# Design Estimates Estimation of Initial Conditions in Ungauged Catchments

The estimation of the initial as rural soil moisture content C_{ini} (mm) and BF_0 (m³/s) at the start of the design complete the design package. 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 a core part of the design package for ensuring the frequency of the rainfall event and frequency of corresponding flow event are the same.

# Estimation of Cini

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}.

## FEH13 design package

The two FEH13 packages (the water balance and non water balance options) 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 Flows 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 FEH99 package, as will be discussed, is not independent of the FEH statistical method.

The primary FEH13 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 BFIHOST for both versions of the FEH13 design packages, but differ in model coefficients. This is primarily because the water balance version of the FEH13 design package ensures a water balance is closed within impermeable catchments and that the total event runoff depth cannot exceed the rainfall depth in all catchments. This has a moderate influence on the percentage runoff and thus C_{ini}. The development of the water balance option for ReFH2–FEH13 is presented within ReFH2 Science Report: Closing a Water Balance^{R2}. As discussed in this report a larger set of catchments was used within this research and a revised version of BFIHOST, BFIHOST19 , was also adopted for the research. The XML file exports from the FEH Web Service include both the revised BFIHOST19 and the original BFIHOST_{IH126} (Boorman et al, 1994) descriptors. ReFH2 uses BFIHOST19 for the water balance option for the FEH13 design package. The other ReFH2 design packages were developed using the original BFIHOST and thus use the BFIHOST_{IH126} descriptor.

The FEH13 design packages also have 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} estimate. The summer C_{ini} model was originally developed for the non-water balance FEH13 design package under the Environment Agency funded Estimating flood peaks and hydrographs for small catchments: Phase 2 (see Report 6). 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. Testing of the model formulation for the FEH13 water balance model using the same procedure identified that the same relationship holds for this model.

Within the FEH13 design packages the C_{ini} estimates defined for the 1:2 year event have proved to be appropriate for events out to the 1:1000 year event (see ReFH2 Science Report: Evaluation of the Rural Design Event Model^{R3} for further details) and furthermore recent research suggests that these estimates are also reasonable for estimating the 1:10,000 year event.

Intuitively it might be expected that generally more extreme events would correspond to wetter initial conditions. However, in reality, instances of large floods within the observed records correspond to a wide range of initial conditions, thus it is difficult to demonstrate this effect. Very extreme events is an area that warrants future research.

## FEH99 design package

The design package was the first design package developed for ReFH2 prior to the release of the FEH13 DDF model. It is retained as a legacy option within the software but is strongly recommended that the FEH13 DDF model 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 4.

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) were 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}.