The ReFH Rural Model

A schematic of the rural model is presented in Figure 1. The ReFH rural model has three components: a loss model, a routing model and a base flow model. The loss model uses a soil moisture accounting approach to define the amount of rainfall occurring over the catchment that is converted to nett rainfall. The rainfall losses are derived as the event unfolds, rather than being defined by a fixed value of percentage runoff. Nett rainfall is routed to the catchment outlet. The routing component of ReFH uses the instantaneous unit hydrograph concept, adopting a kinked triangle as standard.

Finally, the base flow model is based on the linear reservoir concept with its characteristic recession defined by an exponential decay controlled by the recession constant termed base flow lag. Drainage to baseflow is estimated indirectly from direct runoff; the rationale for this is discussed in the context of design application of the ReFH2 model.

The rural model has four model parameters and two model initial conditions which are presented in Table 1.

Figure 1. Schematic representation of the ReFH model

Table 1. Summary of the ReFH model parameters

Name	Parameter or Initial Condition	Description
$Tp$	Model Parameter	Unit hydrograph time to peak (hours)
$BL$	Model Parameter	Baseflow recession constant or lag (hours)
$BR$	Internal State Variable (or Model Parameter in the legacy ReFH2.2 model)	Baseflow recharge (the ratio of recharge to runoff)
$C$ $_m$ $_a$ $_x$	Model Parameter	Maximum soil moisture capacity (mm)
$C$ $_i$ $_n$ $_i$	Initial Condition	Initial moisture content (mm)
$BF_0$	Initial Condition	Initial baseflow (m³/s)

The ReFH2.3 model is designed to close the water balance over an event in impermeable catchments. Accordingly $BR$ is dynamically calculated as an internal state variable, which is set to ensure that the sum of the baseflow and direct runoff depths modelled for an event is equal to the rainfall depth in the event. This was not the case for the ReFH2.2 and earlier models, where $BR$ is a model parameter. This is discussed further in the section on water balance closure.

The development of the ReFH2 design package model parameter and initial conditions is described within the following supporting ReFH2 Science Reports:

The performance of the ReFH2.3 design packages across catchments in NRFA Peak Flow dataset (including comparison with the FEH statistical methods and earlier versions of ReFH2) is provided in the following ReFH2 Science Reports:

The performance of the legacy ReFH2.2-FEH13 model is provided in ReFH2 Science Report: Evaluation of the Rural Design Event Model $^{R3}$ .

Pucknell et al. (2020) presents a first attempt at formulating a framework for estimating the probable maximum flood (PMF) in UK catchments using ReFH2 model. The model provides comparable and credible results when compared with previously published PMF results for 15 reservoired catchments. This was expanded and further developed in Haxton et al. (2023), which presents a method that could be applied in a future software enhancement.

Closing the Water Balance

Why didn't legacy versions of ReFH close the water balance?

It was originally envisaged that applications of ReFH would commonly involve calibrating the model parameters against a relatively small number of observed events. It is notoriously difficult to reliably calibrate rainfall runoff models on small datasets due to the problems of equifinality. That is, different combinations of model parameters may yield the same quality of fit in calibration but may give very different model outcomes when applied to new events. This problem scales with the number of parameters that have to be calibrated. In the ReFH model structure, baseflow is estimated indirectly from soil moisture status. That is, baseflow is estimated as a function of direct runoff, which is in turn a function of soil moisture status. This counter-intuitive model structure is elegant in that during calibration it facilitates the direct estimation of the base flow parameters from stream flow recession analysis leaving only two model parameters, $C$ $_m$ $_a$ $_x$ and $Tp$ , to be calibrated against observed event hydrographs. This enables reliable calibration of the ReFH model structure against a relatively small number of events. With four parameters in free calibration a much larger number of events would be required to obtain a reliable calibration. This is discussed in more detail in sections 2 and 3.2 of the FEH Supplementary Report No1 (Kjeldsen, 2007).

In design application in ungauged catchments the model parameters are estimated independently from one another and the inherent relationship between the Baseflow Recharge ( $BR$ ) and the $C$ $_i$ $_n$ $_i$ / $C$ $_m$ $_a$ $_x$ ratio implicit within the calibration procedure is not maintained (where $C$ $_i$ $_n$ $_i$ is the initial soil moisture content).

This can result in a counter-intuitive water balance violation in impermeable catchments where the sum of the model generated baseflow and direct runoff depths can exceed the total event rainfall depth over events that are of the recommended duration or relatively close to the recommended duration.

If the model is applied over event durations (design or observed) significantly longer than the recommended duration, the model will over-estimate direct runoff as there is no drainage from the soil store. If $BR>1$ this can also lead to the case where the total runoff depth can exceed the rainfall depth.

For more permeable catchments neither of these cases apply as the catchment descriptor estimate of $BR$ is less than 1 (i.e. the ratio of baseflow depth to direct runoff depth is always less than 1).

Closing a Water Balance in the ReFH2.3 Model

The water balance issue has been addressed in the ReFH2.3 model for the design package and the observed event application by the following:

specifying $BR$ as a model state variable with the objective of closing a water balance over the recommended duration for impermeable catchments, and
dividing a model run into segments each with a maximum length equivalent to the recommended duration for the application and allowing drainage to occur at each segment boundary.

These are summarised below, and described in full in ReFH2 Science Report: Closing a Water Balance and Evaluation of the ReFH2.3-FEH13 Design Event Model $^{R2}$ .

Users can revert back to the legacy ReFH2.2 model, in which $BR$ is estimated as a function of catchment descriptors.

Setting BR as an internal state variable

$BR$ is dynamically calculated to close a water balance over an event in impermeable catchments. In permeable catchments, $BR$ is constrained to ensure the water balance cannot be violated through total runoff exceeding rainfall. Due to the presence of significant recharge to aquifers with long residence times in permeable catchments, a water balance over an event is not a hydrologically realistic objective.

ReFH2 will close a water balance if the catchment estimate of the Base Flow Index ( $BFI$ ) is less than 0.5. As catchments become more permeable, the value of $BR$ approaches the estimate of $BR$ from catchment descriptors, and above a $BFI$ of 0.65 the $BR$ value is based on the catchment descriptor-based estimate. This ensures a water balance is closed within impermeable catchments; and for permeable catchments, where the water balance includes recharge to aquifers, $BR$ is set to a value informed by the model performance in permeable catchments. This is discussed in more detail in the ReFH2 Science Report: Closing a Water Balance and Evaluation of the ReFH2.3-FEH13 Design Event Model $^{R2}$ .

Modelling long duration events

As with any model, ReFH has a calibrated model parameter space in application within a catchment. Significant observed events within a catchment are typically of a similar duration to the recommended duration and the model parameters are calibrated to these observed events. The catchment descriptor equations are based on the calibration of the ReFH model structure over a candidate set of catchments. The candidate parameter sets are entered into a multivariate regression to derive the model parameter catchment descriptor equations. These equations thus reflect the duration of the observed events within the candidate catchments used to define the parameter equations.

The issues with baseflow generation are discussed in the previous section, however running the model for events that are significantly longer than the recommended duration within a catchment inevitably means that the model is being applied outside of the calibrated model range. The application of particularly long observed rainfall sequences can also result in unrealistically high runoff fractions for the latter part of the event. The reason for this is that the water content of the loss model is not adjusted to reflect the baseflow generated within an event. This is an issue for design events of duration much longer than the recommended duration. It is also an issue for long observed events where the duration of the event may be many multiples of the recommended duration with multi-modal peaks in the hyetograph.

This is addressed by subdividing the rainfall hyetograph (whether observed or design) into n segments of maximum length equivalent to the recommended duration and the model being run for each segment.

The first segment is run using the initial $C$ $_i$ $_n$ $_i$ and $BF_0$ conditions (design, or specified through DAYMOD in the case of an observed event). At the end of each segment the baseflow depth generated within the segment is calculated. The soil moisture depth ( $SM$ ) of the loss model at the end of the segment is then depleted by the depth of baseflow and a revised $C(t)$ value calculated. This value of $C(t)$ forms the $C$ $_i$ $_n$ $_i$ value for the second segment, and so forth for the remaining segments. The segmenting of the model run and updating of the soil moisture store is presented in detail within ReFH2 Science Report: Closing a Water Balance and Evaluation of the ReFH2.3-FEH13 Design Event Model $^{R2}$ .

This updating of the soil moisture store at increments of the recommended duration ensures that the model cannot generate unfeasibly high runoff rates for event that very significantly exceed the recommended duration. The corrections for events that are less than or equal to twice the recommended duration are minimal as the baseflow depth generated during the first segment is generally relatively small and the model is generally operating within the calibrated model parameter space.