Probabilistic ReFH modelling in Flood Modeller

Overview

One of the greatest sources of uncertainty in hydrodynamic model outputs is associated with estimating hydrological inputs in ungauged locations. With modern drives to apply a probabilistic approach to modelling results, as opposed to the more traditional deterministic approach, a tool has been developed in version 3.5 to quantify uncertainty in ReFH hydrological boundaries. This uses a stratified uncertainty analysis applied to ReFH model parameters to illustrate how uncertainty in inputs propagates through to uncertainty in model outputs, in this case hydrograph generation. In a staged analysis, a sub-set of ReFH hydrographs are then typically routed through a hydrodynamic model in order to quantify uncertainty in the model results.

A stratified, staged approach

The concept of the Latin-Hypercube (LH) simulation² is based on Monte Carlo procedures but uses a stratified sampling approach that allows efficient estimation of the output hydrographs. This is used here to estimate uncertainty in ReFH model generated hydrographs, in preference to a pure Monte Carlo approach. The latter is a robust technique but may require a large number of simulations and consequently large computational resources. Orthogonal Latin Hypercube (OLH) sampling further ensures an equally probable subspace, resulting in a reduced sample space and lessening the chance of correlation between parameters.

The application of probabilistic ReFH modelling in Flood Modeller adopts a staged approach whereby uncertainties in input hydrographs can be screened according to use-specific parameters (that is, peak flow, hydrograph volume and / or timing of the peak). A sub-set of generated hydrographs are typically selected (for example, at selected probabilities of exceedance) and propagated through hydrodynamic models capturing uncertainty in model outputs such as stage and flood extent.

Input uncertainties

The ReFH model parameter input uncertainties have been defined through reference of the ReFH Technical Report¹ which describes the uncertainty in predictor equations for time-to-peak (T_p), the maximum soil moisture capacity (C_max), baseflow lag (BL) and baseflow recharge (BR) through use of a log-normal distribution and factorial standard error (fse). As such, this definition of uncertainty is directly linked to use of FEH catchment descriptors for parameterising the ReFH model. Specifying input uncertainty in these model parameters where they have been defined through use of local data (either directly or through use of a donor adjustment) may require use of an alternative description of uncertainty (for example, normal or uniform distributions)[1].

Additionally, input uncertainty in the initial soil moisture storage (Cini), initial baseflow (BF0) and storm duration (D) associated with use of the ReFH model have been included in the schema, as these are generally recognised to be important parameters in terms of generating ReFH hydrographs.

The table below lists the input parameters and definition of uncertainty for each of the selected input parameters.

For parameters T_p, C_max, BL and BR, a log-normal distribution is assumed, with the mean obtained from the catchment descriptors, and standard deviations from the fse values published in the ReFH technical report¹.

The remaining parameters, C_ini, BF₀ and D are assumed to be described by a uniform distribution between a factor of half and two of their catchment descriptor value.

Sampling strategy

The Orthogonal Latin Hypercube (OLH) method is a sampling strategy used to construct an equally probable sample space from distributions of multiple parameters. To generate a set of n samples, the distribution of each selected model input parameter is subdivided into n strata with a probability of occurrence equal to 1/n. Random values of the parameters are generated such that for each of the input parameters, each interval is sampled only once.

Here, each distribution is sampled individually, typically at 33 evenly spaced sample points (each bounding a probabilistic interval of 2.94%). This ensures a single distribution is produced which is representative of real variability.

Thus for the seven parameters chosen to represent uncertainty in the ReFH method, 33 combinations are produced using an OLH to represent the evenly distributed sample space. A set of 33 hydrographs can then be produced from these combinations, which can be ranked in order of peak flow, total volume or time to peak, hydrograph output parameters which describe the variation of inputs to the hydrodynamic model. Since all these hydrographs are equally probable, ranking these in order is used to derive a percentage exceedance. The user may then select the required input hydrographs, e.g. based on the 10%, 50% and 90% exceedance, from which to run the full hydrodynamic simulations.

Application within Flood Modeller

Running the ReFH uncertainty analysis is a two-stage process within the Flood Modeller interface. The first consists of the generation of a number of hydrographs (defaulting to 33) which can then be ranked in order of peak flow, time to peak or flow volume in order to determine exceedance.

From this, the modeller may then select a smaller subset, for example based on required exceedance percentiles, for which to run a full hydrodynamic simulation.

Uncertainty is therefore estimated by relating a percentage exceedance to model output.

References

1. Revitalisation of the FSR/FEH rainfall runoff method , Joint Defra/EA Flood and Coastal Erosion Risk Management R&D Programme R&D Technical Report FD1913/TR (2005)
2. McKay, M.D., Beckman, R.J. and Conover, W.J. (1979). A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code, Technometrics (American Statistical Association) 21 (2): pp239-245.

_[1]Future releases of Flood Modeller are likely to develop this functionality to better support application where local data has been used.