# Design Estimates Estimation of Initial Conditions in Ungauged Catchments

The estimation of the initial as rural soil moisture content C_{ini} (mm) and initial baseflow BF_0 (m³/s) is necessary at the start of the design complete the design package.

## Estimation of Cini

The event percentage runoff for the rural catchment (the ratio of direct runoff depth to rainfall depth) is given by: $$ PR_{rural} = {C_{ini}\over C_{max}}+{P\over C_{max}} $$

The estimate of peak flow estimated by ReFH2 is sensitive to the percentage runoff and hence the value of the ratio of C_{ini} to C_{max}. The correct estimation of C_{ini} is therefore a core part of the design package for ensuring the frequency of the rainfall event and frequency of corresponding flow event are the same.

The design packages for each of the rainfall models (the water balance and non-water balance FEH13 design packages and the legacy FEH99) have separate initial condition models for C_{ini}.

### ReFH2.3 design package

The ReFH2.3-FEH22 and ReFH2.3-FEH13 design packages have similar C_{ini} models. These are predicated on ensuring that ReFH2 yields an estimate of the QMED flow event corresponding to the RMED recommended duration design rainfall hyetograph. These C_{ini} relationships were optimised for NRFA Peak Flow catchments, where the influence of urbanisation is not significant and the catchments are classified as suitable for QMED estimation.

The RMED storm can be reliably estimated directly from the observed rain gauge records and QMED event can be reliably estimated from the annual maxima with the observed flood series for the catchments considered. Thus these C_{ini} models are optimised on observed data and independent of the FEH statistical method. In contrast the legacy ReFH2.2-FEH99 package, as will be discussed, is not independent of the FEH statistical method.

The primary C_{ini} model is one predicated on the winter storm, C_{iniW}, as the annual maxima flood series within rural catchments tend to be dominated by winter events in the UK.

The estimation of C_{iniW} model is a function of BFIHOST19. Further details are provided in ReFH2 Science Report: Closing a Water Balance and Evaluation of the ReFH2.3-FEH13 Design Event Model^{R2}.

The ReFH2.3 design package also has summer C_{ini} models to enable ReFH2 to be applied with summer storms under the urbanisation rules discussed under Application of ReFH2 within urbanised catchments. As for the winter storm, the C_{ini,summer} are predicated on the relationship between the values of C_{ini} required to reconcile RMED using a summer storm profile and QMED. In application C_{ini,summer} is estimated as a function of the C_{ini, winter}.

The equation relating the summer C_{ini} to the winter C_{ini} is given by:

By inspection it can be seen that the summer value is lower than the winter value by a greater amount in permeable, climatically dry catchments, as would be expected. The summer C_{ini} model was originally developed for the ReFH2.2-FEH13 (non-water balance) design package under the Environment Agency funded Estimating flood peaks and hydrographs for small catchments: Phase 2 (see Report 6). The parameters were updated for the ReFH2.3-FEH13 and ReFH2.3-FEH22 models.

### Legacy ReFH2.2-FEH99 design package

The design package was the first design package developed for ReFH2 prior to the release of the FEH13 and subsequent FEH22 DDF models. It is retained as a legacy option within the software but is strongly recommended that the later DDF models and associated design packages are used in preference. There is only a winter C_{ini} model available within the FEH99 design package.

The estimation of C_{ini,W} for the FEH99 winter storm followed the same basic approach as used for the original ReFH1 research (Kjeldsen, 2006). That is the design C_{ini,W} was optimised for the 1:5 year event to reconcile the 1:5 year storm with the 1:5 year estimated using the FEH statistical method. The model was developed for the 101 catchments used in the original ReFH research. Whereas the original research used the original FEH statistical method, as published in 2009 the REFH2 FEH research used the Enhanced Single Site estimates using the 2008 revised statistical estimation procedures, as implemented within WINFAP.

The best ReFH2 FEH99 C_{ini,W} model fits were obtained by partitioning the data set into permeable catchments and impermeable catchments giving two C_{ini,W} equations for application. Noting that it was not necessary to partition catchments in this way for the later FEH13 design package and noting the differences between the two DDF models over the chalk outcrops of southern England it can be concluded that this pragmatic decision is a function of the choice of DDF model.

There is a larger concern regarding the estimation of C_{ini,W} for the FEH99 design package. As in the original ReFH research, another variable, Alpha (α), is required to ensure correspondence between the rainfall frequency and corresponding flood frequency for more extreme events than the 1:5 year event. This parameter effectively reduces the C_{ini,W} parameter for more extreme events, that is antecedent soil conditions are assumed to be drier for more extreme events.

The outcome that C_{ini} decreases with increasing event extremity is hydrologically counter-intuitive. Furthermore, α for more extreme events is identified through calibration against the FEH statistical estimates for these more extreme events. Thus, in application, the ReFH and statistical method estimates of peak flow are not independent of one another. This is a cause for concern when seeking the best estimate in ungauged catchments.

For the ReFH2 FEH99 design package the estimation of α for use in conjunction with the 1:5 year C_{ini,W} model was updated to include a dependency on average annual rainfall in addition to return period, addressing concerns identified by Faulkner and Barber (2009). It was also optimised against the Enhanced Single Site estimates of peak flow under the current generation of the FEH Statistical Methods for AEPs out to the 1:1000 year return period event. Further details of the FEH99 design equation are provided in ReFH2 Science Report: The ReFH2-FEH99 initial conditions and the alpha parameter^{R5}.

## Estimation of BF0

There are two BF_0 equations used within ReFH corresponding to the winter and summer storm cases. These are the equations originally published by Kjeldsen (2005) and have been adopted for use within ReFH2 for England, Wales and Northern Ireland. Revised BF0 equations were developed for Scotland using the same approach but focused on the catchments used for calibration of ReFH2 within Scotland. The equations are presented in ReFH2 Science Report: Model Parameters and Initial Conditions for Ungauged Catchments^{R1}.