Communicating (nature‐based) flood‐mitigation schemes using flood‐excess volume

As interest mounts in nature‐based solutions (NBS) for flood mitigation as complementary options to civil‐engineering measures, possible flood‐protection strategies have become more diverse and hence complicated to assess. We offer a straightforward and educational protocol targeted for effectiveness analysis and decision making involving stakeholder participation. It is based on the concept of flood‐excess volume (FEV), the volume exceeding a threshold and generating flood damage, and explores what fraction of FEV is reduced, and at what cost, by particular flood‐mitigation measures. Quantification and interpretation of cost scenarios are facilitated using a graphical display that is easy to understand and encapsulates concepts of flood magnitude, FEV and protection‐measures efficacy. It is exemplified for two recent extreme‐flood events on the River Calder in Mytholmroyd (Yorkshire, United Kingdom) and the River Brague in Biot (Alpes‐Maritimes, France). Each case has different flood‐mitigation measures such as natural water‐retention measures, tree planting, river‐bed widening, or use of reservoirs and floods walls. Our straightforward protocol enables fast, quantifiable and easy‐to‐understand exploration of protection strategies using multiple measures, and in doing so highlights the issue of NBS scalability.

or widening and deepening a flood plain, including reopening old river meanders. Evidence for the effectiveness of NFM is case-specific, given the diversity of contributing factors.
Collective slow down of flood peaks in tributaries may not necessarily lower the main flood peak since synchronisation of flood peaks can lead to adverse effects with increased flood peaks and flooding further downstream (Cabaneros et al., 2018). Moreover, in wet periods before large flood events, soil can become oversaturated such that the run-off of extra rainfall is nearly instantaneous (Environment Agency, 2016a). Most of the literature on NFM provides qualitative descriptions of flood-reduction effects and cobenefits, but seldom quantitative assessments and clear recommendations on how to quantify reductions on a catchment scale (Environment Agency, 2017b; Stosser et al., 2015). Because decision making in land-use planning and flood-protection management involves participatory approaches and stakeholder involvement having multifaceted effects, there is a growing need for approaches to educate people in understanding the volumes and costs involved in floods and their mitigation.
A sometimes-neglected aspect of NFM is that proposed methods are difficult to scale up in a robust manner, for example, attenuation features such as leaky dams tend to be small individually, implying that many features are required before effects are significant for larger floods on a catchment scale. Such upscaling also increases and complicates maintenance. Tree and peat planting requires enormous areas before mitigation effects may become significant for larger flood events. Even then, there is a lack of evidence to suggest that increased tree coverage reduces flood risks (Carrick et al., 2018).
Besides showing beneficial effects in small-scale NFM pilot studies (Environment Agency, 2017b), it is important to obtain first estimates of NFM's effectiveness and reliability on grander catchment scales for extreme-flood events. Contrastingly, GRR programmes generally concern larger volumes.
For both NFM and civil-engineering measures, first assessments of the effectiveness of flood-mitigation strategies can be quantified straightforwardly using flood-excess volume (FEV; see Hui & Lund, 2015;Schneider, 2015;Bokhove et al.,2018a). FEV concerns the fraction of the total volume of river discharge, over a certain period, which caused flooding at a certain river location during an extreme event. It implies that in situ river flood levels have exceeded a certain threshold river level h T yielding associated excess discharge rates. FEV is the flood volume one wishes to reduce to zero in flood-mitigation approaches in order to avoid flood damage, for example, by raising flood-defence walls along a river, effectively raising h T ; increasing the river-bed section, enabling more discharge to be conveyed under a similar water depth; or, holding back water or slowing down the flow which flows into the Mediterranean Sea near Antibes. Our analysis is intended to provide an insightful and accessible protocol for assessing flood mitigation, aimed at increasing public understanding and assisting policy makers in decision making. We build upon the analysis of a (hypothetical) flood-alleviation scheme in Bokhove et al. (2018a), in which more available physical and economic data is incorporated. The paper outline is as follows. In Section 2, FEV and available flood-storage volume are revisited to allow use in subsequent sections.
In Section 3, FEVs are calculated for the River Calder and the River Brague floods of 2015. Subsequently, several flood-mitigation measures proposed within these river catchments are analysed in Section 4, with conclusions drawn in Section 5.

TOOL: FLOOD-EXCESS VOLUME
The FEV of a river in flood is defined as the water volume causing flood damage at a certain station due to river levelsh exceeding a relevant threshold h T . This choice of flood quantification is such that, forh > h T , some or major flooding occurs. Conceptually, FEV offers a straightforward and comprehensible means of quantitatively underpinning flood-mitigation strategies. It is presently assumed that the stage-discharge relationship of the river is known in terms of a recorded and accurate time-series (see also Bokhove et al., 2018a).
Given an in situ rating curve Q = Q(t) explicitly as function of time t over a flood duration T f , or implicitly as a function Q = Q(h) of the in situ river levelh =h(t), the approximation of FEV used (which comprises the shaded and hatched areas in Figure 1a) is in which Q T = Q(h T ). In the limit of an infinite number of river-level data h k , that is, when t = T f ∕N m → 0 as N m → ∞, FEV approximation (1) becomes an integral. In general, rating curves are not exact due to errors in the relation Q(h) and measurement errors inh k . Generally, river levelsh are measured and, using theoretical or phenomenological rating curves (Bokhove et al., 2018a;Environment Agency, 2016b), ) .

Available flood-storage volume
Available flood-storage volume is the extra flood-storage volume gained above the flood capacity that a flood-storage site has for a flood of a particular return period. This available flood-storage volume changes as a function of the return period of the flood event; for floods with higher river levels and longer duration, it is less than for floods with lower river levels and shorter duration. Leaky dams slow the flow and increase the upstream water level; their flood-storage capacity is , displayed vertically as function of time horizontally, and a chosen threshold discharge Q T = Q(h T ) with exceedance time T f , involving in situ temporal river levelsh =h(t). (b) FEV square-lake representation as a D = 2 m-deep square lake, with side-length L = √ FEV∕D, to facilitate visualisation of FEV ''size.'' (c) FEV-effectiveness assessment computed for each measure as equivalent FEV fraction, represented as side l i of the square lake

Square-lake representation: a cost-effectiveness communication tool
The three-panel graphs in Figure 1 readily illustrate the FEV concept and offer an approximate sense of the water volume responsible for flooding, though this might not be sufficient to facilitate discussion with all stakeholders and decision makers sufficiently aware of the ''problem size''. Because water volume can be hard to appraise quantifiably, the ''square-lake'' representation has been proposed (Bokhove, Kelmanson & Kent, 2018b) as a conceptual object facilitating such appraisal: FEV volume is represented as that of a square lake of depth 2 m, that is, a ''buildable-scale'' reservoir required to store the FEV; such a visualisation is more meaningful than a volume in comparison with a typical river's length or valley width.
The analysis is further refined by splitting the square-lake FEV into a set of protection measures. The capacity of each measure to store, or deal with a volume by conveying it, is finally computed, whence the cost of each measure can be estimated. A straightforward cost-effectiveness assessment is then performed by considering the ratio between the FEV assigned to a measure and its costs.
The cost-effectiveness is then measured in terms of percentage costs of FEV. This approach is encapsulated in the graphical representation of the flood process exemplified in Figure 1, which comprises: • a three-panel graph displaying the stage-discharge relationship along with the discharge (hydrograph) and measured water-stage-time series highlighting flood duration, peak discharge, water depth, threshold values, and FEV (with corresponding error); and, • a square-lake representation as a 2 m-deep basin (i.e., approximately human height) with the same capacity as the FEV,  Boxing Day Flood-whose severity elicited high-profile coverage in national media (e.g., Gayle & Gunter, 2015). The Boxing Day 2015 events had approximately 1:200 + -and 1:100 +year return periods for the River Aire and Calder respectively (i.e., we consider these return periods as those given in Environment Agency, 2016a). Although both events were extreme and outside the range of data records, their return periods could be estimated using extreme-value theory (Coles, 2001 , displayed vertically as function of time t (day in December 2015) horizontally, and chosen threshold discharge Q T = Q(h T ). It involves in situ temporal river level h =h(t). The rectangle (top-right) represents a mean (approximation of the) FEV based on mean and threshold discharges and a flood duration

River Calder Boxing Day 2015 flood
FEV is first exemplified for the River Calder Boxing Day 2015 flood, using stage data from the Mytholmroyd gauge (located just downstream of Hebden Bridge in Figure 2a). The river level (at 15-min intervals), rating curve, and discharge data are given in Figure 3 with h T = 4.5 m estimated to be the threshold for heavy property flooding. The corresponding FEV and its error (using error-propagation or the capacity of a square lake of depth 2 m and side-length 908 m. For the first stage of the rating curve, error bars of 84.9% for Q are reported; for the second stage, 13.6% for Q, and for the last stage and beyond, error bars are not available (Environment Agency, 2017a).
The error estimate in (2)

River Brague 2015 flood
Following the River Brague 2015 flood, the regional authority, assisted Flood walls were raised to the height necessary to contain the remaining discharge not stored in retention areas. In such a scenario, the is computed by Manning's equation, adding its discharge to current bed capacity to compute the total discharge This uncertainty in the water-level data (±14%) or hydrology (±23%) greatly exceeds that in the rating curve, which is 5% according to the source website, leading to appreciably higher error estimates than those for the River Calder at Mytholmroyd.   with an objective value. Cost-effectiveness then analyses costs to reach a given objective.

River Calder Boxing Day 2015 flood
For the River Calder, we consider three different types of NBS separately, using its FEV of V e (h T = 4.5 m) ≈ 1.65 Mm 3 given in (2)   Pickering (Harrabin, 2016). What is less well-known is that 10% of that ''NFM'' scheme consists of leaky-woody-debris dams, with 90% of the enhanced storage created behind a large controlled cement bund (Potter, 2016). We therefore introduce the small-scale pilot project of the citizens' action group ''Slow-the-Flow-Calderdale,'' upscaling of which to the River Calder catchment scale we consider later. The pilot consists of creating and maintaining run-off attenuation features and restoring old mill ponds to slow the flow so as to increase water-storage capacity (Bradshaw, 2017). An estimate was made of the (effective) attenuation volumes obtained by these interventions, which included ∼120 plate weirs, leaky (small-and large-woody-debris) dams and strategically placed logs as well as restoring plantations on ancient woodland sites. Using two approaches to facilitate estimation, an available (effective) flood-storage volume of ∼7, 000 m 3 was foreseen at a project cost of ∼£50,000 to £72,000 (Bradshaw, 2017).
Here, we have assumed that the aforementioned cumulative volume Mm 2 . Assuming a best-case scenario in which all water is held back on A b = 1.03 Mm 2 , and for uniform rainfall over the catchment, the contribution of these measures is at best (see Table 2) Mytholmroyd lies further upstream in the River Calder catchment, the relevant area A t considered should be smaller and concern only the fraction (∼20% and thus leading to an adjusted V * t ≈ 72, 887 m 3 ) of the total catchment area draining water into Mytholmroyd, cf. the update in Table 2. On the one hand, this would lead to a larger percentage than the 0.84% estimate above; on the other hand, that increase is likely to be offset by the absorption being much less than the assumed 100%.
Again, the contribution of this NFM measure is small and difficult to quantify. FEV reduction. Leaky-woody-debris dams degrade over time and so, to illustrate our protocol, we assume features have an average life span of 25 years. Over 50 years, these need to be constructed twice, leading to twice the base costs, that is, a total of £2.88M, excluding maintenance. These 2,400 features should be replaced using a smart, staggered-replacement scheme to reduce serial failure of dams at certain times, which can lead to devastating flood-wave damage, cf. Cabaneros et al. (2018): this is an interesting optimisation problem not considered here. Over 50 years, 2 × 2, 400∕50 = 96 features p.a. require replacement. We employ one person at ∼£50 k p.a. to carry out maintenance, resulting in £2.5 M employment costs over 50 years (again, as illustration), yielding a total cost of £(2.88+2.5) M =£5.38 M over 50 years, £0.634 M per 1% of flood protection (note that we ignored inflation and rising costs of living). It is neither clear whether full available flood-storage volume is reached nor the extent to which this capacity is reached under varying spatial rainfall distributions. Hence, we introduce this uncertainty via an ad hoc sliding scale of coverage between 50% and 100% of the above capacity.

Cost-effectiveness analysis for hypothetical flood-alleviation scheme
Final ranges for the available flood-storage volume, its FEV fraction, and costs therefore become [0.07, 0.14] Mm 3 or In Leeds' flood-alleviation scheme (Leeds City Council, 2018), it is proposed to increase tree coverage in the River Aire valley from 8% to 15%. We therefore assume another type of NFM by increasing the area of tree and peat coverage to 6% instead of the 0.84% estimated above. Again, using a sliding-scale approach, this yields an increase of flood-storage volume by [0.0495, 0.099] Mm 3 , so [3, 6]% at an estimated cost of £6 M including maintenance costs over 50 years, giving a window of £[1, 2] M per 1% of flood mitigation.
We assume that the above NFM measures are distributed uniformly across the catchment section influencing river flow in Mytholmroyd.
Without further information on the spatial and temporal distributions of rainfall during extreme rainfall and flood events, we assume that the flood mitigation offered varies linearly between the most adverse case with minimum 33.90% flood-mitigation coverage offered by combined measures and maximum 67.81% coverage with mean 50.86% at a cost of £41.38 M for £41.38∕50.86 = £0.8136 M per 1% of flood mitigation. The above cost-effectiveness analysis is visualised in Figure 7. It becomes apparent that reservoir usage (blue shaded area in Figure 7) is the largest and most cost-effective fraction of FEV. Increased tree coverage (green area in Figure 7), a small but not insignificant fraction, offers less value for money. Major upscaling of flood-attenuation features (brown area in Figure 7) has a considerable cost-effective impact.
Unmitigated FEV parts can be covered by other mitigation measures: by building higher flood-defence walls than currently in place or further increasing the reservoir volumes via dynamic control (Breckpot, 2013;Breckpot et al., 2013;Vermuyten et al., 2017). However, optimisation of the draw-down of reservoirs would involve cost functions with the opposite demands of drinkwater maximisation, volume min-FIGURE 7 Graphical overview of flood-excess volume (FEV) fraction captured by three flood-mitigation measures and associated costs for the River Calder at Mytholmroyd. FEV ≈ 1.65 Mm 3 is represented as a 2 m-deep square lake of side-length 908 m, illustrating the flood's magnitude, partitioned here by each measure. Overall flood mitigation ranges from 33.90% to 67.81% at a cost of £41.38 M. The mean of each measure is represented by corresponding quadrilateral areas, partitioning the overall square-lake area with the same FEV-capacity requiring mitigation. Sloping lines reflect the sliding scale between quoted ranges, owing to storage-capacity uncertainty imisation, dam safety, and controlled water release with minimal flood, erosion, and environmental damage.
Alternatively, more attenuation features can be built, or more trees and peat restoration in combination with controlled ponding.
Co-benefits of tree planting and peat coverage can be taken into account in cost assessments, such as carbon sequestration and recreational value (Denjean et al., 2017), but all these measures and decisions demand a clear quantification, especially given the weak evidence for the effect of tree coverage on channel discharge (Carrick et al., 2018). The above analysis and in particular its graphical presentation in Figure 7 suggests that our FEV-based protocol can aid in more quantifiable and rational decision making.

River Brague 2015 flood
The following flood-mitigation analysis for the River Brague is exploratory and distinct from the actual Brague protection scheme to date under study. We highlight a long-term plan to protect the Brague floodplain. Investments and consequences on land use along the river corridor are sufficiently high to warrant time to implement any plan.
We pursue this analysis because discussions with various stakeholders highlight that orders of magnitude of discharge, volume, water depth, effectiveness, and costs of protection measures are generally unknown, resulting in less-informed debates on relevant protection schemes.
Economic 3 data were gathered from local past works, land-acquisition operations, and existing literature (Aerts, 2018;CASA, 2013;Igigabel et al., 2014;Langumier et al., 2014) to instigate quantification of cost-effectiveness analyses (Table 3). In the absence of data, this analysis cannot yet be done on Biot's downstream section threatening Antibes. GRR seems however a good option downstream too since riparian areas are either natural or abandoned following business closures due to excessive flood risks.     Disappointing may be that the NFM measures undertaken in the River Calder catchment to date will contribute only a fraction (about 1% or less) towards the FEV required to mitigate against an extreme flood with a 1∶100-year return period. Upscaling of tree planting is difficult: the flood-mitigation achieved risks facing not only much uncertainty but also requires vast and suitable (i.e., good for absorbing and holding water) areas to be covered by trees. Despite its popularity in the United Kingdom and media, flood mitigation by NFM is prone to an apparently undervalued scalability problem. Although NFM can often reduce flooding locally, for low return-period events, it is much more difficult to scale up NFM as a flood-mitigation measure for large-scale and extreme floods (cf. Lane, 2017;Salazar et al., 2012). The benefit of using our FEV analysis is that it quantifies in an easy-to-understand way this (lack of) scalability and potential (or lack thereof) for upscaling. Three flood-mitigation measures are highlighted in that they account for major fractions of FEVs in the cases studied.

SUMMARY AND DISCUSSION
• Major upscaling of flood-attenuation features, such as leaky dams, can have a significant and cost-effective impact provided long-term maintenance costs are taken into account. Educated guesses were made for the latter maintenance costs and showed that a robust flood-mitigation protocol can be established.
• Draw-down and control of drinkwater reservoirs for flood mitigation were shown to have great flood-mitigation potential, for the River Calder catchment.
• River-bed widening led to a major and cost-effective reduction of the FEV, thereby lowering the need for high flood-defence walls. Whereas the flood-attenuation and tree-planting NFM schemes considered for the River Calder have less potential in the River Brague catchment, river-bed widening seems to be an underexplored flood-mitigation measure in the Calder catchment.
Across all river catchments, emphasis has been placed on the errors inherent in calculating FEVs, ranging from ∼36% for the River Calder to 65% for the River Brague; it stresses the inherent uncertainty in any flood-mitigation planning. However, FEV offers a means of realistically incorporating and quantifying the intrinsic errors into the protocol, over and above other approaches.