We have considered first four French basins (Table 1): two stations on the Middle Loire, operated by the Electric Power Authority Utility (EDF, Avoine station) and by the Loire Basin Authority (AELB, Orleans station), and three stations in the Seine River basins (Oise, Marne and Middle Seine), operated by the Mega Paris water-supply institution, SEDIF). For the Middle Loire, we used quasi-daily records taken during a pilot study from 1981 to 1985 for nitrate, orthophosphate, and total P at Orleans [26] and hourly records of conductivity and water discharge at the Avoine Nuclear Power Plant. For the Oise, Marne, and Seine Rivers, water quality (NO₃⁻, NH₄⁺, and conductivity) is monitored daily (1995–2004) at the Méry-sur-Oise, Neuilly-sur-Marne, and Choisy-sur-Seine water treatment stations, respectively. All these stations have a pluvio-oceanic water-discharge regime with winter high flows and summer low flows.

Conductivity is used here at a given station as a proxy for the total dissolved solids (TDS), as demonstrated for French rivers [20], and commonly used in water quality surveys [4]: as the ionic assemblage does not vary much at our stations, the correlation between conductivity at 20 °C (in μS cm⁻¹) and the sum of major ions (Σ = Ca²⁺ + Mg²⁺ + Na⁺ + K⁺ + Cl⁻ + SO₄²⁻ + HCO₃⁻), expressed in mequiv l⁻¹, is linear and constant. For instance, for the Loire at Orleans: conductivity = 95,52 Σ (r² = 0.9, n = 36) for a range of conductivity between 180 and 330 μS cm⁻¹. We therefore consider throughout this study that conductivity at stations can be assimilated to the TDS (mg l⁻¹), which we processed in the same way as nutrients and SPM.

Two US tributaries to Lake Erie are also selected to test our findings in river basins of different sizes (area < 2000 km²) and hydrological regimes (nivo-pluvial regime). The Cuyahoga and Grand Rivers are surveyed by the Ohio tributary Monitoring Program, operated by the Water Quality Laboratory of the Heidelberg College, Ohio [32]. Six water quality variables are considered: SPM, total phosphorus (TP), soluble reactive phosphorus (SRP), total Kjeldahl nitrogen (TKN), nitrate + nitrite nitrogen (‘NO23’) and chloride (Cl⁻). Records are daily and sub-daily during floods, up to three samples per day.

Altogether, the database of daily records represents about 315 yearly records for eight water quality variables at seven different stations. We have systematically applied the same data processing for each of the water quality variables, according to the following steps. Full details can be found in [27].

2.2.1 Preliminary step: data selection and completion

2.2.2 Step 1: Calculation of a set of statistical indicators designed to characterize the variability of concentration, water discharge and river material flux [25,27]

2.2.3 Step 2: Calculation of the reference fluxes (F_ref) from daily data of annual fluxes

2.2.4 Step 3: Simulation of regular surveys using Monte Carlo techniques

2.2.5 Step 4: Estimation of fluxes for simulated surveys by one averaging method: product of discharge-weighted mean concentration and mean annual discharge

2.2.6 Step 5: Determination of riverine flux errors

2.2.7 Step 6: Relationship between errors and duration indicators

The Loire, Seine, Marne, and Oise Rivers, at their selected stations, are typical of medium-sized Western European Rivers, with pluvio-oceanic river regime (winter high waters, summer low waters), low SPM concentration, high nutrients and high total dissolved salts, expressed by electrical conductivity. The Grand and Cuyahoga Rivers, located in the western part of Lake Erie basin (Michigan, Indiana, Ohio), have a snowmelt discharge regime with higher SPM concentrations, lower phosphate and nitrate, but higher ammonia or total Kjeldahl nitrogen (TKN). Their levels of total dissolved salts is lower than in the rivers of the limestone-dominated Paris Basin, but their chloride levels are somewhat higher than those found at the Loire and Seine stations, probably due to the use of de-icing road salts. Both river basins sets are very much impacted by human activities [32].

Median concentrations (C₅₀, Table 1) represent the general levels. In the Loire and Seine Rivers, nitrates and ammonia levels are among the highest in the world, well above the natural ranges observed in some other rivers in the world [24]. In the Loire River, high nitrogen levels are related to the livestock farming (52 bovine heads of cattle per square kilometre for the catchment at the Orleans station), to the fertiliser use on arable land (23% of the basin area at Orleans), and to a medium–low population density (58 inhab/km²). In the Seine River, nitrates originate mostly from fertiliser use and ammonia from urban sources (population density ranging from 100 to 150 inhab/km² at the selected stations) [2].

Lake Erie tributaries catchments, located in northeastern Ohio, are more forested. The Cuyahoga River basin is also heavily populated and industrialized [32]. In the Seine as a great effort has been done to collect and treat domestic sewage, the major sources of nutrients are now diffuse natural and agricultural ones [21].

The major origin of chloride is de-icing salts, ahead of other anthropogenic sources. In the Seine River, Cl⁻, Na⁺ and SO42− originate mostly from anthropogenic sources both domestic and industrial; de-icing salts are secondary sources. In the Loire River, water quality is relatively less impacted by industrial activity [9]. When compared to a global river set in which SPM range from 5 to more than 5000 mg/l ([25]), the median SPM levels (C₅₀), reported here, are within the low values for the Seine stations (10 to 20 mg/l) and within the median values for Lake Erie's tributaries (15–35 mg/l).

The variability of concentrations and fluxes is illustrated by the ratio of two percentiles of concentrations (C₉₉/C₅₀) and by the ratio (C*/C₅₀) between the discharge-weighted concentration (C*) and the medians (Table 1). Based on C₉₉/C₅₀, the order of variability is the following (from the highest to lowest):

SPM Erie tributaries > SPM Loire–Seine > SRP Grand > TP and Ppart all > NH₄⁺, TKN all ≥ Cl⁻ Grand–Cuyahoga = NO₃⁻ Grand–Cuyohaga > NO₃⁻ Loire–Seine > ∑⁺ Loire–Seine

The order is not quite identical for the C*/C₅₀ indicator: the SRP figure is less variable than for TP in the Grand River, and C*/C₅₀ for NH₄⁺ in the Loire and Seine Basins is much less variable than for the Grand and Cuyahoga Rivers. It is important to note that C*/C₅₀ is inferior to unity for the Grand River's chlorides and for the Loire and Seine Rivers’ conductivities. This is an indication that high flows are characterized by lower ionic contents, a near-universal relationship [40]. The origins of water quality temporal variations are multiple and encompass both natural and anthropogenic factors, physical control, as dilution, and biogeochemical controls, as seasonal phytoplankton uptake, and they very much depend on the size of the catchment [21,22].

Duration curves give the proportion of water volume or riverborne material fluxes discharged in a given proportion of elapsed time [25,27,38]. They are evaluated as cumulative percentage flux vs. cumulative percentage of time, based on ranked daily fluxes, beginning with the highest values. In this paper, we distinguish duration curves evaluated for each individual year and for the whole period of record. The proportion of the highest riverborne material fluxes and water volume reached in 1, 2, 5, 10, 25, 50% of time (M₁, M₂, M₅, M₁₀…, V_w1, V_w2, V_w5, V_w10…) are key indicators of riverborne material and water flux patterns at a given station. For river particulates, the proportion of fluxes transported during a given percentage of time is always much higher than the proportion of river flow. M₂ has already been applied to river basins from 10³ to 10⁶ km² [25], with very different hydrological regimes, including the Mediterranean and monsoon ones. M₂ is more accurately defined than M₁. M₅ would not be convenient with values rapidly exceeding 99% for smaller basins (<1000 km²) and/or flashy flux regimes.

We have chosen here, as a duration indicator, the cumulated highest fluxes carried during 2% of time (M₂, expressed in% of flux during the period of record), as suggested in [25] and used in a previous paper [27] for the suspended particulate matter. The M₂ indicators are calculated here for all water quality variables (Table 1) and compared to the water flux transported in 2% of time (V_w2, Table 1). Lowest figures of M₂ and V_w2 correspond to the least variable river fluxes, while the highest figures describe the most variable ones.

For the Loire and Seine Rivers, V_w2 = 10 and 7%, respectively. These figures are close to the lowest variability observed at the global scale for such river basins [25], while those of the Cuyahoga (12%) and Grand (17%) Rivers are within the medium variability in a global V_w2 range from 4.5% to 31% for basins of similar sizes (5000 to 50 000 km²) [25]. For the SPM fluxes, M₂ duration indicator (M_{2 SPM}) are in the medium part of the global range for the Loire and Seine stations (M₂ from 18 to 25%), while those of the Grand (M₂ = 57%) and the Cuyahoga (M₂ = 47%) Rivers are in the higher values observed at the global scale for such basins: the minimum (7.9%) is recorded in the Somme River (France) and the maximum (66%) for the Eel River (California) [25].

At a given station, the flux duration indicators determined for dissolved nutrients (NO₃⁻, NH₄⁺, and SRP), major ions, particulate phosphorus, total phosphorus (TP), and total Kjeldahl nitrogen (TKN) are always inferior to the SPM flux duration indicator, for the same stations, and generally superior to the water flux indicator (V_w2) (Table 1). In some cases, they can be inferior to V_w2; this is the case for TDS at the four Loire and Seine stations and for nitrate in the Cuyahoga (Table 1). This reflects the prevalence of relatively low TDS at the highest flow, which is also characterized by C*/C₅₀ < 1.0.

When flux duration curves are represented in a double probability scale, they are linear [24]. Two examples are featured in Fig. 1 for the Grand River (Q, SPM, TP, TKN, SRP, NO3−, and Cl⁻) and for the Oise River (Q, NH4+, NO3−, and conductivity) (Fig. 2). In general, these duration lines are parallel to the water-flow duration line. Duration lines related to total concentrations, i.e., combining dissolved and particulate matter, are closer to the SPM lines (see TP and TKN for the Grand River) compared to the duration lines of dissolved material, which are closer to the water flow line (see SRP for the Grand River, NO3− and NH4+ for the Oise River). Chloride in the Grand River and conductivity (CDV, i.e., TDS) in the Oise River are both represented by duration lines below the flow duration ones. The nitrate duration for the Grand River is awkward and cannot be explained at this stage. These differences between river basins and between water quality descriptors are now considered for the flux errors analysis resulting from the simulation of various sampling frequencies.

The flux method based on discharge-weighted concentration (M18, [29]) is recognized to be less biased than many other procedures, for example, methods based on the time-weighted values of mean sample concentration (M14 and M17, [29]), but results in a large variability in flux estimations [39]. However, the M18 method is specified as the first choice of the OSPAR Convention, which deals with fluvial inputs to the North Sea [19]. We present here the flux estimate errors by the M18 for our dataset on SPM, TDS, nutrients, and chloride; then we test if the M_{2 SPM} error nomograph, previously generated for the SPM fluxes [27], can be used to predict flux estimate errors for any riverborne material.

3.3.1 Biases and imprecision on annual fluxes for monthly surveys

In Fig. 3, flux estimates for a one-month sampling interval are compared for all water quality variables, at four selected rivers stations.

• • Bias and imprecisions are important for SPM, TP, TKN, NH₄⁺, and minor for nitrate, TDS, SRP, and chloride. For example, for the Grand River, biases are −46% (SPM), −22% (TP), −13% (TKN), which is consistent with the decrease of the M₂ duration indicator for these materials: 57% for M_{2 SPM}, 36% for M_{2 TP}, 27% for M_{2 TKN} (Table 1).
• • For a given constituent, errors depend on the variability of concentrations with river discharge, and seem also related to the M₂ indicator. For example, the imprecisions of annual nitrate fluxes are between 40 and 30% for the Grand and Cuyahoga Rivers (M₂ around 24%), and only 8% for the Seine and Oise Rivers (M₂ in the range 6 to 7%).
• • Biases and imprecisions are therefore very much station-dependent and river material-dependent for a given sampling frequency.

3.3.2 Biases and imprecision on annual fluxes according to sampling interval

The influence of the sampling interval on the interannual bias and imprecision of annual flux estimates for the whole period of record can be seen in Fig. 4 for the Seine and Grand Rivers and for the eight water quality variables. We are using here the bias and imprecision representation with a variable sampling frequency as used by [17] and already presented for SPM [27]. Biases are generally negative ones and increase with the sampling interval, as already noted in previous studies. For a given sampling interval (e.g., 20 days), the bias order is:

SPM>NH4+>NO3−=TDS=0% for the Seine River (Fig. 4a) and SPM>TP>TKN>Cl−=SRP=NO3−=3% for the Grand River (Fig. 4b).

3.3.3 Influence of the number of simulated surveys

The influence of the number of replicated surveys on flux errors is negligible. We are showing in Fig. 5 as an example the related interannual errors vs. sampling interval curves for SPM in the Grand River for four simulations: 50, 100, 500 and 1000 surveys for each sampling intervals. This is consistent with previous results obtained by [16] (50 simulations), by [15] (100 simulations) and by [3] (at least 150 simulations).

Altogether, we have a set of 23 flux error values, defined on e₅₀ for biases, e₁₀, and e₉₀ for imprecision, for eight different water quality variables: SPM, TP, TKN, SRP, NO3−, NH₄⁺, Cl⁻ and TDS (conductivity), from simulated surveys over at least 10 years. All these errors are determined for the whole period of record and associated with the flux duration indicator for 2% of time (M_{2 SPM}, M_{2 TP}, etc, reported in Table 1) in Fig. 6 for two sampling frequencies: monthly (Fig. 6a and c), weekly (Fig. 6b and d). For nutrients and SPM, the three curves e₁₀, e₅₀, and e₉₀ are those of the nomograph previously set up for the SPM only [27]. For Cl⁻ and TDS, which are diluted at high flows, the bias could be slightly positive (zero to few percents for M₂ < 10%) compared to SPM and nutrients biases, which are negative. For the imprecision (e₁₀ and e₉₀), the TDS data points also fit with the general nomograph previously established for SPM. We present here error nomograph parameters for monthly and weekly sampling intervals [27]: monthly, e50=−0.01 M22−0.095 M2, e₁₀=−1.36 M₂, e₉₀ = −1.50 M₂, weekly, e50=−0.004 M22+0.053 M2, e10=−0.004 M22−0.65 M2, e90=0.004 M22+0.71 M2.

For river SPM, there is a very good fit between the new set of measured errors (n = 5) and those predicted by the general M_{2 SPM} nomograph that was expected. It is striking to note that biases and imprecisions for nutrients can also be well predicted by the same nomograph: the regressions for the new dataset are not significantly different from those of the nomograph.

The comparison between Fig. 6a–d confirms the major gains of accuracy and precision when sampling frequency increases from monthly to weekly. For a given M₂ indicator, as an example 40%, the e₉₀–e₁₀ error range decreases from ±60% to ±30%, and the bias is even more reduced, from −22% to −5%. Such M₂ values are commonly observed for river SPM in river basins between 1000 and 100 000 km² [25,27].

Errors on fluxes of dissolved matter (TDS, Cl⁻, NO₃⁻, NH₄⁺, PO₄³⁻) are generally lower at our stations than those of total nutrients (TKN, TP) and particulate matter (SPM). These clusters are identified in Fig. 6.

The precision of riverine fluxes for a given riverborne material is very similar for the Oise, Seine, and Marne Rivers, which have similar hydrological regimes and concentration vs. discharge relations and basin size. Therefore, we believe that M₂ indicators established at one pilot station for various types of river material can be used at the regional level for stations with similar hydrological and water quality regimes.

Interstation flux error variability is important for SPM, as already observed for a much larger sample of river stations [27], and decreases from SPM to TDS in the following order: SPM > total P and total N > Cl⁻ > dissolved nutrients (N, P) > TDS. Since total P and total N are mixing both dissolved and particulate nutrients, their intermediate rank was expected. Chloride flux errors in Lake Erie's tributaries are here superior to those of dissolved nutrients due to the anthropogenic inputs of de-icing road salts that increase flux variability. This order of flux errors variability is similar to the one noted for the C*/C₅₀ ratio determined on the long-term record at each station (see Table 1).

These types of interstation variability and inter-riverine material variability should now be confirmed on a larger dataset.

So far, most authors dealing with flux duration patterns have considered long-term series. This is particularly needed for daily suspended matter fluxes, which amplify the river flow variability, reaching three to four orders of magnitude [25]. For some rivers, the long-term average SPM flux can only be determined over longer periods, e.g., 20 years, including very rare floods that carry in a few days as much particulates as in a few years. The Eel River in California is one of the few documented examples of such a type [35]. In many rivers, annual fluxes are still reported on an annual basis, in order to comply with regional or international agreements. This is for instance the case of the OSPAR convention in the North Sea. Moreover, this annual reporting is also required for civil years that correspond to basin-management time steps, not for hydrological years, splitting winter high-water periods and their related fluxes into two successive civil years. Therefore, we are here testing the flux generalized duration nomograph for individual civil year and various riverborne materials based on the annual duration descriptors (m₂).

We first consider the m₂ indicators and their year–year variations for two-dozen surveys lasting 10 to 22 years (six river stations, eight riverborne materials, see Table 1). The upper and lower hand-fitted ranges of m₂ include more than 95% of the data points. The m₂ vs. M₂ regression is: m₂ = 0.922 M₂ (r² = 0.81, N = 280). For each riverborne material and station, the year-to-year m₂ range is plotted against the long-term flux duration indicator (M₂, see Fig. 7): the m₂ range is very limited for the lower M₂ and extends with increasing M₂.

The dispersion of annual flux duration (m₂) observed at a given station and for a given material is directly linked to the annual water flow indicator (v_w2), i.e. the proportion of water volume discharged during the seven highest ranked days of each civil year. Fig. 8 features such relationship for the total dissolved solids (three stations), TKN (two stations) and SPM (five stations). The m₂ vs. v_w2 correlation is excellent for TDS, fair for TKN and acceptable for SPM. It should now be confirmed for higher v_w2 figures, particularly for dissolved solids.

We then test if the annual flux duration indicator (m₂) can be used to describe the flux error distribution at the annual level. An example is given on Fig. 9 for the monthly sampling frequency. As for the interannual level, flux duration indicators at the annual level can be used to predict the lowest errors (e^j10) and the biases (e^j50). The prediction of the highest errors (e^j90) is less outstanding, particularly for the SPM fluxes when m₂ > 40%, but is acceptable for most dissolved fluxes with m₂ < 20%. It must be noted that at such temporal scales, slightly positive biases can be observed for m₂ < 20%. The general interannual M₂ nomograph presented in Fig. 6 is also applicable at the annual level.