that the discharge Q is equalled or exceeded in any given year (see Chapters 11 and 13).

For a flood of annual probability p, assume that a corresponding value of flood damage D(p) can be estimated. In the definition of methods used by the USACE, this was based on the depth of inundation of the floodplain and on the value of the inundated structures. Regardless of exactly how the damages are calculated, the EAD is the notional long-run average value of such damages taken over floods of all different AEPs. Mathematically this is the integration of the economic damages D(p) over all exceedance probabilities,

The calculation of EAD can be conveniently split into parts. For the original USACE procedures, there were three parts, as follows:

(1) calculate flood frequencies, which describe the probability of floods equal to or greater than some discharge Q occurring within a given period of time;

(2) calculate stage discharge rating curves to 'convert' the discharge into water level;

(3) calculate relationships between economic damage and water level.

The calculations are a mix of stochastic (probability distributions for flood frequency) and deterministic (rating and damage curves) relationships. A Monte Carlo procedure can be used to generate a large number, N, of random pseudo 'observations' of flood flow, each one leading to a damage value. The EAD is then calculated from the simulated damage values by approximating (16.2) as where p is randomly sampled from a uniform distribution between 0 and 1.

Fig. 16.3 illustrates the procedure, tracing the calculation of damage for one sample value (dashed line).

The risk calculation described above had only one probabilistic component, the flood frequency distribution representing the 'source' of the risk. However, it is generally unrealistic to regard all other links in the chain as being completely deterministic, if only because of the uncertainties in the specified relationships. Instead, there may be uncertainties about each component of the analysis, as illustrated in Fig. 16.4.

The uncertainties can be incorporated in the risk analysis by generating Monte Carlo samples from each of conditional distributions The outcome is a value for EAD that inherently includes the specified uncertainties in the component relationships. This capability to build the uncertainty into the overall measure of risk is an advantage of the risk-based methods, although it also means that there is a danger of regarding the whole, complex, calculation as something of a closed box.

A similar probability integration can be structured from the outset using the 'S-P-R' concept, as illustrated in Fig. 16.5.

The method contains the same essential elements as the USACE approach. The probability integration for EAD is now

probability (p)

Fig. 16.3 Simplified basis for calculation of expected annual damage (EAD). (Adapted from National Research Council, 2000.)

probability (p)

Fig. 16.3 Simplified basis for calculation of expected annual damage (EAD). (Adapted from National Research Council, 2000.)

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Exceedance probability, p Exceedance probability, p Q* Discharge z* Water level, z Fig. 16.4 Uncertainties in the discharge (Q), water level (z) and damage (D) relationships within a risk analysis calculation. Left panel shows the relationship between discharge and exceedance probability (p). The solid curve is the best estimate of this relationship, the dotted lines show confidence intervals of the function f (Q|p), which is the probability density function describing uncertainty in discharge for a given exceedance probability. The second panel shows the relationship between discharge and water height (i.e. the rating curve). The solid line is the best estimate and f (z|Q) represents uncertainty in the water level estimate for a given discharge. The third panel shows the relationship between damage and water height, again with confidence intervals and an illustration of the conditional probability density function for a given water level. Adapted from National Research Council (2000). Q* Discharge z* Water level, z Fig. 16.4 Uncertainties in the discharge (Q), water level (z) and damage (D) relationships within a risk analysis calculation. Left panel shows the relationship between discharge and exceedance probability (p). The solid curve is the best estimate of this relationship, the dotted lines show confidence intervals of the function f (Q|p), which is the probability density function describing uncertainty in discharge for a given exceedance probability. The second panel shows the relationship between discharge and water height (i.e. the rating curve). The solid line is the best estimate and f (z|Q) represents uncertainty in the water level estimate for a given discharge. The third panel shows the relationship between damage and water height, again with confidence intervals and an illustration of the conditional probability density function for a given water level. Adapted from National Research Council (2000). Fig. 16.5 'Source-pathway-receptor' concept for flood risk analysis for an idealised system made up of a river channel ('source'), defence system ('pathway') and floodplain containing capital assets, property, people or other 'receptors' of the risk. The same concept is applicable for a coastal flood risk system. AEP(X) denotes the exceedance probability of X (the probability of observing a value of X greater then some specified threshold value), P(fail) denotes the probability of a food defence failure. The 'pathway' is typically regarded as being the flood defence assets, such as embankments as shown here, but may also include flow routing over the floodplain or other processes that can modify the risk. 'Receptors' may be represented in terms of an economic cost function, as shown here, but could also include other generic 'costs' such as measures of social consequence. Fig. 16.5 'Source-pathway-receptor' concept for flood risk analysis for an idealised system made up of a river channel ('source'), defence system ('pathway') and floodplain containing capital assets, property, people or other 'receptors' of the risk. The same concept is applicable for a coastal flood risk system. AEP(X) denotes the exceedance probability of X (the probability of observing a value of X greater then some specified threshold value), P(fail) denotes the probability of a food defence failure. The 'pathway' is typically regarded as being the flood defence assets, such as embankments as shown here, but may also include flow routing over the floodplain or other processes that can modify the risk. 'Receptors' may be represented in terms of an economic cost function, as shown here, but could also include other generic 'costs' such as measures of social consequence. where D(y) is the flood damage for a given flood depth y and fy(y) is the probability density function of flood depths. The integration may be bounded in practice by a maximum flood depth ymax corresponding to an extremely rare event. The calculation of fy (•) can include treatment of defence system performance using the fragility curve concept. The methods discussed above have been implemented in national scale risk assessments and for analysis of specific flood defence systems (see, e.g. Hall et al., 2003; Gouldby et al., 2008). Large-scale flood risk models now rely on two-dimensional (2D) modelling of flow routing on the floodplain. This can take advantage of readily available digital terrain model (DTM) data, typically a grid of elevation values, combined with automated flood mapping methods that allow many kilometres of floodplain length to be modelled efficiently (Bradbrook, 2006). An example of this type of regional flood risk analysis is a model for the north-east of England including 19 scenarios representative of different flood risk management measures and a wide range of flood flow probabilities. This strategic investigation of flood risk was primarily aimed at providing a consistent evidence base for catchment flood management plans, such as the Tyne CFMP, which is discussed later. The hydrological analysis involved generating hydrographs at hydrological 'inflow nodes' located approximately every 200 m along each watercourse. This step made use of Flood Estimation Handbook (FEH) methods for river flow frequency analysis (see Chapter 13). The natural conveyance capacity of all channels was assumed equal to the estimated median annual flood flow (QMED, i.e. a 2-year return period or 1/2 AEP) to avoid the (very expensive) requirement to have detailed channel cross-section surveys everywhere. To represent different scenarios about flood defence systems, only the volume of water in excess of this channel capacity was then considered as an input for routing over the flood plain. Using this method, the nominal design performance of the defences can be treated as a pseudo-channel capacity and a volumetric adjustment made to the inflow hydrographs to account for the increased effective capacity. The modelling procedure is illustrated in Fig. 16.6, where each hydrograph is plotted along with a dotted line showing the flow rate corresponding to the effective channel capacity and the volume in excess of this capacity. Over 1.25 million independent flood flow hydrographs were generated in this study to represent each of the 19 scenarios at every inflow node. At each node, and for each scenario, the relevant flood hydrograph was routed over a 10m x 10m horizontal resolution DTM using JFLOW-GPU, a 2D diffusion wave approximation to the shallow water flow equations adapted to run on fast parallel processing hardware (Lamb et al., 2009). Inflows are added to the DTM grid over a cross-section set up perpendicular to main flood flow direction (see Fig. 16.6) and routed downstream for at least 1 km until a grid of maximum depths is attained. For each scenario, the maximum depth grids were merged together to form a single regional depth map, using geographical information system (GIS) tools. Fig. 16.6 Large-scale automated flood mapping procedure. Overlapping boxes highlight three individual two-dimensional hydraulic model runs where flows enter along a cross-section aligned perpendicular to predominant flow direction; dots indicate the centre of the cross-section at the river centre line. Feint lines and dots show other 'inflow' cross-sections. Each inflow comprises a flood hydrograph, displayed as inset plots, which show the total hydrograph, the assumed channel capacity (dotted line) and the resulting inflow hydrograph. Fig. 16.6 Large-scale automated flood mapping procedure. Overlapping boxes highlight three individual two-dimensional hydraulic model runs where flows enter along a cross-section aligned perpendicular to predominant flow direction; dots indicate the centre of the cross-section at the river centre line. Feint lines and dots show other 'inflow' cross-sections. Each inflow comprises a flood hydrograph, displayed as inset plots, which show the total hydrograph, the assumed channel capacity (dotted line) and the resulting inflow hydrograph. Table 16.3 Simulation matrix for scenario construction, north-east Region. 'ST' represents an assumed defence system design standard of protection against the T year flood, 'QT' the design flow having a return period of T years. In both cases the corresponding annual exceedance probability (AEP) is 1/T. Shadow indicates cells left blank because they cannot logically contain a simulation Table 16.3 Simulation matrix for scenario construction, north-east Region. 'ST' represents an assumed defence system design standard of protection against the T year flood, 'QT' the design flow having a return period of T years. In both cases the corresponding annual exceedance probability (AEP) is 1/T. Shadow indicates cells left blank because they cannot logically contain a simulation
Fig. 16.7 Expected annual damage (EAD) estimates for one administrative area as a function of the assumed standard of protection of flood defences (SoP). Example from the River Wear catchment. Standard of protection expressed as 1/AEP (years) Fig. 16.7 Expected annual damage (EAD) estimates for one administrative area as a function of the assumed standard of protection of flood defences (SoP). Example from the River Wear catchment. Each entry in Table 16.3 represents one regional grid of modelled flood depths corresponding to a given standard of protection ('S' prefix) and severity of hydrological event ('Q' prefix). Economic damages were estimated based on (deterministic) depth-damage relationships (Penning-Rowsell et al., 2005) and spatial information about properties and agricultural land use. For each of the defence performance scenarios, the simulation matrix therefore provides estimates of quantiles of the probability distribution of depth or damages, allowing the EAD to be computed. The example calculations in Fig. 16.7 show the EAD estimated for each defence performance scenario in a single administrative area. The data show little difference in EAD between defence design standards of 1/75 and 1/100 AEP, suggesting that, in this area at least, the additional damage caused by very infrequent larger events (up to 1/1000 AEP) does not add greatly to the long-term average damages brought about by more frequent, but less severe flooding. |

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