203 1.24 1.15 0.52 1.32× 10203 1.32 0.39 21.5 2.00× 10203 1.27 0.39 911Summer 1.57× 10201 0.68 1.06 0.28 3.15× 10202 0.90 0.31 16.9 3.10× 10205 1.69 0.33 915Fall 4.54× 10203 1.17 1.09 0.46 9.80× 10203 1.09 0.35 10.9 1.20× 10204 1.55 0.33 1416Rise 2.40× 10202 0.91 1.13 0.38 1.30× 10202 1.01 0.32 244.2 7.11× 10201 0.63 0.36 410Winter-rise 2.79× 10203 1.24 1.17 0.47 1.67× 10202 1.04 0.31 2132.5 1.44× 10101 0.33 0.33 238Winter-fall 5.68× 10203 1.09 1.12 0.49 8.98× 10204 1.35 0.44 9.8 4.47× 10206 1.97 0.68 673Summer-rise 1.71× 10202 0.99 1.07 0.46 5.81× 10203 1.14 0.55 6.8 1.31× 10203 1.31 0.55 171Summer-fall 5.77× 10201 0.50 1.05 0.18 5.00× 10201 0.53 0.14 10.1 3.81× 10202 0.81 0.14 744All 71-90 5.35× 10202 0.85 1.16 0.30 1.89× 10202 1.01 0.25 7.8 2.60× 10203 1.23 0.25 7305

Andernach Weissent All 1.61× 10202 0.99 1.09 0.49 1.72× 10203 1.29 0.44 6.6 4.24× 10204 1.44 0.44 1826Winter 4.32× 10203 1.15 1.12 0.59 6.46× 10204 1.41 0.44 1.7 4.58× 10204 1.45 0.44 911Summer 1.87× 10201 0.67 1.07 0.28 9.22× 10203 1.07 0.50 22.1 8.54× 10206 1.94 0.54 915Fall 5.34× 10202 0.83 1.09 0.41 8.34× 10203 1.07 0.41 11.4 3.60× 10204 1.42 0.42 1376Rise 3.52× 10203 1.20 1.13 0.56 2.38× 10203 1.27 0.45 220.3 2.10× 10202 1.03 0.45 450

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Table 2 (continued)

Location Data Log-transformed datalog C � log a 1 b × log Q

NonlinearC � a × Qb

NonlinearC � p 1 a × Qb

N

10a b CF r2 a b r2 p a b r2

Winter-rise 1.21× 10203 1.34 0.14 0.61 1.71× 10203 1.31 0.42 229.1 3.02× 10202 1.01 0.43 260Winter-fall 1.52× 10202 0.98 1.10 0.55 2.58× 10203 1.22 0.44 5 6.87× 10204 1.37 0.44 651Summer-rise 4.64× 10202 0.86 1.09 0.64 4.21× 10203 1.17 0.63 21.6 2.70× 10205 1.72 0.64 190Summer-fall 4.59× 10201 0.55 1.06 0.20 4.36× 10202 0.87 0.36 21.7 2.10× 10205 1.71 0.41 725All 75-90 3.08× 10202 0.91 1.13 0.40 2.99× 10203 1.22 0.40 9.6 2.89× 10204 1.48 0.64 5844

Rees All 3.36× 10100 0.32 1.08 0.11 7.93× 10201 0.51 0.16 34.3 4.00× 10206 1.86 0.20 1826Winter 1.64× 10100 0.53 1.09 0.28 1.89× 10201 0.69 0.27 21.2 1.22× 10203 1.24 0.27 911Summer 6.90× 10101 20.07 1.05 0.01 1.66× 10101 0.12 0.10 41.5 2.90× 10213 3.61 0.12 915Fall 1.39× 10101 0.13 1.06 0.02 8.51× 10100 0.20 0.04 37.4 5.20× 10208 2.25 0.06 1392Rise 2.84× 10201 0.64 1.10 0.31 1.08× 10201 0.78 0.31 12.5 1.82× 10202 0.96 0.31 434Winter-rise 1.26× 10201 0.75 1.10 0.38 1.17× 10201 0.77 0.31 218 5.85× 10201 0.61 0.32 270Winter-fall 2.15× 10100 0.36 1.07 0.19 9.70× 10201 0.47 0.18 26.4 1.80× 10204 1.40 0.20 641Summer-rise 2.36× 10100 0.37 1.08 0.13 1.25× 10201 0.74 0.27 35.2 1.00× 10206 1.99 0.32 164Summer-fall 3.06× 10102 20.26 1.04 0.09 2.48× 10102 20.23 0.06 35.7 5.15× 10106 21.79 0.07 751All 75-91 2.62× 10100 0.33 1.12 0.09 6.85× 10201 0.52 0.15 29.3 1.96× 10206 1.93 0.44 6148

without subdivision after season or discharge stage.The steepest rating curves are observed in the tribu-taries of the river Rhine. Along the main channel ofthe river Rhine the steepest rating curve is found nearRheinfelden. The steepness decreases in a down-stream direction. The flattest rating curve is observednear Rees.

5.2. Model efficiency of the fitted rating curves

Although the scatter plots in Fig. 2 suggest that

nonlinear least squares regression yields the best esti-mates of suspended sediment concentrations (curves 3and 4), the distribution of the concentration data indi-cate that linear regression on logarithmic transformeddata might produce better results. It can easily be seenin the scatter plots of Fig. 2 that the concentration datais not normally distributed and that the variance is notconstant with discharge. Also, analyses of the resi-duals of these models indicate that they are notnormally distributed either. Best ‘unbiased’ estimatesof a and b would thus be obtained by least squares


Fig. 2. Sediment rating curves established for gauging stations along the river Rhine, using different fitting procedures.

regression on logarithmic transformed data in combi-nation with the correction factor Eq. (4). However,when the data distribution is studied in more detailwith the Kolmogorov–Smirnov test (Blalock, 1981),it appears that the data is not log-normally distributedeither. Thus, as statistical considerations indicate thatboth fitting procedures are inappropriate for these datasets, it can be suggested that the rating curve with thehighest model efficiency should be used for furthercomputations.

The efficiency of the different rating curve modelsin estimating time series of suspended sediment loadswas evaluated using the model efficiency criterion(R2) as defined by Nash and Sutcliffe (1970).R2

is a criterion that determines the efficiency of amodel in comparison with the average value. It iscomputed as:

R2 � F20 2 F2

F20

�7�

in which F02 is the initial variance, andF2 is the

residual variance given by

F20 �

XNt�1

�Ct 2 �C�2 �8�

F20 �

XNt�1

�Ct 2 C 0t�2 �9�

in which Ct and C0t are measured and computedconcentrations at timet, and �C is the averagemeasured concentration. The model efficiencyR2

should not be confused with the correlationcoefficient, which indicates the degree of inter-relation between two or more variables or betweenestimated and measured values of a singlevariable. Instead, it shows whether the appliedmodel provides better estimates than applicationof the average value. Values ofR2 range betweenminus infinity and plus 1, where plus 1 indicatesperfect agreement between measured and computedsediment transport rates. Negative values indicatethat the average measured concentration value is a


Fig. 3. Sediment rating curves for locations along the Rhine and its main tributaries fitted using nonlinear least squares regression.

better estimate of instantaneous suspended sedimentconcentrations than the estimates obtained by themodel. As R2 depends on the initial variance inmeasured sediment concentrations, which may bequite different for different time periods or differentgauging stations, values ofR2 can only directly becompared for different model results of sedimenttransport time series, obtained for thesamegaugingstation, and for thesamevalidation period.

Evaluation of theR2-factors (Table 3) shows thatfor most gauging stations the rating curves based onleast squares linear regression of the logarithmictransformed data have the lowest efficiency. The effi-ciency increases when the correction factor is applied.The best results are obtained when sediment-ratingcurves in the form of a power function with additionalconstant term are used. The efficiency also increaseswhen the data is subdivided after season and dischargestage. This, however, is to be expected as addition ofparameters to a model usually improves the fit. Acriterion for selecting reasonable models in regressionanalysis that embraces a penalty for increasedcomplexity is Akaike’s Information Criterion (AIC).It is estimated as (Webster and McBratney, 1989):

A� n ln2pn

� �1 n 1 2

� �1 n ln R1 2m �10�

in which n is the number of observations,m thenumber of model parameters, andR the residualsum of squares. Differences in model performancedepend on differences in the residual sum of squaresand the number of model parameters. The number ofmodel parameters only has a significant effect on theAIC when regression is carried out on a limitednumber of observations. In this study, however,more than 1800 observations were used to fit the

regression model. Hence, differences in the residualsum of squares predominate over the penalty for thenumber of model parameters. Consequently, the bestrating curve model is a power function with additionalconstant term, based on nonlinear least squares regres-sion on concentration and discharge data, subdividedafter season and discharge stage.

5.3. Model bias and accuracy

Validation of the fitted rating curves also indicatesthat rating curves fitted using nonlinear least squaresregression yield the most accurate estimates of long-term average suspended sediment loads. The modelvalidation results of 4 locations along the river Rhineand 1 location along a major tributary are summarisedin Table 4. When no statistical bias occurs, sedimentloads computed for the calibration period shouldequal measured sediment loads derived by multiplica-tion of daily discharge and suspended sedimentconcentration values. As variations in discharge andsuspended sediment concentrations during a singleday are small, it is expected that errors in calculatedmeasured loads are small. When the rating curvesfitted for Rees are applied to compute sedimentloads over the period used for model calibration itappears that the rating curve based on logarithmictransformed data underestimates the long-term sedi-ment load up to 12% (Table 4). When a power func-tion based on nonlinear least squares regression isapplied an error of less than 1% is attained. At loca-tions that are characterised by steeper rating curvesthe differences in bias are even larger. At Kalkofen forexample, rating curves fitted on logarithmic trans-formed data tend to underestimate sediment loads by56%, whereas the rating curves based on nonlinear


Table 3Model efficiencyR2 (Nash and Sutcliffe, 1970) of different sediment rating curves in estimating suspended sediment transport rates (datameasured during the period 1979–1983 were used for model calibration)

Rees Andernach Kaub Maxau Rheinfelden

Log, all data 0.50 0.56 0.60 0.57 0.32Log × CF, all data 0.52 0.59 0.63 0.60 0.36Power, all data 0.57 0.64 0.63 0.61 0.41Power1 C, all data 0.58 0.64 0.63 0.62 0.51Power1 C, sum/win 0.59 0.65 0.65 0.69 0.53Power1 C, fall/rise 0.65 0.68 0.68 0.65 0.45Power1 C, sum/win1 fall/rise 0.66 0.69 0.71 0.72 0.57

least squares regression overestimate sediment loadsby 0.2% or less (Table 4). When sediment-ratingcurves are used to compute sediment transport duringsingle years-larger errors may occur. At Andernachfor instance (not shown in Table 4) errors in annualsediment loads computed with a rating curve based onnonlinear least squares regression, varied from133%in 1983 to 239% in 1976, whereas at Kaub errorsranged from261% in 1971 to123% in 1983.

6. Discussion

6.1. Interpretation of sediment rating curves

As discussed previously, the values of the regres-sion coefficients of sediment rating curves areassumed to depend on the erosion severity, or theavailability of sediment in a certain area, the powerof the river to erode and transport the available mate-rial, and on the extent to which new sediment sourcesbecome available in weather conditions that causehigh discharge. According to Peters-Ku¨mmerly(1973) and Morgan (1995), the erodibility of thesoils is represented by thea-coefficient Eq. (1). Highvalues of thea-coefficient occur in areas characterisedby intensively weathered materials, which can easilybe eroded and transported. Theb-coefficient repre-sents the erosive power of the river, with large valuesbeing indicative for rivers with a strong increase inerosive power and in sediment transport capacitywhen discharge increases. According to Walling(1974) b-values are also affected by the grain sizedistribution of the material available for transport,i.e. in streams characterised by sand sized sedimentsthe power of the stream to transport sediment will bemore important than in streams that mainly transportsilt and clay. This will result in highb-values.

However, as thea- andb-coefficients of sedimentrating curves are inversely correlated (Rannie, 1978;Thomas, 1988) it seems more appropriate to use thesteepness of the rating curve, which is a combinationof thea- andb-values, as a measure of soil erodibilityand erosivity of the river. Steep rating curves, i.e. lowa- and highb-values, should thus be characteristic forriver sections with little sediment transport takingplace at low discharge. An increase in dischargeresults in a large increment of suspended sediment

concentrations, indicating that either the power ofthe river to erode material during high discharge peri-ods is high, or that important sediment sourcesbecome available when the water level rises. Flatrating curves should be characteristic for riversections with intensively weathered materials orloose sedimentary deposits, which can be transportedat almost all discharges.

When this line of reasoning is accepted for the riverRhine, the following interpretation can be assigned tothe rating curves shown in Fig. 3. The steepest ratingcurves are found in the tributaries of the river Rhine.This suggests a limited amount of fine sediment,


Table 4Differences (%) between estimated and measured sediment loadsfor different types of sediment rating curve (types of rating curve:(1) logarithmic; (2) logarithmic with correction factor; (3)nonlinear; and (4) nonlinear with constant. Rating curves are cali-brated and validated using the same data series (1979–1983))

Data Types of rating curve

1 2 3 4

Rees (Rhine)All 212.1 25.2 20.4 4.1Subdivision after season 211.5 25.0 20.7 0.6Subdivision after stage 29.0 21.8 0.5 1.0Subdivision after seasonand stage

28.7 22.1 0.1 0.1

Andernach (Rhine)All 218.5 211.4 20.1 0.2Subdivision after season 218.3 29.7 20.0 2.6Subdivision after stage 214.3 25.0 1.4 1.2Subdivision after seasonand stage

214.1 25.2 22.0 0.8

Kaub (Rhine)All 213.6 24.2 0.5 20.0Subdivision after season 213.8 24.5 20.1 0.2Subdivision after stage 24.1 5.8 10.5 3.7Subdivision after seasonand stage

29.0 0.7 2.7 2.4

Maxau (Rhine)All 213.1 27.1 20.1 20.5Subdivision after season 213.8 23.2 20.8 20.1Subdivision after stage 211.2 20.3 0.9 1.6Subdivision after seasonand stage

211.9 21.3 0.1 20.8

Kalkofen (Lahn)All 256.0 240.0 20.0 0.2

which can be picked up from the bed at low discharge.Once a certain discharge threshold is exceeded, sedi-ment supply to the river increases, and sediment canbe picked up from the riverbed, resulting in a rapidincrease in suspended sediment concentrations. Thepresence of weirs in most tributaries will also resultin steep rating curves. During low discharge, muchsuspended sediment will settle behind the weirs.When discharge increases this temporary depositedsediment will be flushed out, resulting in a largeincrease in suspended sediment concentrations.Along the river Rhine, the steepness of the ratingcurves decreases in a downstream direction. The flat-test rating curve is found near Rees, indicating thatnear the German–Dutch border large quantities of finematerial are available for transport at low discharge.

This argument leads to the hypothesis that steeperrating curves are indicative of rivers, or river sections,where most of the sediment transport takes place athigh discharge. To test this hypothesis, the sedimenttransport regime of the river Rhine and its tributarieswas studied. To describe the regime,Qs,avgandQs,10%

were computed.Qs,avg is defined as the percentage ofthe total annual sediment load that is transported atdischarges below the average discharge (Qavg). Qs,10%

represents the percentage of the total annual sedimentload that is transported at very high discharges that areexceeded 10% of the time, i.e. 36 days per year. As

shown in Table 5 the sediment transport regime varieslittle between different locations along the main chan-nel of the river Rhine. At all locations along the mainstream, about 25–30% of the annual sediment load istransported at discharge levels below the long-termaverage discharge. Little over 35% of the annual sedi-ment load is transported at very high discharges,which occur only 10% of the time (i.e. about 36days per year). In the tributaries low discharges areless important in transporting suspended sediment,while high discharges are more important. In theMain and the Mosel, about 10–20% of the annualload is transported at discharges below the annualaverage discharge, whereas more than 60% of theannual load is transported at high discharges whichoccur only 10% of the time. At Rockenau (Neckar)and Kalkofen (Lahn) about 77% of the annual load istransported during these extreme and infrequentdischarges. High discharge can thus be consideredmore important in the tributaries than in the Rhineitself.

Although the steepness of the rating curvesdecreases along the Rhine in a downstream direction,no differences in the sediment transport regime areobserved. This contradicts the previous interpretationof the sediment rating curves. It can thus be concludedthat differences in steepness of sediment rating curvesdo not necessarily indicate differences in the sediment


Table 5Importance of high and low discharges in transporting suspended sediment at several locations along the Rhine and its main tributaries

Qavg (m3/s)a Qavg% (% time)b Qs,avg (% Qs)c Q10% (m3/s)d Qs,10%(%Qs)

e

Rockenau (Neckar) 142 69 9 264 77Kalkofen (Lahn) 46 71 10 98 77Hauconcourt (Mosel) 131 67 12 297 66Cochem (Mosel) 384 68 10 830 70Schweinfurt (Main) 108 68 24 204 53Kleinheubach (Main) 191 70 20 371 69Rheinfelden (Rhine) 1100 57 23 1708 35Maxau (Rhine) 1300 58 26 2020 36Worms (Rhine) 1450 60 26 2300 37Kaub (Rhine) 1688 60 27 2630 38Andernach (Rhine) 2219 61 23 3569 46Rees (Rhine) 2386 61 34 3896 34

a Qavg, annual average discharge.b Qavg%, percentage of time during which discharges are lower thanQavg.c Qs,avg, percentage of the total annual sediment load transported at discharges belowQavg.d Q10%, discharge that is exceeded 10% of the time, i.e. about 36 days per year.e Qs,10%, percentage of the total annual sediment load transported at discharges higher thanQ10%.

transport regime. Instead, the difference in steepnessof the rating curve observed in Fig. 3 may be related toscale. For instance, an increase in discharge of100 m3/s can be regarded as a relatively largerincrease near Rheinfelden than near Rees, as the aver-age discharge is much higher at the latter location. Asa result, a stronger increase in suspended sedimentconcentrations will be observed near Rheinfeldenthan near Rees. High discharges can however, stillbe of equal importance in transporting suspendedsediment at both locations. The question that remainsis whether another method exists to determine theimportance of high discharges in transportingsuspended sediment using the shape of sedimentrating curves.

6.2. Correlation between the regression coefficients

To study the effect of the sediment transport regimeon the shape of the sediment rating curve, the regres-sion coefficientsa andb Eq. (1), obtained by nonlinearleast squares regression, were plotted in a graph(Fig. 4). Per location up to 10 combinations ofa-andb-values are plotted, each of which corresponds

with a rating curve fitted through a different part orsubset of the data (see also Table 2).

It is striking that all coefficients obtained for loca-tions along the main channel plot as a single line,whereas the coefficients obtained for the tributariesplot above this line. Three parallel lines can bedrawn. The lower line coincides with the coefficientvalues obtained for the main channel. The upper linecoincides with sample locations in the river Neckar atRockenau, and in the river Lahn at Kalkofen. Inbetween these lines a third line can be drawn formeasurement locations in the Mosel at Hauconcourtand Cochem, and in the Main near Schweinfurt andKleinheubach.

A negative correlation between the regression coef-ficients has been observed more often (e.g. Rannie,1978; Thomas, 1988). Thomas (1988) states that thisis due to the fact that a linear regression line fitsthrough the average values of both variables, herelog-discharge (Q) and log-suspended sedimentconcentration (C). Data sets that have similar meansof log-Q and log-C will result in rating curves thathave this one point in common. The slope of the fittedcurves can then be expressed as a linear function ofthe intercepts with parameters that depend on the


Fig. 4. Correlation between slope/intercept values of sediment rating curves fitted using nonlinear least squares regression.

co-ordinates of the common point. In slope/interceptplots these points will result in a straight line.

Thomas (1988) sub-sampled a data record ofdischarge and suspended sediment concentrations ata single gauging station and fitted rating curvesthrough the simulated data sets. However, as all ratingcurves are based on the same period of record it can beexpected that the simulated data sets will have similardischarge and suspended sediment concentrationvalues. Hence, the rating curves fitted through thedata sets will have one point in common and theslope/intercept parameters will plot on a straight line.

For the river Rhine different data sets were used,measured at different locations. Still the slope inter-cept values plot on a single straight line, indicatingthat the rating curves have one common discharge-concentration value, which can be regarded as afulcrum around which the rating curves rotate. Tostudy this common point in more detail, sediment-rating curves were developed per calendar year. Theregression coefficients obtained by least squaresregression on the logarithmic transformed variables,were plotted in a graph similar to the graph in Fig. 4,and a logarithmic relation was fitted. For the 16 ratingcurves established for Andernach the followingrelation was obtained (Fig. 5A):

b� 0:46682 02963 loga �r2 � 0:997� �11�

When no scatter around this regression line occurs, allpossible combinations ofa- and b-values result inrating curves that plot through a communal point,indicated by�Q01; C 01� in Fig. 5B. However, as inreality some scatter is present, true-rating curveswill plot through a communal area rather than througha communal point. Measurement locations that plot ona different line in the graph in Fig. 4 will have adifferent point in common. For Rockenau, the idea-lised relation established between thea- andb-coeffi-cients result in rating curves that plot through thecommon point�Q02; C 02� (Fig. 5B).

The value of the common discharge/concentrationpoint is related to the position of the line in Fig. 4. Itsvalue is different for the different tributaries, i.e. itvaries spatially, but it appears to be relatively constantin time. Hence, it can be suggested that the commonpoint, or the position of the line in Fig. 4, is a functionof watershed characteristics that do not vary in time,

i.e. relief, drainage area, and drainage density. Theposition of the rating curve coefficients on this line,however, varies in time and therefore must mainly berelated to watershed characteristics that also vary intime, such as average or maximum discharge, andsediment availability. However, no such relation-ships could be identified in this dataset. For instance,drainage area increases along the Rhine in a down-stream direction, while relief and channel slopedecrease, still all measurement locations plot on thesame line.

As the sample locations along the main channel ofthe river Rhine are characterised by a similar sedimenttransport regime, it can be argued that the commonpoint, and hence the position of the line plotted inFig. 4, provides an indication of the sediment trans-port regime. In other words, all locations that plot onthe same line have a similar sediment transportregime, which can be characterised by the communalpoint. The locations that plot on the upper left side ofthis line have a steeper rating curve than the locationsthat plot on the lower right side. This probably isrelated to the size of the river, or the distribution ofthe samples taken at high and low discharge. Loca-tions that plot on different lines not only are charac-terised by steeper or flatter rating curves, but also by adifferent sediment transport regime. It can thus beassumed that the lower line in Fig. 4 represents riversections where a large part of the annual sedimentload is transported at relatively low discharge. Linesthat have a higher position in this graph, such as theline plotted through the coefficients obtained forRockenau and Kalkofen, would then be representativefor river sections where high discharges are moreimportant in transporting suspended sediment.

The hypothesis that information on the sedimenttransport regime of a river section can be obtainedby from slope/intercept plots is based on a limitedamount of data and therefore remains debatable.However, when the results presented in Table 5 areused to verify the hypothesis, no contradictions orinconsistencies are found. In the main channel about35% of the total annual sediment load is transportedduring infrequent high flows. In the Mosel and theMain this percentage increases to more than 60%,and in the Neckar and the Lahn about 77% of theannual load is transported during these extreme andinfrequent discharges.


6.3. Comparison with the literature

To assess the validity of the above hypothesis,rating curve coefficient values published in the litera-ture were plotted in slope/intercept plots (Fig. 6). Fennet al. (1985) established sediment rating curves for aproglacial stream of the glacier de Tsidjiore Nouve inSwitzerland, using hourly instantaneous streamdischarge and suspended sediment concentrationdata. The data were subdivided using meteorologicaland discharge data. Although differences in suspended

sediment concentrations were observed, all ratingcoefficients plot on a single line (Fig. 6A).

Schulte (1995) derived sediment rating curves forthe river Elsenz, a tributary of the river Neckar, usingdifferent intervals of measurements carried out duringa flood in March 1988. The resultinga- andb-coeffi-cient values plot on a single line (Fig. 6A). The posi-tion of the line above the lines obtained in the presentstudy implies that during this particular flood a verylarge part of the sediment (more than 80%) must havebeen transported at very high discharge.


Fig. 5. Sediment rating curves fitted for Andernach and Rockenau (A) slope/intercept relations, (B) fulcrum of the fitted rating curves as derivedfrom the slope/intercept relations.

Kern (1997) fitted rating curves using data of 3measurement locations within a 11 km river sectionof the river Neckar, about 65 km upstream ofRockenau, and 1 location at the confluence of theriver Enz with the river Neckar. Thea- andb-valuesderived from the rating curves plot on a single lineslightly above the line obtained for the river Neckar at

Rockenau (Fig. 6A). This implies that high dischargesare more important in transporting suspended sedi-ment at the locations in the river Neckar studied byKern than at Rockenau. As the latter gauging station islocated downstream of the confluence of the Neckarwith the rivers Kocher and Jagst (Fig. 1), a decreasingimportance of very high discharge can be expected.


Fig. 6. Correlation between slope/intercept values of sediment rating curves obtained from the literature. Slope/intercept relations establishedfor the rivers Rhine and Neckar are shown for comparison.

Walling (1977b) established sediment rating curvesfor three small rivers in England, i.e. the riversCreedy, Exe, and Dart. Data were grouped accordingto season and discharge stage to obtain several ratingrelationships. Thea- and b-values of these relation-ships are plotted in Fig. 6B. All values plot well abovethe lines found for the river Rhine and its tributaries.This suggests that in these rivers most sediment istransported at extreme and infrequent discharges andthat theQs,10% value as defined in Table 5 exceeds80%. According to the suspended sediment load dura-tion curves published by Webb and Walling (1984)Qs,10%for the river Creedy is in the order of 95%. Forthe river DartQs,10% is even higher.

Finally, rating curves developed by Kesel (1989)for the Mississippi River at New Orleans were usedto assess the validity of the hypothesis. As the ratingcurves are based on annual concentration anddischarge data the results can not directly becompared with the results of other studies. Kesel(1989) identified four groupings in the data: a prehis-toric period (1851–1952), a pre-dam period (1930–1952) and two post-dam periods (1953–1962 and1963–1982). For each period he derived a rating rela-tionship. According to Kesel (1989) a 43% decrease insediment load occurred between the prehistoric andthe pre-dam period, which was mainly caused bychanges in land use practice, whereas river dischargeremained almost constant. The regression coefficientsof both periods plot close to each other (Fig. 6C).Thus, although the sediment load decreased, the sedi-ment transport regime probably did not change much.During the first dam period (1953–1962) the annualsediment load declined with 51%. The position of theregression coefficients in the plot is significantlylower, which suggests a change in the sediment trans-port regime, with a decreasing importance of years ofhigh discharge. This can be expected as reservoirs andlakes tend to trap large amounts of sediment, whichwill not be flushed out during high discharge events,decreasing the importance of high discharge in trans-porting suspended sediment. During the next period(1963–1982) the sediment load decreased by another27%. A line can be drawn through the coefficients ofthe first and the second post dam periods. As the lineruns more or less parallel to the lines observed forother rivers (see the line of the river Rhine forcomparison) it can be assumed that the sediment

transport regime remained the same even though thetotal load decreased.

Although none of the examples cited aboveprovesvalidity of the hypothesis that the relationshipbetween the regression coefficients provides infor-mation on the importance of extreme discharge stagesin transporting suspended sediment, they do notcontradict it either.

6.4. Implications and applicability

The advantage of the sediment rating curve techni-que is that once a rating curve has been developed itcan be applied to past streamflow data to reconstructlong-term sediment transport records or to fill gaps inexisting sediment transport records. However, one ofthe major shortcomings of the application of ratingcurves in the extrapolation of sediment transporttime series is that the requisite assumption of statio-narity is often questionable. To judge the validity ofthe stationarity assumption, we need to know moreabout the factors that determine the shape of the sedi-ment-rating curve. In other words, we can only judgethe stationarity of one or both of the rating coefficientsif we have more insight into the physical meaning ofthese coefficients. For instance, can we expect one orboth of the rating coefficients to remain constant whenriver discharge increases due to a change in climate,or when flood discharge decreases due to the construc-tion of dams? If not, can we predict in which way thecoefficients will change? In the previous discussion onthe interpretation of sediment rating curves it wasshown that a logarithmic relation exists between therating coefficients which seems to be related to thesediment transport regime of the river section underinvestigation. For larger rivers, such as the riverRhine, variations in annual discharge or sedimentload do not seem to have a major effect on the sedi-ment transport regime as the coefficient valuesobtained for periods of low and high discharge ploton a single line. This implies that gaps in sedimenttransport records can be filled using the relationbetween the rating coefficients, the daily dischargerecord, and an estimate of the total sediment load.Also, as variations in the sediment transport regimealong the river Rhine are small this method could beapplied to estimate sediment concentrations at loca-tions where only water discharge has been measured.


Data of the gauging station Andernach were used toillustrate how the logarithmic relation between therating coefficients can be applied to extrapolate thesediment transport record. The measurement recordat Andernach covers a period of 16 years, 1975–1990. When no sediment concentrations would beavailable for the period since 1986 the followingprocedure can be used to obtain an estimate of dailysediment transport at Andernach since 1986.

First, sediment rating curves are established for allyears between 1975 and 1985. The regression coeffi-cients are plotted in a graph similar to Fig. 5A and theslope/intercept relation is determined:

b� 0:47012 0:2948 loga �r2 � 0:998� �12�As data from the measurement period 1986–1990 areomitted, the relation is slightly different from therelation given in Eq. (11).

Second, the total sediment load during the period1986–1990 is estimated. Four methods can beapplied: (1) The annual sediment load equals the aver-age annual sediment load measured between 1975 and1985. (2) The annual sediment load equals the sum ofthe loads measured at Kaub (Rhine), Cochem (Mosel)and Kalkofen (Lahn). (3) The sediment load is a func-tion of the load measured near Rees. During the period1975–1985 the load at Andernach was 1.07 times theload measured near Rees, hence the annual sedimentload can be estimated as 107% of the sediment loadmeasured at Rees. (4) The sediment load is a functionof the annual discharge. The relationship betweenannual discharge and annual sediment load can bedetermined from the 1975–1985 data.

Third, the daily discharge frequency distributionover the period 1986–1990 is determined. Combina-tion of this daily discharge frequency distribution withthe slope/intercept relation given in Eq. (12) and theestimated annual sediment load provides the opportu-nity to deduce a sediment rating curve and to estimatesediment transport rates as a function of dischargeduring the period 1986–1990. This is done iterativelyusing:

Qs �Xni�1

QiCi

( )86:4 �13�

with

Ci � aQ�0:470120:2948 loga�i �14�

in which Qs is the total sediment load (kg/5 year),Qi

the discharge of dayi (m3/s), Ci the estimated sedi-ment concentration of dayi (mg/l), a the rating curvecoefficient determined by iteration and 86.4 theconversion factor (86400 s/day× 0.001 g/kg)

Sediment transport rates estimated using a sedi-ment-rating curve always differ from measured sedi-ment loads. These errors are shown by ‘method I’ inFig. 7. The sediment-rating curve was fitted ondischarge and suspended sediment data measuredbetween 1986 and 1990. This sediment-rating curveproduces reasonable estimates of long-term total sedi-ment loads. For the period 1986–1990, the estimatedtotal sediment load is 6% less than the measured sedi-ment load. For the different discharge intervals esti-mation errors are larger and range from240% toabout 120% (Fig. 7).

When sediment transport at the different dischargeintervals is calculated with the slope/intercept relationdetermined at Andernach in combination with esti-mated long-term total sediment loads, similar errorsare obtained. The long-term total sediment load can beestimated using any of the four methods describedpreviously. An example is given by method II(Fig. 7). Depending on the method applied to estimatethe long-term sediment load, errors in estimated totalsediment load vary from25% to 14%. Estimationerrors for the different discharge intervals are largerbut in the same order of magnitude as those obtainedwith the sediment rating curves fitted on measureddata of the period 1986–1990 (method I, Fig. 7).

Method III (Fig. 7) presents the errors that resultwhen the slope/intercept relation determined atotherlocations along the river Rhine is applied. Since theerrors fluctuate around zero it can be concluded thatthe steepness of the rating curve, which is determinedby this slope/intercept relation, is correct (method III,Fig. 7). When an inappropriate slope/intercept relationis applied the errors exhibit a clear trend. This effectbecomes apparent when the slope/intercept relationobtained for Rockenau (Fig. 5) is combined withdischarge data from Andernach (Fig. 7, method IV).Because the total sediment load is set equal to themeasured sediment load, errors are only the result ofinaccuracies in the shape of the sediment-rating curve.This method (IV) overestimates sediment loads at lowdischarge and underestimates sediment loads at highdischarge. The resulting trend in the errors indicates


that the shape of the applied rating curve is wrong, andthat only the slope/intercept relations of gaugingstations that plot on the same line in Fig. 4 shouldbe used to fill gaps in the sediment transport record.

Finally, the method can be applied to estimate sedi-ment transport at locations where only waterdischarge is measured. As all gauging stations alongthe main channel of the river Rhine are characterisedby a similar sediment transport regime, it can beassumed that a location in between two measurementlocations will have a similar sediment transportregime. Hence, it can be expected that the slope/inter-cept relation established for the river Rhine is valid forall other locations between Rheinfelden and Rees, sothat sediment transport rates can be estimated fromdischarge measurements only.

7. Conclusions

Sediment rating curves were fitted for different

locations along the river Rhine and its main tribu-taries. Rating curves obtained by least squares regres-sion on logarithmic transformed data tend tounderestimate sediment transport rates by about10% to more than 50%. The degree of underestima-tion decreases when a correction factor is applied.Better estimates are obtained when rating curves arefitted in the form of a power function, based onnonlinear least squares regression. For most locations,computed long-term average sediment transport ratesdiffer less than 20% from measured sediment loads.

The steepness of the fitted rating curves decreasesalong the main channel of the Rhine in a downstreamdirection. The steepest rating curves are found for thetributaries. It is generally believed that the steepnessof the rating curve is related to the availability ofsuspended sediment in a certain area, in combinationwith the erosive power of the river to transport thismaterial. Steep rating curves are expected to be char-acteristic for rivers where most sediment is trans-ported at high discharge. However, although the


Fig. 7. Differences between measured and computed sediment transport. Sediment loads are computed as a function of discharge, using fourmethods. Method I: a sediment rating curve based on sediment concentrations measured at Andernach between 1986 and 1990. Method II:combination of the slope/intercept relation determined at Andernach, and the long-term average sediment load measured between 1975 and1985. Method III: combination of the slope/intercept relation determined at other locations along the Rhine, and the measured sediment load.Method IV: combination of the slope/intercept relation determined at Rockenau (river Neckar) and the measured sediment load.

steepness of the rating curve decreases along the riverRhine in a downstream direction, no changes in sedi-ment transport regime occur. Thus, rating curve steep-ness does not provide information on the sedimenttransport regime.

A better indication of the sediment transport regimeof a river section is obtained when the slope/interceptpairs of the fitted rating curves are plotted in a graph.All locations that plot on the same line have acommon discharge-concentration point and appearto be characterised by a similar sediment transportregime. Locations that plot on higher lines, i.e. thathave higher slope values at a given intercept value,have a different discharge-concentration point and arecharacterised by a sediment transport regime where alarger part of the annual sediment load is transportedduring high discharge.

For the river Rhine all slope/interval pairs plot on asingle line. This suggests that for any location alongthis river a rating curve can be established using therelationship between the slope/interval values deter-mined at other locations along the Rhine, in combina-tion with water discharge data, and an estimate of thelong-term sediment load.

Acknowledgements

The author thanks the Bundesanstalt fu¨r Gewasser-kunde (Germany) for providing the discharge andsuspended sediment data.

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Fitting and Interpretation of Sediment Rating Curves

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