Uncertainty estimates prior to the incorporation of local data

The model uncertainty for Mean Flow estimation is expressed as an FSE and is 1.15.

The 68% confidence interval for an estimate of mean flow ( $MF$ ) is therefore $MF / FSE$ to $MF \times FSE$ . Note that the interval is biased and the upper half of the interval is larger than the lower and thus the use of FSE in formulating confidence intervals is commonly approximated (in longhand) to $MF ( 1 \pm (FSE - 1))$ .

FSE are available for key flow percentiles. The raw estimates of FSE are for the normalised FDC. The model (adjusted for gauging station uncertainty) and unadjusted FSE for the estimation of the normalised FDCs are presented in Table 1 together with the assumptions used for gauged station uncertainty.

Table 1 - FSE for the estimation of key FDC statistics expressed as a %MF

FSE	Q95	Q70	Q50	Q10
Unadjusted	1.37	1.32	1.19	1.09
Model	1.31	1.32	1.18	1.07

In application, the final estimate of error for the FDC when multiplied by the local estimate of mean flow can also be approximated through pooling of error variances. These pooled estimates of FSE are presented within Table 2.

Table 2 - FSE for the estimation of key FDC statistics expressed in m³/s

FSE	Q95	Q70	Q50	Q10
Gauged	1.20	1.05	1.05	1.05
Unadjusted	1.40	1.36	1.25	1.18
Model	1.35	1.35	1.24	1.17

We do not have the equivalent uncertainty estimates for monthly or seasonal flow statistics.

Uncertainty estimates following the incorporation of local data

The FSE for the $ROMF_{ug}$ and $Q(x)_{ug}$ model error in m³/s are presented above together with the FSE used for the gauged data to estimate model error.

Upstream

For the upstream case, where the estimates are a simple addition, the error in the estimate is given by:

$\Delta Q = \sqrt{( Q_{inc} \times FSE_{Q_{ug}})^2 + \sum (Q(i)_{us} \times FSE_{Q_{g}})^2}$

Where the suffix $ug$ refers to the ungauged model error and $g$ refers to the gauging station error used to estimate the ungauged model error.

Downstream

For the downstream case estimate is the sum of the ungauged estimate and a correction term based on the gauged estimate, the ungauged estimate and the inferred incremental catchment estimates. This can be set up algebraically but it is overly complex and thus the downs stream case is approximated as

$\Delta Q = Q \sqrt{ \Big( (1 - CWF) (FSE_{Q_{ug}} - 1)\Big) ^2 + \Big( CWF (FSE_{Q_{g}} - 1)\Big) ^2}$

Both Upstream and Downstream

And for the case of both upstream and downstream gauges:

$Q_{final} = \sqrt{ \bigg({ \bigg({ 1 - \frac{ ROMF_{ug} - \sum_{i=1}^n ROMF(i)_{us} }{ ROMF_{ds} - \sum_{i=1}^n ROMF(i)_{us} } }\bigg) \Delta Q_{us} }\bigg)^2 + \bigg({ \bigg({ 1 - \frac{ ROMF_{ds} - ROMF_{ug} }{ ROMF_{ds} - \sum_{i=1}^n ROMF(i)_{us} } }\bigg) \Delta Q_{ds} }\bigg)^2 }$