Probable maximum flood: the potential for estimation in the UK using ReFH2

Symbol

Meaning                        Units

BFIHOST19

BFI (baseflow index) estimated using HOST (Hydrology of Soil Types) classification

BL

Baseflow recession constant (or lag)           hours

BR

Baseflow recharge

C_ini

Initial soil moisture depth                mm

C_max

Maximum soil moisture depth              mm

CWI

Catchment wetness index                mm

DPLBAR

Mean drainage path length                km

DPR_CWI

Dynamic percentage runoff dependent on CWI      %

DPR_RAIN

Dynamic percentage runoff dependent on P        %

DPSBAR

Mean drainage path slope                km

EM-2h

Estimated maximum 2-h rainfall             mm

EM-24h

Estimated maximum 24-h rainfall            mm

P

Total design storm depth                mm

PMF

Peak flow of a PMF event                m³/s

PMP

Total depth of a design PMP storm           mm

PR

Percentage runoff                   %

PROPWET

Index of proportion of time that soils are wet

SAAR

Standard Annual Average Rainfall            mm

SPR

Standard percentage runoff         %

SPRHOST

SPR estimated using HOST (Hydrology of Soil Types) classification

Tp

Unit hydrograph time to peak       hours

URBEXT

FEH index of fraction urban extent

Reservoir safety in the UK is regulated through the Reservoirs Act 1975 (RA75). The safety regulations require the estimation of the probable maximum flood (PMF) for reservoirs which fall within category A, where failure of a reservoir can result in loss of life. The ICE (2015) states that the PMF represents ‘the flood hydrograph resulting from PMP [probable maximum precipitation] and, where applicable, snowmelt, coupled with the worst flood-producing catchment conditions that can be realistically expected in the prevailing meteorological conditions’. Current guidelines for estimating the PMF are summarised by Pether & Fraser (2019) and detailed within the fourth edition of the Floods and Reservoir Safety publication (ICE 2015). These guidelines stipulate that the PMF is estimated using the method outlined in Flood Estimation Handbook (FEH) volume 4 (Houghton-Carr 1999); a restatement of the original method described in the Flood Studies Report (FSR) (NERC 1975). While the original FSR method has been replaced by the revitalised flood hydrograph (ReFH) method for design flood estimation (Kjeldsen et al. 2005; WHS 2019), the estimation of PMF still relies on the original FSR method.

Depending on the category of dam, flood hydrographs (and peak flows) are required for the 150-, 1,000- and 10,000-year events as well as the PMF. For each dam category, a different combination of design rainfall and rainfall-runoff models may be recommended. A subset of these is presented in Table 1.

While the ReFH2 model is not cited within Pether & Fraser (2019) for use in 10,000-year return period events, simulation of design events up to a return period of 10,000 years was tested and enabled within the ReFH2.3 software released in November 2019 (WHS 2022). Thus, the PMF event is the only return period where the FSR/FEH rainfall-runoff model is still required to be used. Many of the issues relating to the current estimation of PMF within the UK are summarised in Faulkner & Benn (2019) and included in a recent review of current methods by the Environment Agency (EA 2023). Many of the areas highlighted for improvement require substantial investment and further research. The aim of this study is not to resolve the larger issues but to investigate whether it is feasible to use a consistent rainfall-runoff model (ReFH2) for all return periods, up to and including the PMF event. Notably, Pucknell et al. (2020) present a framework for estimating PMF using the ReFH2 model, by translating the FSR/FEH procedure into an equivalent ReFH2 procedure. Here, we develop these methods further to show that PMF peak flows (and hydrographs) can be estimated using the PMP rainfall event, the ReFH2 rainfall-runoff model and the assumptions associated with the current PMF method. Updates can be incorporated within the framework without recourse to older methods.

The FSR/FEH and ReFH2 models are conceptual unit hydrograph rainfall-runoff models and are described in subsequent sections. Both can be utilised in ungauged catchments as parameters can be estimated from catchment descriptors. This is a requirement of the method as many reservoired catchments (or those where reservoirs may be planned) are ungauged.

The estimation of the PMP event is independent of that for design rainfall events of lower return periods. Details are provided by Houghton-Carr (1999) and only a summary provided here. The baseline data for the method uses the FSR estimated maximum (EM) rainfall depths for the 2-h and 24-h events (EM-2h and EM-24h) which are interpolated or extrapolated for different duration events. A ‘nested’ approach is used in which, for each subsequent larger duration, the shorter duration event PMPs are retained. Areal reduction factors and seasonal correction factors are also applied. For the winter event, the 100-year snowmelt event may be added to both the PMP and antecedent conditions. In the past, there has been confusion on how to apply snowmelt and a generic 42 mm/day has often been used. Recent guidance (DEFRA 2022) has clarified that the Hough & Hollis (1997) method, based on observed snowmelt records, should be applied.

The PMP event is used as input data to the FSR/FEH rainfall-runoff model. This is an update of the FSR rainfall-runoff model, utilising catchment descriptors released in the FEH, Volume 5 (Bayliss 1999). The model consists of three main components: a loss model, a routing model and baseflow component model.

Routing is based on a unit hydrograph, with time-to-peak Tp, which can be estimated from catchment characteristics (DPSBAR, PROPWET, DPLBAR and URBEXT).

Baseflow is constant and can be estimated using the CWI and catchment descriptors (AREA and SAAR; the Standard-period i.e. 1961–1990, Average Annual Average Rainfall).

To reflect the ‘ultra conservative assumptions’ (NERC 1975) required for PMF estimation, adjustments are made to the rainfall and rainfall-runoff model. These adjustments are summarised in Table 2.

As summarised by the Environment Agency (2023), many of these adjustments are somewhat arbitrary and have not been updated since the FSR (NERC 1975).

The revitalised flood hydrograph rainfall-runoff model (ReFH) was first developed by Kjeldsen et al. (2005). The ReFH conceptual model has a number of improvements over the existing FSR/FEH rainfall-runoff model, summarised in Table 3. In addition, the development used more calibration data and higher-resolution soils data.

The ReFH loss model has one static parameter, ⁠, which represents the maximum soil moisture depth and an initial soil moisture depth (⁠⁠), which can vary between (observed) events.

The first term on the right-hand side relates to the antecedent conditions, while the second part represents the dynamic rainfall effects. This form is similar to the FSR/FEH loss model, presented in Equation (1). Unlike the FSR/FEH loss model, the losses in the ReFH model are calculated for each time step of the simulation to account for the wetting-up of the soil during the flood event.

Subsequently, there have been a number of additional updates including the incorporation of the FEH13 rainfall model (Stewart et al. 2013), improved parameterisation (as well as a bespoke calibration for Scotland) and, more recently within ReFH2.3, inclusion of water balance features. The latest release also increased the maximum return period, such that the 1 in 10,000-year event can now be estimated.

The ReFH2 model is recommended for use, and widely utilised, within flood risk assessments where return periods up to 1,000 years are required. It is widely accepted that the form of the ReFH rainfall-runoff model offers considerable improvements over the FSR/FEH rainfall-runoff model and the ReFH2 rainfall-runoff model is recommended for use within reservoir studies for lower return period estimates. Use of the ReFH2 model for PMF estimation would therefore offer improvement relating to the structure of the model, as well as allowing consistency across all return periods. While by no means the largest issue relating to PMF estimation, consistency will better enable users to make informed decisions relating to differences between lower and higher return period peak flows without the complicating factor that these have been estimated using different rainfall-runoff models.

The resulting PMF peak flows were comparable with those estimated using the FSR/FEH rainfall-runoff method.

The main aim of this study is to develop a framework by which ReFH2 can be used to implement the current PMF methods based on a translation of the assumptions listed in Table 2 from the FSR to the ReFH modelling method. The framework should be sufficiently flexible to ensure that, as further research is completed and any assumptions or datasets are updated, they can be readily translated into operational practice.

Pucknell et al. (2020) illustrated that it was possible to estimate the PMF using the ReFH2 rainfall-runoff model. However, there were a number of limitations to this study, including the small study size (14 catchments), the use of the ‘recommended duration’ only, and the use of the 42 mm/day snowmelt assumption. This study builds on this work by firstly increasing the sample size. Secondly, the ‘recommended duration’ is the duration which, in the absence of any storage, is estimated to produce the highest peak flows. However, other durations may be necessary as part of reservoir design; ICE (2015) states that PMF estimation with a number of different durations may be required, in the event that the ‘recommended’ duration is not the ‘critical’ duration. This study, therefore, aims to develop a method in which any duration can be used. Finally, this study retains the 42 mm/day snowmelt assumption, allowing results from this study to be compared with those reported by Pucknell et al. (2020).

The dataset was maximised to capture a good spatial distribution and cross-section of catchment types (although Northern Ireland was excluded due to a lack of digital EM data). The existence of good quality gauged data at these sites also means that the resulting PMF values can be compared with observed AMAX values.

Different methods have been adopted for incorporating effects of urbanisation on storm runoff within the FSR/FEH and ReFH2 rainfall-runoff models. As the aim is to understand the difference between how the two models estimate the PMF, and given that the incorporation of urban impacts may complicate our understanding of this, the rural estimates of PMF are used.

The EM-2h and EM-24h were obtained from the UKCEH FSR database at the centroids of each catchment; a justified assumption given the comparative aim of the study.

The 100-year snow depth, which limits the snowmelt that may occur, was obtained from a digitised version of Figure 4.7 in the FEH Volume 4 (Houghton-Carr 1999). The mid value of each snow depth contour boundary at the centroid of each catchment was used. Given the resolution of the map and aims of the study, this assumption is justified.

As far as the authors are aware, this dataset represents the largest catchment set for which the FSR/FEH rainfall-runoff PMF has been estimated in the UK.

Three main methods, with a fourth for comparison purposes only, were trialled, and the results compared to ascertain the credibility of the proposed ReFH2-PMF modelling framework:

• 1.

  Replication of the Pucknell et al. (2020) method for a large number of stations. Referred to as the ‘Delta PR Rec Duration’ method.

• 2.

  Extension of the Pucknell et al. (2020) method to include greater flexibility in duration selection. Referred to as the ‘Delta PR’ method.

• 3.

  Development of flexible method with no link to the FSR method. Referred to as the ‘Direct Antecedent’ method.

• 4.

  The C_{ini_PMF} for ReFH2 was increased using the direct PR increase from the FSR/FEH rainfall-runoff model. Referred to as ‘FSR/FEH Percent Diff’, this is for comparison purposes only. Methods 1 and 2 are effectively ‘fitting’ to this dataset.

The results are presented for the recommended duration at each catchment. The recommended duration is based on the Tp and SAAR, hence these are different for the FSR/FEH and ReFH2 rainfall-runoff models. Where the change in PR from the FSR was required (‘Delta PR Rec Duration’, ‘Delta PR’ and ‘FSR/FEH Percent Diff’ methods), this was calculated using the FSR recommended duration. Application within ReFH2 used the ReFH2 recommended duration.

• 1.

  Delta PR Rec Duration

The absolute percentage difference in the PR for the FSR/FEH rainfall-runoff model between the standard design PR and PMF PR was calculated for all stations. The revised C_ini required to produce this percentage difference was then calculated, and the relationship between the C_{ini_PMF}/C_ini and C_max was determined. This was used to derive new coefficients for Equation (5), following Pucknell et al. (2020). The two models start to deviate in more permeable catchments (as C_max increases), with the larger dataset model producing higher C_{ini_PMF}/C_ini ratios in these types of catchments. Application of the two models might therefore result in significant differences to the C_{ini_PMF}/C_ini ratio, thus peak flows, in highly permeable catchments.

The differences highlight the importance of testing methods within large representative datasets. While reservoirs in the past have been predominantly within small upland catchments, this may change in the future if more lower-altitude flood storage schemes are developed.

• 2.

  Delta PR

Within the FSR/FEH application, the EMa represents the depth of rainfall that falls prior to the PMP event, over a period two times the duration of the PMP event. Application of Equation (3) then uses this to estimate the PMF CWI. This process is replicated within ReFH2 by modelling the EMa as a constant-intensity event of 2 times the PMP event duration, with the initial C_ini for this ‘event’ calculated from catchment descriptors. Within ReFH2.3, the ‘drainage’ feature then reduces the total impact that this has on the soil moisture. The soil moisture depth at the end of the EMa event is then used as C_ini for the PMP rainfall event.

For each of the three methods, the ReFH2 rainfall-runoff model was applied in combination with the summer PMP event using the ReFH2 recommended duration, the PMP, the reduced Tp and the relevant C_{ini_PMF}. For the ‘Delta PR Rec Duration’ and ‘Direct Antecedent’ methods, the winter PMP event was also run which included the additional snowmelt and rainmelt added to the PMP and antecedent conditions, and a minimum 53% (to represent frozen ground) PR for every timestep.

Figure 2 shows that the PMF peak flows are of a similar order for all models. The Bias (%, based on ln peak flows), which represents the difference between the models not performance, ranges from 7.59 to 12.7, with the ‘Direct Antecedent’ method having the lowest Bias.

Figure 3 illustrates that, in general, higher peak flows occur in higher SAAR and lower BFIHOST19 catchments. This is confirmed within the Bias which ranges from 24.3 to 27.8 where SAAR is greater than 1,000 mm and from 16.1 to 19.6 where BFIHOST19 is less than 0.65.

There is a greater range of Bias in dry and permeable catchments between the methods with the ‘Direct Antecedent’ method consistently producing, in general, the lowest peak flows. Where SAAR is less than 1,000 mm, the Bias ranges from −9.75 to 0.04 and where BFIHOST19 is greater than 0.65, the Bias is −40.5 for the ‘Direct Antecedent’ method and ranges from −20.6 to −29.6 for the other methods. It is useful to note that over 90% of the permeable catchments (BFIHOST19 > 0.65) have a SAAR less than 1,000 mm.

Figure 4 illustrates that, while there is a large increase in the C_{ini_PMF} at low C_ini values for the ‘FSR/FEH Percent Diff’ method, this is not found for the ‘Direct Antecedent’ method. This large difference occurs in catchments where SAAR is very low and is attributed to the ‘disconnect’ between the FSR/FEH rainfall-runoff model standard and PMF CWI (which then impacts on the PR). For lower return periods, CWI decreases sharply for catchments with SAAR less than 934 mm; above this, the gradient of change is far lower. For the PMF method, the CWI is related to the size of the antecedent PMP event. This can result in large increases in PR for low-SAAR catchments (which in this dataset includes most of the permeable catchments) for the FSR/FEH rainfall-runoff model, which is replicated within the ‘Delta PR Rec Duration’ and ‘Delta PR’ methods.

This illustrates a weakness of the first two methods, where the implementation within the ReFH2 rainfall-runoff method is based on the impacts as modelled within the FSR/FEH rainfall-runoff model. The ‘Direct Antecedent’ method does not use these assumptions, hence that method is the most consistent application of the PMF method within the ReFH2 rainfall-runoff model.

For summer events, the differences between the rainfall-runoff models are generally attributed to the differences between the methods for deriving PR. The differences between the ‘Direct Antecedent’ method and the other methods are driven by the differences in the initial C_ini values, particularly within low-SAAR catchments. As the permeable catchments are dominated by low-SAAR catchments, these differences are marked within this catchment type.

Figure 5 shows a greater agreement between the FSR/FEH and ReFH2 rainfall-runoff model peak flow estimates for winter events than summer events. This is borne out by the statistics where the overall Bias values are 6.2 and −5.16 for the ‘Delta PR Rec Duration’ and ‘Direct Antecedent’ methods, respectively, and the FSE (Factorial Standard Error) is 1.19 and 1.2, respectively; note that the FSE values for the summer events were higher at 1.28 and 1.36, respectively. The similarity between the two models is attributed to the frozen ground component, whereby the minimum PR is set to 53%, producing high percentage runoffs for all catchments.

In general, users apply both the summer and winter events to see which is the critical season for a particular reservoir; it is possible that one may be critical for peak flow and the other for volume.

Within the study dataset, for the FSR/FEH rainfall-runoff model, the winter event peak flows are greater than the summer event within 55% of catchments. For the ReFH2 rainfall-runoff model, the summer event peak flow exceeds the winter event within 71% of catchments. For both the FSR/FEH and ReFH2 rainfall-runoff models, the PMP volume is greater for summer, whereas the PRs are lower for summer events. Whether the summer or winter peak flows are higher is therefore attributed to a balance between the peakier, higher rainfall and the lower PR for the summer event and the less peaky, lower rainfall, but higher PR for winter events. This balance is different between the FSR/FEH rainfall-runoff model and the ReFH2 rainfall-runoff model. This study was completed using a constant snowmelt rate of 42 mm/day, and it is possible that the summer/winter balance would change if the Hough & Hollis (1997) snowmelt methods were used.

A number of studies have sought to determine whether PMFs have been exceeded in the past (Acreman 1989; EA 2023). Potential exceedances have generally been found to occur at ungauged sites, where peak flow has been modelled post-event. However, as this study has produced PMF estimates which represent a large dataset for the UK, it was thought to be advantageous to compare these with the observed AMAX values. Within this dataset, there are no AMAX that are higher than either the FSR/FEH urban winter or summer PMF. This does not necessarily mean that no events have exceeded the PMFs at these stations but that no quality-controlled AMAX values within the NRFA Peak Flow dataset have exceeded PMF at present. The winter PMF results may also differ if the H&H snowmelt method is used in the future. A similar assessment for the ReFH2 rainfall-runoff model rural PMF estimates (which may be an underestimation of the PMF) shows similar results, although the variability of the PMF for summer events is greater.

The 10,000-year return period peak flow from ReFH2 (rural) was estimated for each of these catchments. For the FSR/FEH rainfall-runoff model, the median ratios of the PMF to the 10,000 year peak flow are 2.5 and 2.1 for winter and summer, respectively. These ratios are related to both SAAR (lower ratios for higher rainfall) and BFIHOST19 (higher ratios for more permeable catchments). The median ratios for the ReFH2 rainfall-runoff model are 2.4 and 2.5 for winter and summer, respectively, with a similar relationship to SAAR and BFIHOST19.

This study has illustrated that the ReFH2 model can be used to estimate the PMF. The ‘Delta PR Rec Duration’ and ‘Delta PR’ methods utilise the outputs of the FSR/FEH rainfall-runoff method for determining how the PR changes under PMF conditions. This can result in very large PR increases in low-SAAR conditions. This is avoided with the ‘Direct Antecedent’ method, resulting in lower initial conditions (hence lower resulting PR) within these catchments. The ‘Direct Antecedent’ method does not rely on the outputs of the FSR/FEH rainfall-runoff model, which means that any future improvement to the data/assumptions can be directly applied within ReFH2, without recourse to the FSR/FEH rainfall-runoff model.

We have presented a methodology for implementing PMF events within the structure of the ReFH2 rainfall-runoff method which:

• 1.

  Is consistent with the current PMF assumptions implemented within the FSR/FEH rainfall-runoff model.

• 2.

  Does not require recourse back to the FSR/FEH rainfall-runoff model and the way in which this responds to the PMF event.

• 3.

  Is consistent with the rainfall-runoff model used within current design methods in the UK.

In addition, this study has illustrated the importance of testing methods with large datasets representative of the variability of catchment type/climate across the UK.

The dataset produced has been compared with gauged data from the NRFA Peak Flow dataset and has shown that PMFs have not been exceeded at present within this dataset. The median ratios between the FSR/FEH or ReFH2-PMF peak flow estimates and the ReFH2 rural 10,000-year peak flow estimates are between 2.1 and 2.5.

The dataset and methods offer opportunities for further analysis of catchments where current PMF estimates are close to the maximum AMAX or the 10,000 year peak flow estimates. The sensitivities of PMF peak flows to the assumptions within the PMF method (particularly snowmelt) could also be investigated further.

This study has illustrated that the ReFH2 rainfall-runoff model can be used for PMF estimation and the framework is such that, as aspects of the PMF modelling are improved (for example, the PMP or our understanding of how assumptions might be applied) that these can be easily incorporated.

Many thanks to those at UKCEH, particularly the FEH team and the NRFA, for providing datasets and advice on implementation to allow this work to be possible. Many thanks also to the anonymous reviewer who provided invaluable feedback to improve the clarity of the manuscript.

This work was funded by WHS and UKCEH, through reinvestment of FEH Software and FEH Web Service income, respectively.

Data cannot be made publicly available; readers should contact the corresponding author for details.

Tracey Haxton is an employee of WHS who develop and distribute the ReFH2 software.