2.1 Hydraulic configuration

There appears to be a low weir or riffle downstream of the bridge and the station. A floodplain extends on the right bank about 100 m wide, very roughly. We therefore consider three hydraulic controls: a rectangular weir, a rectangular channel representing the main channel, and another rectangular channel representing the floodplain:

\[ \begin{array}{|c|c|c|} \hline \text{Control} & \text{Nature} & \text{Type} \\ \hline 1 & \text{Weir / riffle} & \text{section} \\ \hline 2 & \text{Main channel} & \text{channel} \\ \hline 3 & \text{Floodplain} & \text{channel} \\ \hline \end{array} \]

The hydraulic operation of this station can be summarized as follows: at low flows, the stage-discharge relationship is controlled by the geometry of a critical cross-section near the rectangular weir. When the water height increases, the weir is submerged and the stage-discharge relationship is then controlled by the average geometry and roughness of a rectangular channel corresponding to the main channel. For an even greater water height, the stage-discharge relationship is controlled by two rectangular channels, corresponding to the main channel and the floodplain.

The control matrix corresponding to this configuration is given below.

\[ \begin{array}{|c|} \hline &\text{control 1} & \text{control 2} & \text{control 3}\\ \hline \text{segment 1} &\color{lime}{1} &\color{darkslategray}{0} &\color{darkslategray}{0}\\ \hline \text{segment 2} & \color{darkslategray}{0} & \color{lime}{1} &\color{darkslategray}{0}\\ \hline \text{segment 3} & \color{darkslategray}{0} & \color{lime}{1} & \color{lime}{1} \\ \hline \end{array} \]

2.2 Specification of priors

The priors defined for these three controls are deduced from Géoportail:

• Weir (low flow control):

  • \(\kappa = 0~\mathrm{m} \pm 0.5~\mathrm{m}\) (deduced from the lowest waters of the stage series)
  • \(B_w = 14~\mathrm{m} \pm 4~\mathrm{m}\) (derived from distance measurements on the Géoportail aerial view, see Fig. 1)
• Main channel (medium and high flow control):
  • \(\kappa = 1~\mathrm{m} \pm 1~\mathrm{m}\) (roughly estimated from Géoportail altitudes)
  • \(K_S = 25~\mathrm{m}^{1/3}\mathrm{/s} \pm 10~\mathrm{m}^{1/3}\mathrm{/s}\) (typical value for this type of main channel, cf. Chow’s table)
  • \(B_w = 12~\mathrm{m} \pm 5~\mathrm{m}\) (deduced from distance measurements on the Géoportail aerial view, see Fig. 1)
  • \(S = 0.001 \pm 0.001\) (estimated by default for a lowland river)
• Floodplain (high flow control):
  • \(\kappa = = 2.5~\mathrm{m} \pm 0.5~\mathrm{m}\) (roughly estimated from Géoportail altitudes)
  • \(K_S = 20~\mathrm{m}^{1/3}\mathrm{/s} \pm 10~\mathrm{m}^{1/3}\mathrm{/s}\) (typical value for this type of floodplain, cf. Chow 1959 table recalled in the BaRatinAGE documentation)
  • \(B_w = 100~\mathrm{m} \pm 50~\mathrm{m}\) (roughly estimated from distance on the Géoportail aerial view, see Fig. 1)
  • \(S = 0.001 \pm 0.001\) (estimated by default for a lowland river)

With these specifications, we obtain the prior rating curve shown below.

Figure 2. Prior rating curve for the Aisne River at Verrières.

2.3 Gaugings and first posterior rating curve

The “normal” rating curve is estimated using the gaugings associated with the existing rating curve without the single gauging of the July 2021 flood measured on 07/16/2021 (Fig. 2). No conflict between the posterior results and the prior distributions is detected (however, some prior distributions such as that of \(k_1\) are very wide). The rating curve estimated with BaRatin is very close to the existing rating curve for most of the discharge range.

However, the computed rating curve does not agree well with the lowest five gaugings, which have significantly higher discharges than the MaxPost rating curve discharges. This discrepancy suggests that a control for very low flows is certainly missing in the assumed hydraulic configuration. There is likely a notch in the weir, or some lower part of the weir, that would explain higher discharges than expected for the lowest stages. Therefore, our first hydraulic assumptions must be revised.

Figure 2. The Aisne River at Verrières (H6021020), “normal” rating curve: a) prior and posterior distributions of parameters, b) BaRatin rating curve with parametric (pink) and total (red) uncertainty envelopes, c) close-up view on lowest lows (discharge in log scale).

3.1 Hydraulic configuration

The weir assumed in our first hydraulic configuration is now split into two additive weirs: the low-flow notch and the rest of the weir. We therefore consider four hydraulic controls: a rectangular weir representing the notch, another rectangular weir that adds to the first one, a rectangular channel representing the main channel that takes over the weirs (when their fall disappears), and another rectangular channel representing the floodplain, that adds to the first one:

\[ \begin{array}{|c|c|c|c|} \hline \text{Control} & \text{Nature} & \text{Type} \\ \hline 1 & \text{Notch in the weir / riffle} & \text{section} \\ \hline 2 & \text{Rest of the weir / riffle} & \text{section} \\ \hline 3 & \text{Main channel} & \text{channel} \\ \hline 4 & \text{Floodplain} & \text{channel} \\ \hline \end{array} \]

The control matrix corresponding to this configuration is given below.

\[ \begin{array}{|c|} \hline &\text{control 1} & \text{control 2} & \text{control 3} & \text{control 4}\\ \hline \text{segment 1} &\color{lime}{1} &\color{darkslategray}{0} &\color{darkslategray}{0} &\color{darkslategray}{0}\\ \hline \text{segment 2} &\color{lime}{1} &\color{lime}{1} &\color{darkslategray}{0} &\color{darkslategray}{0}\\ \hline \text{segment 3} & \color{darkslategray}{0} & \color{darkslategray}{0} & \color{lime}{1} &\color{darkslategray}{0}\\ \hline \text{segment 4} & \color{darkslategray}{0}& \color{darkslategray}{0} & \color{lime}{1} & \color{lime}{1} \\ \hline \end{array} \]

3.2 Specification of priors

The priors of the section controls are updated as follows, estimating that an unknown part of the weir is lower:

• Notch in the weir (very low flow control):

  • \(\kappa = 0~\mathrm{m} \pm 0.5~\mathrm{m}\) (deduced from the lowest waters of the stage series)
  • \(B_w = 6~\mathrm{m} \pm 2~\mathrm{m}\) (assumed as realistic)

• Rest of the weir (low flow control):

  • \(\kappa = 0.5~\mathrm{m} \pm 0.5~\mathrm{m}\) (assumed as realistic)
  • \(B_w = 8~\mathrm{m} \pm 3~\mathrm{m}\) (total width derived from distance measurements on the Géoportail aerial view, see Fig. 1)

The priors for the channel controls (main channel and floodplain) remain the same as previously defined.

3.3 Gaugings and posterior rating curves

The “normal” rating curve is estimated using the gaugings associated with the existing rating curve without the single gauging of the July 2021 flood measured on 07/16/2021 (Fig. 2). We do not see any conflict between the posterior results and the prior distributions, with a more precise calibration of the main channel parameters. The rating curve estimated with BaRatin is very close to the existing rating curve. Thanks to the addition of the notch in the controls matrix, the posterior rating curve now agrees reasonably well with the very low flows gaugings.

Figure 2. The Aisne River at Verrières (H6021020), “normal” rating curve: a) prior and posterior distributions of parameters, b) BaRatin rating curve with parametric (pink) and total (red) uncertainty envelopes, c) close-up view on lowest lows (discharge in log scale).

The “July 2021” rating curve is estimated using the same gaugings except those with a stage greater than 2.4 m (before overbank flow) and this time with the gauging of the July 2021 flood (Fig. 3). The previous priors are kept except for the prior overbank flow stage set as previously (\(\kappa_3 = 2.4~\mathrm{m} \pm 0.12~\mathrm{m}\)). Indeed, we assume that the overbank flow stage of the floodplain does not change depending on the season, and we transfer this information that the only gauging after overbank flow in July 2021 cannot provide.

Figure 3. The Aisne River at Verrières (H6021020), “July 2021” rating curve: a) prior and posterior distributions of parameters, b) BaRatin rating curve with parametric (pink) and total (red) uncertainty envelopes.

We do not see any conflict between the posterior results and the prior distributions, but the uncertainty of the rating curve is greater than previously, especially for flood discharges. The rating curve estimated with BaRatin deviates from the existing rating curve, which falls on the upper edge of the uncertainty envelope (not shown).

3.4 Stage series and discharge series

As an example of handling various data, two stage series extracted from the Hydroportail (instantaneous stages in mm) are offered in several versions:

• Stages recorded at variable time intervals from 06/01/2022 to 07/31/2022, in three versions:

  • limni with uncertainty: with uncertainties defined equal to 1 cm (non-systematic errors) and 1 cm (systematic errors, 1 single period)
  • error free limni: without uncertainties
  • limni with missing values: same version as the first one (with uncertainties) but with gaps (missing data)

• Stages recorded at variable time intervals from 04/01/2022 to 05/31/2022, in two versions:

  • April-May 2022: with defined uncertainties equal to 2 cm (non-systematic errors only)
  • april-may 2022 error free: without uncertainties

The “normal” rating curve is applied to these five stage series to generate the corresponding discharge series with uncertainties (example Fig. 4).

Figure 4. The Aisne River at Verrières (H6021020): discharge series estimated by BaRatin for the stage series limni with uncertainty and the “normal” rating curve, with envelope of stage (yellow), parametric (pink) and total (red) uncertainty at the 95% probability level.