Introduction

Flood risk for an area may be informally defined as a combination of the annual inundation likelihood with the impact that this causes, resulting from the depth and velocity of flood flow and its interaction with buildings and other infrastructure. Computational models of flood inundation allow a detailed assessment of flood risk for an area, when used in conjunction with spatial analysis of infrastructure in a GIS and a statistical assessment of flood exceedance probabilities (return periods). Generally, flood models for an area are developed based on a past event for which observational data are available, allowing model validation. Once developed, it is possible to assess a range of “what-if” scenarios including, for example, for testing of mitigation scenarios or storm design hydrographs. This generalized modeling structure is illustrated in the diagram below:

Data flow in an idealized flood modelling system for flood hazard mapping using GIS

Over the last two decades, one of the primary drivers of research in the development of flood inundation models is the advent of fine-spatial resolution topographic and flood extent data acquired using airborne remote sensing – particularly those from airborne LiDAR (Light Detection and Ranging). These topographic data are much more detailed and accurate than other sources of data which may be available, such as stereo photogrammetry:

Comparison of elevation data for the River Avon, Hampshire, U.K. Left: data from airborne stereo photogrammetry at a spatial resolution of 10 m; Right: data from LiDAR, at a spatial resolution of 2 m.

LiDAR data have been shown to be ideal for the development and assessment of flood models, allowing far greater detail and accuracy than was possible previously, facilitating two-dimensional representations of flood inundation, and leading to a push for high-resolution simulations of greater complexity. This is a significant advance over one-dimensional models which are most suited to situations where detailed floodplain topography is not available. However, greater data complexity brings with it a high computational burden: in order to be able to fully utilize these data, reduced complexity hydraulic models have been developed.

Reduced complexity flood inundation models: LISFLOOD-FP

The LISFLOOD-FP model code, developed at the University of Bristol (Bates and De Roo, 2000), represents the “next generation” of inundation model and it has been adopted for research use worldwide and within industry (for example, the model was adapted and used for producing extreme flood outline maps on behalf of the UK government). As with other models, LISFLOOD-FP usually takes a hydrograph (stream flow over time) as input at the upstream end of a channel reach and routes this downstream. The hydrograph may be derived through measurement, or from the output of hydrological catchment models.

LISFLOOD-FP uses the 1D Saint-Venant equations of open channel flow within the channel (if present), and on the floodplain is a 2D raster model, which changes in cell depth (\Delta h^{i.j}) with each time-step (\Delta t) governed by the mass continuity equation:

 

(1)   \begin{equation*}  \frac{\Delta h^{i.j}}{\Delta t} = \frac{Q^{i-1,j}_x \; - Q^{i,j}_x \; + Q^{i,j-1}_y \; - Q^{i,j}_y}{\Delta x^2} \end{equation*}

lisflood_continuitytran

Flows in each direction (Q_x and Q_y) are obtained using the Manning equation:

 

(2)   \begin{equation*}  Q = \frac{h^{5/3}_{\mathrm{flow}}}{n} \; \(\frac{\Delta (h+z)}{\Delta x}\)^{1/2} \Delta x \end{equation*}

 

where h is depth, z is the bed surface elevation, n is the Manning friction coefficient and \Delta x is the cell size. h_{\mathrm{flow}} represents the depth through which water can flow, defined as the maximum free surface height, \mathrm{max} \{h^{i-1,j}+z^{i-1,j}, \; h^{i,j}+z^{i,j}\}, minus the maximum bed surface height, \mathrm{max} \{z^{i-1,j}, \; z^{i,j}\}, as illustrated below:

 

 

While numerically very simple, the original formula for Q used in LISFLOOD-FP (Eqn. 2), had issues with stability with flows tending to develop a “chequerboard” patterns unless very small time-steps were used, particularly for areas of high flow, when the Courant–Freidrichs–Lewy (CFL) condition was not likely to be satisfied (i.e. when flood wave speed would propagate flow across more than one cell per time step). Hunter et al. (2005) developed an adaptive time-step version of the code in order to overcome this limitation – however, at fine spatial resolutions, extremely small time-steps were needed, leading to computationally very demanding simulations. This issue was overcome by Bates et al. (2010), with development of a new formula which incorporates a simple inertia term:

 

(3)   \begin{equation*}  Q = \frac{q - g \; h_{\mathrm{flow}} \; \Delta t \; \frac{\Delta (h+z)}{\Delta x}}{(1 \; + g \; h_{\mathrm{flow}}\; \Delta t \; n^2 \; |q|\; /\; h^{10/3}_{\mathrm{flow}})} \; \Delta x \end{equation*}

 

Here, q is the calculation of Q for the previous iteration of the model and g represents acceleration due to gravity. By incorporating flow from the previous iteration, model stability issues are effectively dampened.  In order to estimate a suitable model time-step, Bates et al. (2010) define the model timestep as:

 

(4)   \begin{equation*}  \Delta t_{\mathrm{max}} = \alpha \frac{\Delta x}{\sqrt{gh}} \end{equation*}

 

where \sqrt{gh} is the celerity of a long wavelength, small amplitude wave and \alpha is a model stability parameter ranging between 0.2 and 0.7 (Bates et al. 2010). This final formulation of LISFLOOD-FP is both computationally efficient and numerically stable, making it possible to tackle increasingly challenging modelling problems in terms of the level of detail included, the simulation duration or the spatial extent.

Flood modelling case study: Upton-upon-Severn, U.K.

Perhaps one of the best research datasets for (non-urban)flood inundation modelling exists on the lower River Severn, United Kingdom, at Upton-upon-Severn. This site has a combination of high quality input data and excent data for model validation. This page provides an overview of a modelling study using LISFLOOD-FP based on the Autumn 2000 flood events – this study was published in the Journal of Hydrology – see Bates et al. (2006) for more details.

Upton-upon-Severn study site

The site is situated in Worcestershire, UK, and runs from Severn Stoke to Mythe Bridge on the River Severn, incorporating the rural village of Upton-upon-Severn (see here for a detailed map/ satellite imagery). This village is situated on a floodplain “island” and becomes surrounded by water during flood events, with lower parts of the village inundated.

The Environment Agency are now use of temporary flood barriers around Upton-upon-Severn and other locations during flood events – see images of these here. These have caused some controversy among residents since they were not deployed during the major floods of July 2007 due to logistical difficulties in reaching the site – however, during that flood event, the River Severn reached record levels and the barriers would not have been sufficient to protect properties in Upton from flooding (Environment Agency, 2008).

Model input data

Data used in a flood inundation model are: floodplain topography, river flow rates, channel geometry and estimates of channel and floodplain friction. A good representation of floodplain topography is crucial for the accurate prediction of inundation extent and floodplain flows, since it controls strongly where water is able to flow, and the speed at which it does so. High-quality topographic data are essential – LiDAR (Light Detection and Ranging) data are probably the best suited for this purpose since they have a high vertical accuracy (around 10 cm). These data are obtained using a laser scanning device on an aircraft which, coupled with a differential GPS, provides a series point measurements with a dense spatial sampling (roughly 1 point every square metre). In the UK, LiDAR data are flown by the Geomatics Group of the Environment Agency.

 

Digital Elevation Model (DEM) generated from LiDAR data for the Upton-upon-Severn site.

 

LiDAR point data must be interpolated onto a raster grid for use in a flood inundation model. A LiDAR DEM may have a spatial resolution of around 3 m, however, for this modelling study, the DEM was coarsened to a spatial resolution of 18 m to reduce computation time. In addition, the DEM was processed for vegetation effects (to avoid hedges and trees acting as barriers to flow), and to preserve river embankments (to maintain embankment heights in the resampling process).

River flow rates were obtained from the Saxon’s Lode gauging station near Upton-upon-Severn. During the simulation, these flow rates are put into the top end of the river channel, making this a dynamic simulation of an historic event. It is also possible to use hypothetical flow rates either in a steady-state simulation, or using output from a hydrological model – these allow scenario flood events to be assessed.

 

Flow rates obtained from Saxon’s Lode gauging station, indicating the timing of available flood imagery for model validation.

 

Other important data are those for the channel geometry. Since LISFLOOD-FP uses a one-dimensional channel representation, we only need generalised information rather than detailed cross-sections: a location vector including width, river bed elevation and an estimate of channel friction. In practice, channel friction (along with floodplain friction) is often used as a calibration parameter, whereby the model is run a few hundred times with different friction parameters in order to find the optimum (based on the accuracy of the model, assessed using validation data such as flood extent imagery).

Model outputs

The outputs from LISFLOOD-FP include depth on the floodplain throughout the simulation, which may be used to obtain statistics such as maximum depth, time of maximum depth, time of initial inundation etc. Below is an animation of flood depth throughout the flood event. Note that this was a double-peaked flood event, and this can be seen clearly in the animation.

 

 

Model validation

The Upton-upon-Severn site has a good multi-temporal validation dataset for the Autumn 2000 flood event, comprising four airborne SAR (ASAR) images. It is highly unusual to have  multi-temporal validation data such as these – more often, we are lucky to obtain one image for a flood event, and this may only be a coarse spatial-resolution satellite image. These images allow the spatial dynamics of the model to be assessed – usually we are limited to either spatial (comparison with one image), or dynamic (comparison with time-series stage data), but rarely do we have data for a concurrent assessment.

 

Airborne SAR imagery for the River Severn at Upton: (a) 8, (b) 14, (c) 15, and (d) 17 November 2000.

 

The standard way to compare model output with flood imagery is to convert each to binary (flood/ non-flood) imagery, then work out the percentage accuracy, excluding correctly predicted dry areas (which would positively skew results, particularly for large model domains). In this case, friction parameters were calibrated to the image at the flood peak, and the most accurate simulation is shown below.

 

Comparison of model output with inundation extent obtained from ASAR imagery: (a) 8, (b) 14, (c) 15, and (d) 17 November 2000. Light blue is correctly predicted flood inundation; Red is model over-prediction; Yellow is model under-prediction; Dark blue indicates areas which were predicted as flooded but where imagery was not obtained. Click image for a larger version.

 

Accuracy at the flood peak was very high (89%), and reduced as the flood receeded (79% on 14 November; 78% on 15; and 73% on 17). This indicates that the model did not capture the drainage dynamics correctly, most likely due to an insufficient representation of floodplain drainage networks, but possibly also due to the simplistic way in which friction was represented (one value for the whole floodplain).

References

Bates, P.D. and De Roo, A.P.J., 2000. A simple raster-based model for flood inundation simulation. Journal of hydrology, 236(1), pp.54-77.

Bates, P.D., Wilson, M.D., Horritt, M.S., Mason, D.C., Holden, N. and Currie, A., 2006. Reach scale floodplain inundation dynamics observed using airborne synthetic aperture radar imagery: data analysis and modelling.Journal of Hydrology, 328(1), pp.306-318.

Bates, P.D., Horritt, M.S. and Fewtrell, T.J., 2010. A simple inertial formulation of the shallow water equations for efficient two-dimensional flood inundation modelling. Journal of Hydrology, 387(1), pp.33-45.

Hunter, N.M., Horritt, M.S., Bates, P.D., Wilson, M.D. and Werner, M.G., 2005. An adaptive time step solution for raster-based storage cell modelling of floodplain inundation. Advances in Water Resources, 28(9), pp.975-991.