A reducedcomplexity model of ﬂuvial inundation with a subgrid representation of ﬂoodplain topography evaluated for England, United Kingdom
 ^{1}UK Centre for Ecology and Hydrology, Maclean Building, Crowmarsh Gifford, Wallingford, OX10 8BB, United Kingdom
 ^{2}School of Geography and the Environment, University of Oxford, OX1 3QY, United Kingdom
 ^{3}Met Ofﬁce, FitzRoy Road, Exeter EX1 3PB, United Kingdom
 ^{1}UK Centre for Ecology and Hydrology, Maclean Building, Crowmarsh Gifford, Wallingford, OX10 8BB, United Kingdom
 ^{2}School of Geography and the Environment, University of Oxford, OX1 3QY, United Kingdom
 ^{3}Met Ofﬁce, FitzRoy Road, Exeter EX1 3PB, United Kingdom
Abstract. Timely predictions of fluvial flooding are important for national and regional planning and realtime flood response. Several new computational techniques have emerged in the past decade for making rapid fluvial flood inundation predictions at time and space scales relevant to early warning, although their efficient use is often constrained by the tradeoff between model complexity, topographic fidelity and scale. Here we apply a simplified approach to largearea fluvial flood inundation modelling which combines a 5 solution to the inertial form of the shallow water equations at 1 km horizontal resolution, with two alternative representations of subgrid floodplain topography. One of these uses a fitted subgrid probability distribution, the other a quantilebased representation of the floodplain. We evaluate the model's performance when used to simulate the 0.01 Annual Exceedance Probability (AEP; 100year
) flood and compare the results with published benchmark data for England. The quantilebased method accurately predicts flood inundation in 86 % of locations, with a domainwide hit rate of 95 % and 10 false alarm rate of 10 %. These performance measures compare with a hit rate of 71 %, and false alarm rate of 9 % for the simpler, but faster, distributionbased method. We suggest that these approaches are suitable for rapid, widearea flood forecasting and climate change impact assessment.
Simon J. Dadson et al.
Status: closed

CC1: 'Comment on hess202160', Oliver Wing, 05 Mar 2021
The comment was uploaded in the form of a supplement: https://hess.copernicus.org/preprints/hess202160/hess202160CC1supplement.pdf

AC3: 'Reply on CC1', Simon Dadson, 27 Apr 2021
Author reply to comment by Wing et al.
We thank Oliver Wing and colleagues for their short comment. The comment is helpful because it allows us to justify the scale of the approach that we have taken in this study. We take this opportunity to restate that the principal use case for this model is as a component within a landsurface climate modelling system where there is a pressing need to simulate flooding at our chosen resolution to correctly specify hydrological fluxes at the landatmosphere boundary. Despite the many highresolution datasets available in the UK, which of course underpin finerresolution limitedarea modelling applications, there remains a need for rapid, largescale assessments for situational awareness in times of major flood.
The datasets used to constrain hydraulic geometry and for flood depth estimation were chosen for consistency with the benchmark validation data. In parallel work we are collating a large quantity of more recent data into a usable opensource form to support future analyses.
Performance metrics are presented in good faith at the scale of the analysis. We acknowledge the helpful suggestion to compare flooded fraction directly and will include such a comparison in a subsequent revision.
Below, in bold, we respond in more detail to the various points raised.
The Bates & Neal Flood Lab, part of the Hydrology Group in the School of Geographical Sciences at the University of Bristol, reviewed this HESS Discussion paper during one of our meetings and provide the following comments that we hope are useful to the authors.
General comments
The general framing of the paper does not justify what place a model of this fidelity has in a country like the UK. Where metricresolution inundation models with gaugebased flows, better parameterised channels, lidar terrain, and flood defences are already available, what is the need for a steadystate, 1 km, undefended model? Observations of flow, channel properties, elevation, and flood defences that are more accurate than the components used here are readily available for the UK.
The main motivation for this work is to test the ability of a newlyconfigured landsurface model component to simulate fluvial flooding. Its central purpose is to serve as one option within the JULES landsurface model, ultimately for coupled landatmosphereocean simulation of flood inundation at 1 km resolution. Correct specification of the land boundary is important in such models because it controls the partitioning of water and energy fluxes at the surface.
Of course, we acknowledge the many excellent highresolution datasets in the UK. We also acknowledge the existence of many important problems in flood hazard modelling at finer resolution. But, as we state in the paper, those are not the target applications for this study. We have taken great care in the manuscript to note that we do not expect our modelling approach to take the place of traditional finescale flood inundation modelling (p.2 lines 2025) and will take the opportunity to strengthen those statements in revision.
The justification that simulations are quick does not outweigh the need for accurate models.
We agree. The purpose of our study is to ensure that our landsurface flood parametrisation is fit for its stated purpose. The tradeoff in the present case is between speed of execution and spatial precision, not accuracy. Our study is intentionally structured to test whether such an approach provides the necessary accuracy at the scale for which it is designed.
The authors fail to discuss the merits of sampling from presimulated libraries of more accurate flood inundation maps when time is at a premium, for instance, or downscaling the 1 km model back to the native resolution of the DEM. The other justification that simplified models are yet to be evaluated fully is not the case. There is already a wealth of literature on the general inability of coarse and/or physicslite models to replicate detailed validation data, some of which the authors themselves cite.
We appreciate the utility of simulation libraries and will include relevant references in revision. Such an approach would be unsuitable for use in a coupled landatmosphereocean model though because the land boundary fluxes need to be updated at all model timesteps as part of the coupled simulation.
Channel and boundary condition configuration
We were unable to understand the treatment of channels in the model. In particular, whether estimated bankfull depths are “burned” into the DEM or retained subgrid. It appears that all channel variables are based upon a linear regression of ~30 observations collected 30 years ago. We question how representative such poorly constrained equations are for applications at national scales, particularly since there is no consideration of slope or discharge (the ultimate determinants of hydraulic geometry).
The extreme boundary condition is again based on a simple uplift of the same measurements from the 1991 paper, rather than any understanding of growth curves that the authors’ own organisation sets out in the Flood Estimation Handbook. It is also unclear how this boundary condition is input to the model: by being steadystate, the model would struggle to simulate nonvalley filling floods.
In a country as datarich as the UK, there is little need to estimate these properties in the way the authors describe. The use of its rich network of river gauges and channel approximations based on discharge and slope would undoubtedly be a more justifiable approach than the one taken: it would not “introduce additional uncertainty” (P3/L15), it would decrease it. If the model could readily receive flows as input, as seems to be suggested through its intended coupling to JULES, how then would channels be parameterised?
Our rationale for using the original flood depth estimation procedure referred to in the 1991 study is to remain as close as possible to the method used to construct the benchmark. This is explained on p3. lines 1314. The justification for this approach is so that errors and uncertainties diagnosed from our comparison can then be attributed to the structure of our model rather than to differences in the driving data used. During the development of this work, we did also use a later dataset produced by estimating flood discharges from flow records across the United Kingdom as part of the Flood Estimation Handbook. We will include a substantial section showing results from that analysis in the revised manuscript.
In parallel work we have collated additional width observations and we will include them in a revised version of this manuscript given the interest in updated width observations shown here and in RC1.
The model is not a steadystate model. It is a timedependent model (see Eq 5), which is here evaluated for the steadystate case associated with the 100year flood. Transient evaluation is amongst our planned future work pending acquisition of widearea validation data.
To calculate river levels using statisticallyestimated flows requires use of an additional flow resistance equation which adds uncertainty to the calculations. Channels remain fixed within the 1 km grid box and their properties are parametrised as described above.
Model validation
The model validation is questionable. Scaling up the highresolution benchmark data to that of the coarse model is not a fair test of its skill – the appropriateness of low model resolution is partly what should be tested. Most channels and floodplains in the UK are <1 km wide, meaning (as is shown in Figure 9) the validation procedure simply discriminates whether channels exist in broadly the correct locations and whether water is input to them. To suggest that “these performance metrics are comparable with those obtained in previous studies” is disingenuous when such exercises exhausted the utility of the validation data rather than degrading it to fit the model. For the highlevel conclusion to be 86% similarity to EA maps is patently false.
We disagree that the validation protocol is inappropriate for this study, given the 1 km resolution of our intended application. Validation metrics have been calculated according to equations given in the text and are presented in good faith at the scale of the study. We take seriously the important comment about the difference in resolution between ours and others’ studies and will include clarification in revision.
The translation of inundated cell fractions to binary wet/dry grids with a very low threshold of detection (ε) is, again, a very forgiving comparison. It is not clear why the flood fractions are not just compared directly. A more useful test, however, would be to use the validation data at their native resolution.
Our stated aim was to evaluate a model of flooding at 1 km resolution (for the use cases indicated above). The suggestion to include direct comparison of flooded fraction is a good one which we will introduce in revision.
We are grateful to the discussants for their careful attention to our manuscript and for their helpful suggestions to improve the paper.

AC3: 'Reply on CC1', Simon Dadson, 27 Apr 2021

RC1: 'Comment on hess202160', Anonymous Referee #1, 05 Mar 2021
The objective of the paper is not clear or basically wrong. For England, an accurate appraisal of flooded areas and depths for the 100 year flood is available on a majority of locations so that the results of the calculations are useless. Such calculations as proposed in the paper can be useful for more extreme floods for which the uncertainty is always higher because of both the uncertainty of the flow and the lack of calibration data. They may be also useful for countries in which the data are scarce, maps of historical floods are lacking and for which simplified calculations could permit to obtain a whole coverage of the country without costly studies. If the ultimate objective is one out of the two quoted here above, the text should be reoriented in order to be sure that additional data for extreme floods or raw data would be available.
The structure of the paper is also not clear. First, the organization of the paper is not provided at the end of § 1. Second, the method (the best one) should be first described and second, the validation (or calibration) of the results should follow. Third, the discussion could question some aspects of the method and/or the efficiency of the method comparing it to other methods. However, the paper is not organized like this. It seems to me that the paper begins by discussing te one and then the other component of the method. For instance, if table 1 sums up the comparison between alternatives, a conclusion should be provided just below; in the paper, oppositely, one of the three methods is compared to an other estimate on a data set that is not described and appears to date from before 1991 (30 years ago !). If such data set is a reference, other references should be provided; if not I guess that the conclusions from this data set are questionable similarly to other studies that tried to link bankful depth to drainage area (they are a lot not quoted here!). Similar procedure is used later on for wbf (instead of hbf) without more explanations and any clear justification.
The third parameter to be estimated is the channel roughness: I really do not understand the few lines of explanations (how estimate roughness from a database of river crosssections?) but I retain that it is so difficult that the authors are using a uniform value of 0.04. Similarly, I could not understand what is the relaxation time at the bottom of page 7 but I am quite sure that this parameter is not linked with the there previous ones and thus the explanation is not at the right place.
 2.4 describes two sub grid parametrization; however, because the detailed procedure of the whole method was never presented, the reader cannot understand the advantage of any sub grid parametrization compared to the solution to establish directly a relation between depth and inundated area from the DTM source.
From §2.5, I understand that the authors are not using the maps of the flooded area but a percentage of flooded area per Km2 to validate their model. However, form §2.6 I understood that the plot for comparison is 50 m x 50 m. What is correct? I ask the authors clearly explain what means a hit rate of 71%, a FAR of 9% and a success score of 67% in simple words. The following discussion between the various regions does not interest me because the tables 4 and 5 show similar results from one region to another one. It might be more useful to show quantitative results at smaller regions (subregions) for which the results are very different.
For such a type of model, the sensitivity analysis is a key issue and so a wider sensitivity analysis was expected. Moreover, I could not understand why the results are not sensitive to the Manning coefficient : once the geometry is provide transforming a flow into a flooded area or flood depth should depend on the Manning coefficient if hydraulics equations are used.
The first sentence of the conclusion should be written in a different way in order to avoid confusion: I understood that the authors calculated the percentage of flooded area for their studied area quite accurately but they are not providing a map of flooded areas accurately and are not providing at all the peak water depth of the 100 year flood in any location accurately. The objective (for validating the method) should be to provide such results as the ratio of the water depths: reference over calculation for any location (for instance on the 50 m grid).
As a conclusion, I guess that with a new organization of the paper, adding explanations and real validation of their method, the authors could obtain a valuable paper. However, I am not sure that the method is useful for England and that the method can be extended to other countries because a lot of “morphological” equations are very specific to the local geography.

AC1: 'Reply on RC1', Simon Dadson, 27 Apr 2021
Author response to RC1: 'Comment on hess202160', Anonymous Referee #1
We thank the reviewer for their comments and suggestions to improve the manuscript. In this response we address all the points which the reviewer has made (in bold) and indicate how we plan to address these in a revised manuscript.
The objective of the paper is not clear or basically wrong. For England, an accurate appraisal of flooded areas and depths for the 100 year flood is available on a majority of locations so that the results of the calculations are useless. Such calculations as proposed in the paper can be useful for more extreme floods for which the uncertainty is always higher because of both the uncertainty of the flow and the lack of calibration data. They may be also useful for countries in which the data are scarce, maps of historical floods are lacking and for which simplified calculations could permit to obtain a whole coverage of the country without costly studies. If the ultimate objective is one out of the two quoted here above, the text should be reoriented in order to be sure that additional data for extreme floods or raw data would be available.
The purpose of the study is to test whether a local inertial approximation to the shallow water equations, combined with a subgrid topographic parametrisation, is capable of simulating flood inundation over a large area for use in a landsurface model. The aims of the study are stated on p2. lines 1825 and we will strengthen the statement further in a revised version of the paper.
Landsurface models represent fluxes of water and energy in the atmospheric boundary layer and serve as the lower boundary to weather forecasting and climate models. They therefore require specification of surface properties to calculate latent vs. sensible heat exchange and can be affected significantly by the presence or absence of terrestrial open water on floodplains. Correct specification of the land boundary is therefore essential to calculate these fluxes correctly. As part of weather forecasting and climate models, LSMs are also increasingly used to provide situational awareness of potential flood hazard over large areas, further motivating our study.
To perform this evaluation, we use existing Environment Agency flood inundation data to test whether the model’s predictions match an existing benchmark. There is no suggestion that the results from this evaluation against the benchmark data are intended to replace the benchmark data (as we state explicitly on p.2 lines 2025).
The structure of the paper is also not clear. First, the organization of the paper is not provided at the end of § 1. Second, the method (the best one) should be first described and second, the validation (or calibration) of the results should follow. Third, the discussion could question some aspects of the method and/or the efficiency of the method comparing it to other methods. However, the paper is not organized like this. It seems to me that the paper begins by discussing te one and then the other component of the method. For instance, if table 1 sums up the comparison between alternatives, a conclusion should be provided just below; in the paper, oppositely, one of the three methods is compared to an other estimate on a data set that is not described and appears to date from before 1991 (30 years ago !). If such data set is a reference, other references should be provided; if not I guess that the conclusions from this data set are questionable similarly to other studies that tried to link bankful depth to drainage area (they are a lot not quoted here!). Similar procedure is used later on for wbf (instead of hbf) without more explanations and any clear justification.
We thank the reviewer for their comments and will revise the structure of the paper accordingly. In the circumstances we propose to remove the analysis of the parameterbased subgrid model (which performs less well) from the revised manuscript to make room for additional uncertainty experiments suggested below and by Reviewer 2.
Our rationale for using the original flood depth estimation procedure referred to in the 1991 study is to remain as close as possible to the method used to construct the benchmark data. This is explained on p3. lines 1314. The justification for this approach is so that errors and uncertainties diagnosed from our comparison can then be attributed to the structure of our model rather than to differences in the driving data used. During the development of this work, we did also trial another dataset produced by estimating flood discharges from flow records across the United Kingdom as part of the Flood Estimation Handbook. We will include a substantial section showing additional results from that analysis in the revised manuscript.
In parallel work we have collated additional width observations and whilst we did not intend to use them in the present study, we will include them in a revised version of this manuscript given the interest in updated width observations.
The third parameter to be estimated is the channel roughness: I really do not understand the few lines of explanations (how estimate roughness from a database of river crosssections?) but I retain that it is so difficult that the authors are using a uniform value of 0.04. Similarly, I could not understand what is the relaxation time at the bottom of page 7 but I am quite sure that this parameter is not linked with the there previous ones and thus the explanation is not at the right place.
Channel roughness was calculated using a standard procedure described by Chow et al. together with a newlycompiled database of bed texture observations. We used a reference value of 0.04 only where no additional information was available. The procedure is explained in detail in lines p7 lines 713 and we propose to add further clarification in revision.
The relaxation time computed in Eq 8 is a time for the model to run to reach its steady state. It is not connected to the roughness calculation but determines the length of model integration required to achieve our aims. We will clarify the purpose of this equation in a revised submission and move it to its own section.
2.4 describes two sub grid parametrization; however, because the detailed procedure of the whole method was never presented, the reader cannot understand the advantage of any sub grid parametrization compared to the solution to establish directly a relation between depth and inundated area from the DTM source.
We will add some additional explanation here, including a diagram showing how the subgrid information have been used.
From §2.5, I understand that the authors are not using the maps of the flooded area but a percentage of flooded area per Km2 to validate their model. However, form §2.6 I understood that the plot for comparison is 50 m x 50 m. What is correct? I ask the authors clearly explain what means a hit rate of 71%, a FAR of 9% and a success score of 67% in simple words. The following discussion between the various regions does not interest me because the tables 4 and 5 show similar results from one region to another one. It might be more useful to show quantitative results at smaller regions (subregions) for which the results are very different.
The model uses a subgrid topographic distribution (here at 50 m resolution) to calculate flooded fraction for each 1 km grid box. The evaluation metrics are used with their standard definitions given in the text. Hit rate is the probability of detection of flooding in the 1 km grid box; FAR is the false alarm rate, and the critical success index is a metric which accounts for true positives but penalises for false negatives, as defined in the text. We will amplify this explanation in revision and propose to show an analysis of flooded fraction across the model domain as suggested in SC1. In addition, we will supply model performance metrics for subregions given in Figure 9 as the reviewer suggests.
For such a type of model, the sensitivity analysis is a key issue and so a wider sensitivity analysis was expected. Moreover, I could not understand why the results are not sensitive to the Manning coefficient: once the geometry is provide transforming a flow into a flooded area or flood depth should depend on the Manning coefficient if hydraulics equations are used.
We thank the reviewer for this suggestion and propose to add a broader range of sensitivity analysis to the revised paper both in response to this comment and a similar suggestion from Reviewer 2 to consider topographic uncertainty. It is indeed notable that the sensitivity to Manning’s n is more subdued than initially expected and we will explore this in more detail with a broader range of analysis in revision.
The first sentence of the conclusion should be written in a different way in order to avoid confusion: I understood that the authors calculated the percentage of flooded area for their studied area quite accurately but they are not providing a map of flooded areas accurately and are not providing at all the peak water depth of the 100 year flood in any location accurately. The objective (for validating the method) should be to provide such results as the ratio of the water depths: reference over calculation for any location (for instance on the 50 m grid).
We thank the reviewer for this important point and will add a comprehensive regional analysis comparing flooded fraction between the model and the benchmark to a revised submission. Unfortunately, flooded depth is not given in the benchmark data and so it will not be possible to include that variable in the analysis.
As a conclusion, I guess that with a new organization of the paper, adding explanations and real validation of their method, the authors could obtain a valuable paper. However, I am not sure that the method is useful for England and that the method can be extended to other countries because a lot of “morphological” equations are very specific to the local geography.
We are pleased that the reviewer sees potential value in the paper and grateful for their suggestions to improve it. We will implement all the recommended changes in a revision. Whilst the morphological equations are specific to the study region, they are really only used to ensure that the test case is as close as possible to that used to produce the benchmark data and do not preclude wider application of the model.

RC2: 'Comment on hess202160', Anonymous Referee #2, 08 Mar 2021

AC2: 'Reply on RC2', Simon Dadson, 27 Apr 2021
Author response to Comment on hess202160; Anonymous Referee #2
We thank the reviewer for their positive remarks on the presentation of the paper and for their constructive comments on how the work can be improved. We set out below (in bold) how we propose to make the suggested changes in a revised submission.
The paper is well written, covers a large literature review in the hydrology field, but according to me completely fails in providing a convincing motivation of the hydrodynamic structure of the proposed model. More specifically:
1) It is not clear if the model is a 1D or a 2D model. Eq. (3) is the continuity equation of a 1D model, where the flow along the direction normal to direction xi is zero. The same holds for the momentum equation (4). If Eqs (3) and (4) hold for direction x1, they cannot hold for direction x2. On the other hand, authors adopt a regular structured grid, with grid size of 1 km.
The model is a 2D model and we thank the reviewer for pointing out some oversimplifications in the way in which we described the local inertial approximation in the original submission. The original paper gave the model in abbreviated index notation and referred the reader to several papers in the literature where its details can be found. In the revised version we will include a full derivation in two dimensions including details of the numerical scheme.
2) I assume the water depth is updated at the new time level from the finite difference approximation of the continuity equation (3), but this is not discussed in the paper.
Yes, that is correct and we will add this step to the explanation of the model physics in revision.
3) If Eq. (4) is the momentum equation along the xi direction of a 2D model, the 3rd resistance term on its l.h.s. must be written in the form:
[Equation]otherwise it depends on the grid orientation. If Eq. (4) is the momentum equation of a 1D model, the approximation of the hydraulic radius with the hydraulic depth is a very strong one, also because the channel width is a very arbitrary choice.
Yes, exactly. The resistance term requires the magnitude of the vector water flux to be calculated. This is denoted q in the original paper and the reviewer is correct to note that this has been computed as the vector norm. The approach that we have adopted is explained in Almeida et al., which we cite. We will give full details and a complete derivation in the revised manuscript.
4) The choice of a zero convective inertia model should be discussed against other possible approximations. It is known that the error of the zero convective inertia model is larger than the error of the zero model ([1][4]). A trivial example is the front of a sharp shock wave, where the local inertia is positive, but the convective inertia is negative. In this case to neglect only one of the two components is worse than to neglect both. The advantage of the zero inertia model is that it allows an easy solution in the case of small water depths, but there are also other options that can be applied for fully diffusive models ([5][6]).
We thank the reviewer for these suggested references. We proposed to include an additional paragraph discussing the motivation for our choice of model in the revised submission, in which we will refer to the suggested papers.
5) The authors carry on a sensitivity analysis of the model results for the choice of the Manning coefficient and of the channel width, but they should do the same also for the topographic elevation z. Because the adopted value, in each computational cell, is the mean elevation computed over a 1kmq area, I assume that both the averaging technique and the measurement error lead to a very large uncertainty.
We thank the reviewer for the suggestion to investigate topographic uncertainty  it would be a useful and informative addition to the paper. We therefore propose to add a substantial new section to the manuscript in which we investigate the effect of the fidelity of the subgrid topographic representation alongside the hydraulic parameters and quantify the associated uncertainties.
1) Perumal, M., and K. G. Ranga Raju (1998a), Variableparameter stage hydrograph routing method. I: Theory, J. Hydrol. Eng., 3(2), 109 – 114.
2) Perumal, M., and K. G. Ranga Raju (1998b), Variableparameter stage hydrograph routing method. II: Evaluation, J. Hydrol. Eng., 3(2), 115 – 121.
3) Ponce VM, Li RM, Simons DB. Applicability of kinematic and diffusion models. J Hydraul Div, ASCE 1978;104:353–60.
4) Ponce VM. Generalized diffusive wave equation with inertial effects. Water Resour Res 1990;26:1099–101.
5) Sinagra, M., Nasello, C., Tucciarelli, T., Barbetta, S., Massari, C., Moramarco, T. A selfcontained and automated method for flood hazard maps prediction in urban areas (2020) Water (Switzerland), 12 (5), art. no. 1266.
6) Aricò, C., Filianoti, P., Sinagra, M., Tucciarelli, T. The FLO diffusive 1D2D model for simulation of river flooding (2016) Water (Switzerland), 8 (5), art. no. 200.

AC2: 'Reply on RC2', Simon Dadson, 27 Apr 2021
Status: closed

CC1: 'Comment on hess202160', Oliver Wing, 05 Mar 2021
The comment was uploaded in the form of a supplement: https://hess.copernicus.org/preprints/hess202160/hess202160CC1supplement.pdf

AC3: 'Reply on CC1', Simon Dadson, 27 Apr 2021
Author reply to comment by Wing et al.
We thank Oliver Wing and colleagues for their short comment. The comment is helpful because it allows us to justify the scale of the approach that we have taken in this study. We take this opportunity to restate that the principal use case for this model is as a component within a landsurface climate modelling system where there is a pressing need to simulate flooding at our chosen resolution to correctly specify hydrological fluxes at the landatmosphere boundary. Despite the many highresolution datasets available in the UK, which of course underpin finerresolution limitedarea modelling applications, there remains a need for rapid, largescale assessments for situational awareness in times of major flood.
The datasets used to constrain hydraulic geometry and for flood depth estimation were chosen for consistency with the benchmark validation data. In parallel work we are collating a large quantity of more recent data into a usable opensource form to support future analyses.
Performance metrics are presented in good faith at the scale of the analysis. We acknowledge the helpful suggestion to compare flooded fraction directly and will include such a comparison in a subsequent revision.
Below, in bold, we respond in more detail to the various points raised.
The Bates & Neal Flood Lab, part of the Hydrology Group in the School of Geographical Sciences at the University of Bristol, reviewed this HESS Discussion paper during one of our meetings and provide the following comments that we hope are useful to the authors.
General comments
The general framing of the paper does not justify what place a model of this fidelity has in a country like the UK. Where metricresolution inundation models with gaugebased flows, better parameterised channels, lidar terrain, and flood defences are already available, what is the need for a steadystate, 1 km, undefended model? Observations of flow, channel properties, elevation, and flood defences that are more accurate than the components used here are readily available for the UK.
The main motivation for this work is to test the ability of a newlyconfigured landsurface model component to simulate fluvial flooding. Its central purpose is to serve as one option within the JULES landsurface model, ultimately for coupled landatmosphereocean simulation of flood inundation at 1 km resolution. Correct specification of the land boundary is important in such models because it controls the partitioning of water and energy fluxes at the surface.
Of course, we acknowledge the many excellent highresolution datasets in the UK. We also acknowledge the existence of many important problems in flood hazard modelling at finer resolution. But, as we state in the paper, those are not the target applications for this study. We have taken great care in the manuscript to note that we do not expect our modelling approach to take the place of traditional finescale flood inundation modelling (p.2 lines 2025) and will take the opportunity to strengthen those statements in revision.
The justification that simulations are quick does not outweigh the need for accurate models.
We agree. The purpose of our study is to ensure that our landsurface flood parametrisation is fit for its stated purpose. The tradeoff in the present case is between speed of execution and spatial precision, not accuracy. Our study is intentionally structured to test whether such an approach provides the necessary accuracy at the scale for which it is designed.
The authors fail to discuss the merits of sampling from presimulated libraries of more accurate flood inundation maps when time is at a premium, for instance, or downscaling the 1 km model back to the native resolution of the DEM. The other justification that simplified models are yet to be evaluated fully is not the case. There is already a wealth of literature on the general inability of coarse and/or physicslite models to replicate detailed validation data, some of which the authors themselves cite.
We appreciate the utility of simulation libraries and will include relevant references in revision. Such an approach would be unsuitable for use in a coupled landatmosphereocean model though because the land boundary fluxes need to be updated at all model timesteps as part of the coupled simulation.
Channel and boundary condition configuration
We were unable to understand the treatment of channels in the model. In particular, whether estimated bankfull depths are “burned” into the DEM or retained subgrid. It appears that all channel variables are based upon a linear regression of ~30 observations collected 30 years ago. We question how representative such poorly constrained equations are for applications at national scales, particularly since there is no consideration of slope or discharge (the ultimate determinants of hydraulic geometry).
The extreme boundary condition is again based on a simple uplift of the same measurements from the 1991 paper, rather than any understanding of growth curves that the authors’ own organisation sets out in the Flood Estimation Handbook. It is also unclear how this boundary condition is input to the model: by being steadystate, the model would struggle to simulate nonvalley filling floods.
In a country as datarich as the UK, there is little need to estimate these properties in the way the authors describe. The use of its rich network of river gauges and channel approximations based on discharge and slope would undoubtedly be a more justifiable approach than the one taken: it would not “introduce additional uncertainty” (P3/L15), it would decrease it. If the model could readily receive flows as input, as seems to be suggested through its intended coupling to JULES, how then would channels be parameterised?
Our rationale for using the original flood depth estimation procedure referred to in the 1991 study is to remain as close as possible to the method used to construct the benchmark. This is explained on p3. lines 1314. The justification for this approach is so that errors and uncertainties diagnosed from our comparison can then be attributed to the structure of our model rather than to differences in the driving data used. During the development of this work, we did also use a later dataset produced by estimating flood discharges from flow records across the United Kingdom as part of the Flood Estimation Handbook. We will include a substantial section showing results from that analysis in the revised manuscript.
In parallel work we have collated additional width observations and we will include them in a revised version of this manuscript given the interest in updated width observations shown here and in RC1.
The model is not a steadystate model. It is a timedependent model (see Eq 5), which is here evaluated for the steadystate case associated with the 100year flood. Transient evaluation is amongst our planned future work pending acquisition of widearea validation data.
To calculate river levels using statisticallyestimated flows requires use of an additional flow resistance equation which adds uncertainty to the calculations. Channels remain fixed within the 1 km grid box and their properties are parametrised as described above.
Model validation
The model validation is questionable. Scaling up the highresolution benchmark data to that of the coarse model is not a fair test of its skill – the appropriateness of low model resolution is partly what should be tested. Most channels and floodplains in the UK are <1 km wide, meaning (as is shown in Figure 9) the validation procedure simply discriminates whether channels exist in broadly the correct locations and whether water is input to them. To suggest that “these performance metrics are comparable with those obtained in previous studies” is disingenuous when such exercises exhausted the utility of the validation data rather than degrading it to fit the model. For the highlevel conclusion to be 86% similarity to EA maps is patently false.
We disagree that the validation protocol is inappropriate for this study, given the 1 km resolution of our intended application. Validation metrics have been calculated according to equations given in the text and are presented in good faith at the scale of the study. We take seriously the important comment about the difference in resolution between ours and others’ studies and will include clarification in revision.
The translation of inundated cell fractions to binary wet/dry grids with a very low threshold of detection (ε) is, again, a very forgiving comparison. It is not clear why the flood fractions are not just compared directly. A more useful test, however, would be to use the validation data at their native resolution.
Our stated aim was to evaluate a model of flooding at 1 km resolution (for the use cases indicated above). The suggestion to include direct comparison of flooded fraction is a good one which we will introduce in revision.
We are grateful to the discussants for their careful attention to our manuscript and for their helpful suggestions to improve the paper.

AC3: 'Reply on CC1', Simon Dadson, 27 Apr 2021

RC1: 'Comment on hess202160', Anonymous Referee #1, 05 Mar 2021
The objective of the paper is not clear or basically wrong. For England, an accurate appraisal of flooded areas and depths for the 100 year flood is available on a majority of locations so that the results of the calculations are useless. Such calculations as proposed in the paper can be useful for more extreme floods for which the uncertainty is always higher because of both the uncertainty of the flow and the lack of calibration data. They may be also useful for countries in which the data are scarce, maps of historical floods are lacking and for which simplified calculations could permit to obtain a whole coverage of the country without costly studies. If the ultimate objective is one out of the two quoted here above, the text should be reoriented in order to be sure that additional data for extreme floods or raw data would be available.
The structure of the paper is also not clear. First, the organization of the paper is not provided at the end of § 1. Second, the method (the best one) should be first described and second, the validation (or calibration) of the results should follow. Third, the discussion could question some aspects of the method and/or the efficiency of the method comparing it to other methods. However, the paper is not organized like this. It seems to me that the paper begins by discussing te one and then the other component of the method. For instance, if table 1 sums up the comparison between alternatives, a conclusion should be provided just below; in the paper, oppositely, one of the three methods is compared to an other estimate on a data set that is not described and appears to date from before 1991 (30 years ago !). If such data set is a reference, other references should be provided; if not I guess that the conclusions from this data set are questionable similarly to other studies that tried to link bankful depth to drainage area (they are a lot not quoted here!). Similar procedure is used later on for wbf (instead of hbf) without more explanations and any clear justification.
The third parameter to be estimated is the channel roughness: I really do not understand the few lines of explanations (how estimate roughness from a database of river crosssections?) but I retain that it is so difficult that the authors are using a uniform value of 0.04. Similarly, I could not understand what is the relaxation time at the bottom of page 7 but I am quite sure that this parameter is not linked with the there previous ones and thus the explanation is not at the right place.
 2.4 describes two sub grid parametrization; however, because the detailed procedure of the whole method was never presented, the reader cannot understand the advantage of any sub grid parametrization compared to the solution to establish directly a relation between depth and inundated area from the DTM source.
From §2.5, I understand that the authors are not using the maps of the flooded area but a percentage of flooded area per Km2 to validate their model. However, form §2.6 I understood that the plot for comparison is 50 m x 50 m. What is correct? I ask the authors clearly explain what means a hit rate of 71%, a FAR of 9% and a success score of 67% in simple words. The following discussion between the various regions does not interest me because the tables 4 and 5 show similar results from one region to another one. It might be more useful to show quantitative results at smaller regions (subregions) for which the results are very different.
For such a type of model, the sensitivity analysis is a key issue and so a wider sensitivity analysis was expected. Moreover, I could not understand why the results are not sensitive to the Manning coefficient : once the geometry is provide transforming a flow into a flooded area or flood depth should depend on the Manning coefficient if hydraulics equations are used.
The first sentence of the conclusion should be written in a different way in order to avoid confusion: I understood that the authors calculated the percentage of flooded area for their studied area quite accurately but they are not providing a map of flooded areas accurately and are not providing at all the peak water depth of the 100 year flood in any location accurately. The objective (for validating the method) should be to provide such results as the ratio of the water depths: reference over calculation for any location (for instance on the 50 m grid).
As a conclusion, I guess that with a new organization of the paper, adding explanations and real validation of their method, the authors could obtain a valuable paper. However, I am not sure that the method is useful for England and that the method can be extended to other countries because a lot of “morphological” equations are very specific to the local geography.

AC1: 'Reply on RC1', Simon Dadson, 27 Apr 2021
Author response to RC1: 'Comment on hess202160', Anonymous Referee #1
We thank the reviewer for their comments and suggestions to improve the manuscript. In this response we address all the points which the reviewer has made (in bold) and indicate how we plan to address these in a revised manuscript.
The objective of the paper is not clear or basically wrong. For England, an accurate appraisal of flooded areas and depths for the 100 year flood is available on a majority of locations so that the results of the calculations are useless. Such calculations as proposed in the paper can be useful for more extreme floods for which the uncertainty is always higher because of both the uncertainty of the flow and the lack of calibration data. They may be also useful for countries in which the data are scarce, maps of historical floods are lacking and for which simplified calculations could permit to obtain a whole coverage of the country without costly studies. If the ultimate objective is one out of the two quoted here above, the text should be reoriented in order to be sure that additional data for extreme floods or raw data would be available.
The purpose of the study is to test whether a local inertial approximation to the shallow water equations, combined with a subgrid topographic parametrisation, is capable of simulating flood inundation over a large area for use in a landsurface model. The aims of the study are stated on p2. lines 1825 and we will strengthen the statement further in a revised version of the paper.
Landsurface models represent fluxes of water and energy in the atmospheric boundary layer and serve as the lower boundary to weather forecasting and climate models. They therefore require specification of surface properties to calculate latent vs. sensible heat exchange and can be affected significantly by the presence or absence of terrestrial open water on floodplains. Correct specification of the land boundary is therefore essential to calculate these fluxes correctly. As part of weather forecasting and climate models, LSMs are also increasingly used to provide situational awareness of potential flood hazard over large areas, further motivating our study.
To perform this evaluation, we use existing Environment Agency flood inundation data to test whether the model’s predictions match an existing benchmark. There is no suggestion that the results from this evaluation against the benchmark data are intended to replace the benchmark data (as we state explicitly on p.2 lines 2025).
The structure of the paper is also not clear. First, the organization of the paper is not provided at the end of § 1. Second, the method (the best one) should be first described and second, the validation (or calibration) of the results should follow. Third, the discussion could question some aspects of the method and/or the efficiency of the method comparing it to other methods. However, the paper is not organized like this. It seems to me that the paper begins by discussing te one and then the other component of the method. For instance, if table 1 sums up the comparison between alternatives, a conclusion should be provided just below; in the paper, oppositely, one of the three methods is compared to an other estimate on a data set that is not described and appears to date from before 1991 (30 years ago !). If such data set is a reference, other references should be provided; if not I guess that the conclusions from this data set are questionable similarly to other studies that tried to link bankful depth to drainage area (they are a lot not quoted here!). Similar procedure is used later on for wbf (instead of hbf) without more explanations and any clear justification.
We thank the reviewer for their comments and will revise the structure of the paper accordingly. In the circumstances we propose to remove the analysis of the parameterbased subgrid model (which performs less well) from the revised manuscript to make room for additional uncertainty experiments suggested below and by Reviewer 2.
Our rationale for using the original flood depth estimation procedure referred to in the 1991 study is to remain as close as possible to the method used to construct the benchmark data. This is explained on p3. lines 1314. The justification for this approach is so that errors and uncertainties diagnosed from our comparison can then be attributed to the structure of our model rather than to differences in the driving data used. During the development of this work, we did also trial another dataset produced by estimating flood discharges from flow records across the United Kingdom as part of the Flood Estimation Handbook. We will include a substantial section showing additional results from that analysis in the revised manuscript.
In parallel work we have collated additional width observations and whilst we did not intend to use them in the present study, we will include them in a revised version of this manuscript given the interest in updated width observations.
The third parameter to be estimated is the channel roughness: I really do not understand the few lines of explanations (how estimate roughness from a database of river crosssections?) but I retain that it is so difficult that the authors are using a uniform value of 0.04. Similarly, I could not understand what is the relaxation time at the bottom of page 7 but I am quite sure that this parameter is not linked with the there previous ones and thus the explanation is not at the right place.
Channel roughness was calculated using a standard procedure described by Chow et al. together with a newlycompiled database of bed texture observations. We used a reference value of 0.04 only where no additional information was available. The procedure is explained in detail in lines p7 lines 713 and we propose to add further clarification in revision.
The relaxation time computed in Eq 8 is a time for the model to run to reach its steady state. It is not connected to the roughness calculation but determines the length of model integration required to achieve our aims. We will clarify the purpose of this equation in a revised submission and move it to its own section.
2.4 describes two sub grid parametrization; however, because the detailed procedure of the whole method was never presented, the reader cannot understand the advantage of any sub grid parametrization compared to the solution to establish directly a relation between depth and inundated area from the DTM source.
We will add some additional explanation here, including a diagram showing how the subgrid information have been used.
From §2.5, I understand that the authors are not using the maps of the flooded area but a percentage of flooded area per Km2 to validate their model. However, form §2.6 I understood that the plot for comparison is 50 m x 50 m. What is correct? I ask the authors clearly explain what means a hit rate of 71%, a FAR of 9% and a success score of 67% in simple words. The following discussion between the various regions does not interest me because the tables 4 and 5 show similar results from one region to another one. It might be more useful to show quantitative results at smaller regions (subregions) for which the results are very different.
The model uses a subgrid topographic distribution (here at 50 m resolution) to calculate flooded fraction for each 1 km grid box. The evaluation metrics are used with their standard definitions given in the text. Hit rate is the probability of detection of flooding in the 1 km grid box; FAR is the false alarm rate, and the critical success index is a metric which accounts for true positives but penalises for false negatives, as defined in the text. We will amplify this explanation in revision and propose to show an analysis of flooded fraction across the model domain as suggested in SC1. In addition, we will supply model performance metrics for subregions given in Figure 9 as the reviewer suggests.
For such a type of model, the sensitivity analysis is a key issue and so a wider sensitivity analysis was expected. Moreover, I could not understand why the results are not sensitive to the Manning coefficient: once the geometry is provide transforming a flow into a flooded area or flood depth should depend on the Manning coefficient if hydraulics equations are used.
We thank the reviewer for this suggestion and propose to add a broader range of sensitivity analysis to the revised paper both in response to this comment and a similar suggestion from Reviewer 2 to consider topographic uncertainty. It is indeed notable that the sensitivity to Manning’s n is more subdued than initially expected and we will explore this in more detail with a broader range of analysis in revision.
The first sentence of the conclusion should be written in a different way in order to avoid confusion: I understood that the authors calculated the percentage of flooded area for their studied area quite accurately but they are not providing a map of flooded areas accurately and are not providing at all the peak water depth of the 100 year flood in any location accurately. The objective (for validating the method) should be to provide such results as the ratio of the water depths: reference over calculation for any location (for instance on the 50 m grid).
We thank the reviewer for this important point and will add a comprehensive regional analysis comparing flooded fraction between the model and the benchmark to a revised submission. Unfortunately, flooded depth is not given in the benchmark data and so it will not be possible to include that variable in the analysis.
As a conclusion, I guess that with a new organization of the paper, adding explanations and real validation of their method, the authors could obtain a valuable paper. However, I am not sure that the method is useful for England and that the method can be extended to other countries because a lot of “morphological” equations are very specific to the local geography.
We are pleased that the reviewer sees potential value in the paper and grateful for their suggestions to improve it. We will implement all the recommended changes in a revision. Whilst the morphological equations are specific to the study region, they are really only used to ensure that the test case is as close as possible to that used to produce the benchmark data and do not preclude wider application of the model.

RC2: 'Comment on hess202160', Anonymous Referee #2, 08 Mar 2021

AC2: 'Reply on RC2', Simon Dadson, 27 Apr 2021
Author response to Comment on hess202160; Anonymous Referee #2
We thank the reviewer for their positive remarks on the presentation of the paper and for their constructive comments on how the work can be improved. We set out below (in bold) how we propose to make the suggested changes in a revised submission.
The paper is well written, covers a large literature review in the hydrology field, but according to me completely fails in providing a convincing motivation of the hydrodynamic structure of the proposed model. More specifically:
1) It is not clear if the model is a 1D or a 2D model. Eq. (3) is the continuity equation of a 1D model, where the flow along the direction normal to direction xi is zero. The same holds for the momentum equation (4). If Eqs (3) and (4) hold for direction x1, they cannot hold for direction x2. On the other hand, authors adopt a regular structured grid, with grid size of 1 km.
The model is a 2D model and we thank the reviewer for pointing out some oversimplifications in the way in which we described the local inertial approximation in the original submission. The original paper gave the model in abbreviated index notation and referred the reader to several papers in the literature where its details can be found. In the revised version we will include a full derivation in two dimensions including details of the numerical scheme.
2) I assume the water depth is updated at the new time level from the finite difference approximation of the continuity equation (3), but this is not discussed in the paper.
Yes, that is correct and we will add this step to the explanation of the model physics in revision.
3) If Eq. (4) is the momentum equation along the xi direction of a 2D model, the 3rd resistance term on its l.h.s. must be written in the form:
[Equation]otherwise it depends on the grid orientation. If Eq. (4) is the momentum equation of a 1D model, the approximation of the hydraulic radius with the hydraulic depth is a very strong one, also because the channel width is a very arbitrary choice.
Yes, exactly. The resistance term requires the magnitude of the vector water flux to be calculated. This is denoted q in the original paper and the reviewer is correct to note that this has been computed as the vector norm. The approach that we have adopted is explained in Almeida et al., which we cite. We will give full details and a complete derivation in the revised manuscript.
4) The choice of a zero convective inertia model should be discussed against other possible approximations. It is known that the error of the zero convective inertia model is larger than the error of the zero model ([1][4]). A trivial example is the front of a sharp shock wave, where the local inertia is positive, but the convective inertia is negative. In this case to neglect only one of the two components is worse than to neglect both. The advantage of the zero inertia model is that it allows an easy solution in the case of small water depths, but there are also other options that can be applied for fully diffusive models ([5][6]).
We thank the reviewer for these suggested references. We proposed to include an additional paragraph discussing the motivation for our choice of model in the revised submission, in which we will refer to the suggested papers.
5) The authors carry on a sensitivity analysis of the model results for the choice of the Manning coefficient and of the channel width, but they should do the same also for the topographic elevation z. Because the adopted value, in each computational cell, is the mean elevation computed over a 1kmq area, I assume that both the averaging technique and the measurement error lead to a very large uncertainty.
We thank the reviewer for the suggestion to investigate topographic uncertainty  it would be a useful and informative addition to the paper. We therefore propose to add a substantial new section to the manuscript in which we investigate the effect of the fidelity of the subgrid topographic representation alongside the hydraulic parameters and quantify the associated uncertainties.
1) Perumal, M., and K. G. Ranga Raju (1998a), Variableparameter stage hydrograph routing method. I: Theory, J. Hydrol. Eng., 3(2), 109 – 114.
2) Perumal, M., and K. G. Ranga Raju (1998b), Variableparameter stage hydrograph routing method. II: Evaluation, J. Hydrol. Eng., 3(2), 115 – 121.
3) Ponce VM, Li RM, Simons DB. Applicability of kinematic and diffusion models. J Hydraul Div, ASCE 1978;104:353–60.
4) Ponce VM. Generalized diffusive wave equation with inertial effects. Water Resour Res 1990;26:1099–101.
5) Sinagra, M., Nasello, C., Tucciarelli, T., Barbetta, S., Massari, C., Moramarco, T. A selfcontained and automated method for flood hazard maps prediction in urban areas (2020) Water (Switzerland), 12 (5), art. no. 1266.
6) Aricò, C., Filianoti, P., Sinagra, M., Tucciarelli, T. The FLO diffusive 1D2D model for simulation of river flooding (2016) Water (Switzerland), 8 (5), art. no. 200.

AC2: 'Reply on RC2', Simon Dadson, 27 Apr 2021
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