The initial step consists of processing the digital elevation model (DEM) in order (i) to obtain a flowing drainage direction map, (ii) to identify the subcatchments (corresponding to the river reaches), and (iii) to compute the height above nearest drainage (HAND) in each subcatchment. This initial processing is the basis of the floodplain analysis in MHYST. To compute the drainage direction map, we used the D8 method from the Flow Direction function provided by ArcGIS 10.3. It computes the drainage direction by calculating the steepest slope from the eight possible directions for a given cell.

Figure  shows the procedure used to compute HAND values: for a given floodplain cell, it is the difference between its elevation and that of the closest river cell in terms of drainage direction. For instance, the cell of elevation 25 at the top of the figure is linked to (i.e. flows towards) the most upstream red river cell which has an elevation of 18: thus, its HAND value is 25-18=7. This relative height has been used as a proxy for inundation height by various studies . To derive an inundation map from HAND values, we must define a threshold height HT: the flooded area corresponds to all the cells whose HAND value is strictly lower than HT .

MHYST is mostly based on a DEM and its derivatives (drainage map and drainage areas) and on the hydraulic equations describing a steady uniform flow at the reach scale. This means that for a given time step (day in this case), at a given reach, we make the approximation that the flow is constant over time and space (this is obviously a strong simplification that we will discuss later). Table  sums up the variables used in the following equations as well as their respective units and interpretations. Table  describes the two free parameters of the model. The following equations show the path to build a reach-scale relation between HT and the streamflow Q by calculating, with hydraulic formulas, the discharge value corresponding to a given HT. Once this relation is known, the model can easily simulate a hydrological event by inverting the relation, and by searching for the HT which corresponds to the given Q (Fig. ).

Other variables can be directly calculated from the DEM (Fig. ): for a given threshold height HT at a reach of length L (L is a fixed parameter of the model), V(HT) is the sum of volumes above all flooded pixels and S(HT) is the area occupied by the flooded cells. A(HT), in Eq. (), is the average cross section area of the flooded reach and it depends on V(HT) and on the bankfull cross section area of the channel (Ab=hb⋅Wb, Fig. ). This variable can also be defined as the sum of the channel cross section area Ach=Ab+HT⋅Wb and the floodplain cross section area Afp=A(HT)-Ach. B(HT), in Eq. (), is the average surface width of the flooded reach, defined similarly from S(HT) and L. A(HT)=1L⋅V(HT)+Ab=Ach+AfpB(HT)=1L⋅S(HT)

The only unknown variables in these equations are sub-grid parameters hb (bankfull water level) and Wb (bankfull width), i.e. the bankfull geometry, which cannot be obtained from usual DEMs and are only available from detailed surveys for a small number of rivers. Indeed, in situ bathymetric data are quite scarce and red lasers cannot penetrate the water surface more than a few centimetres, which means that the real elevation of the river bed is mostly not correctly represented in the DEMs. This is why we chose to use downstream hydraulic geometry equations to estimate these geometric parameters, assuming a rectangular channel, the size of which depends on the upstream drainage area (Eqs.  and ). To assess the coefficients α and β, we used satellite images from the French platform Géoportail in order to link observed bankfull widths and drainage areas. The values found for the Loing catchment are α=0.053 and β=0.822. The other coefficients, δ and ω, were taken from a study by , which attempted to regionalise these parameters in the US. We used the general values found for the whole set of catchments: δ=0.27 and ω=0.21. Although these values probably add uncertainties in the model, they are an accessible way to assess bankfull channel geometry and could still be improved by local bankfull studies when available. Wb=α⋅ADβhb=δ⋅ADω

The fundamental equations of the MHYST model come from an experimental study by which defines the DEBORD formulation as in Eqs. () to (). Building on the Manning–Strickler formula, these authors proposed an empirical parameterisation of turbulent momentum exchange between the channel and the floodplain. This formulation expresses the conveyance capacity depending on channel-related and floodplain-related variables. The coefficient C takes into account the interaction of flows between the fast-flowing channel and the slow-flowing floodplain, as well as the corresponding head losses. De=Kch⋅C⋅Ach⋅Rch2/3+Kfp⋅Afp2+Ach⋅Afp⋅1-C2⋅Rfp2/3C=C0=0.9⋅KfpKch1/6ifr=RfpRch>0.31-C02⋅cos⁡π⋅r0.3+1+C02if0≤r≤0.3Q=De⋅If

The streamflow Q is finally defined from the conveyance capacity and the channel slope, since we hypothesise a uniform flow. Rch and Rfp can easily be calculated from the assumed reach geometry (Eqs.  and ), which only leaves the Strickler coefficients as unknown variables. Rch=AchWb+2⋅hbRfp=AfpB(HT)-Wb

The two Strickler coefficients add 2 degrees of freedom, and Kch is additionally used to calculate the bankfull flow from the Manning–Strickler formula (Eq. ).

Qb=Kch⋅Wb⋅hbLb+2⋅hb2/3⋅If⋅hb⋅Wb Here, we sum up the procedure, which operates at the reach scale:

For a given threshold height HT, we use the DEBORD formulation to calculate the corresponding discharge Q.

Although this method and that of were developed independently, they share a lot of similarities, both using HAND to derive a reach-scale geometry which is used as input for a simplified hydraulic model. However, in addition to HAND, MHYST uses downstream hydraulic geometry relationships to evaluate a sub-grid representation of the channel geometry. The hydraulic model is also different: use the Manning–Strickler formula, while MHYST computes streamflow values from the DEBORD formulation.

MHYST can work with either simulated or observed flows. In this paper, observed data from 12 measurement stations of the French HYDRO database were used to create an observed distributed streamflow map by interpolating flows based on drainage area (Eq. ) for river pixels between outlets: Q=Qup+AD-AD,upAD,down-AD,up×Qdown-Qup, where Q and AD are the streamflow and drainage area of any river cell between two outlets, Qup, Qdown, AD,up and AD,down are the direct upstream and downstream outlet discharges and drainage areas. This way, streamflow is coherently interpolated over the network, and then averaged at the reach scale.

In this study, we used a 5m resolution DEM with a vertical resolution of 0.01m covering the Loing catchment (Fig. ) from IGN (the French national institute for geographic information), which was filled and corrected to avoid depressions and to allow a strict coherence of flow directions, meaning that every pixel flows to the sea. Drainage directions and areas were derived from this DEM and used as model inputs along with elevations. The adaptations and modifications of the DEM were conducted using ESRI ArcGIS 10.3.

Daily observed discharges were obtained from the French HYDRO database and the stations were used to delineate the hydrological network over the catchment. Calibration data for the Loing catchment were obtained from the activation EMSN028 of the Copernicus Emergency Management Service (© 2016 European Union). The original Copernicus study covered a small part of the River Seine and half of the Loing catchment (Fig. ). However, since the study area and the defined river network were smaller, we cropped the inundation extent to match the study area (Fig. ). These calibration data are post-processed observed data, meaning that the original maps came from satellite observations but they were then modified to build a more homogeneous inundation extent, i.e. nearby areas whose elevations were below the observed flood level were added to the inundation extent and merged with all the others. The maximum flood extent was then validated by the European Service against reported flood damage and hydrological measurements .

Following an extremely wet month of May (namely the wettest on record for many stations), a heavy rainfall event started on 30 May 2016 over the centre of France, affecting the Upper and Middle Seine basin and the Middle Loire basin. This episode lasted until 6 June and, combined with highly saturated soils due to a series of preceding minor events, led to major flood inundations. Over this period, overall precipitation reached 180mm in Paris and Orléans, while in some tributaries, such as the River Loing, peak flows largely exceeded those of the record 1910 flood event (Fig. ). The flood resulted in 4 deaths, 24 people injured and EUR 1.4 billion worth of damage. A total of 1148 cities were declared to be in a state of natural disaster and insurance companies received about 182 000 claims .

Since calibration data were available for June 2016 event, we chose to use our model to simulate this episode and compare the results with observations. We conducted this study over the River Loing, tributary to the River Seine, with a catchment covering 3900km2, a mean elevation of 148m and a mean slope of 0.03mm-1. This catchment was heavily impacted by the flood event and contains a significant proportion of the inundated area, making it a suitable area to carry out the study. Streamflow data were interpolated from measurements, so no hydrological model is involved in this paper.

To assess the model's performance, we used several criteria based on the contingency table in Fig. . These scores are presented in detail by and are defined as a ratio between members of the table where n1 is the number of hits, i.e. the number of flooded cells correctly forecast; n4 is the number of pixels correctly forecast as dry; n2 is the number of false alarms; and n3 the number of observed flooded cells missed by the model. Table  summarises the formulas and the interpretations of each score used in this study.

The POD (probability of detection), which is also called Correct or M1 , calculates the percentage of observed inundated pixels intersected by the simulation map. Its main drawback is that it does not take into account the false alarms and thus it can give good results for a clearly overestimating inundation extent. On the contrary, the FAR (false alarm ratio) or M2 computes the proportion of cells wrongly flooded by the model. But similarly, if the model does not flood anything, the FAR can reach its optimal value. The critical success index (CSI), also known as Fit, F index or FAI , is a criterion which tries to give an overall performance of the simulation by calculating the percentage of correctly flooded cells above the total number of flooded cells (observed and simulated). In this way, the score is penalised by the over- and underestimation. However, this criterion does not specify if the model is over- or underestimating the observed extent. This is why we also looked at the BIAS, which computes the ratio between the number of simulated and observed flooded cells. If it is above 1, the model overestimates, and if it is below 1, it underestimates. However, a value of 1 does not equal a perfect simulation since there may be a balance between the misses and the false alarms.

These ratios are particularly reliable if they are used to compare simulations and exhaustive observations. This is almost the case with Copernicus calibration data, which represent a “maximum flood extent”. However, MHYST outputs are dated, which is not the case for the observed map. This is why all daily simulated inundation extents were merged into one maximum simulated extent, meaning that we did not try to validate the temporal dynamic of the flood, but only aimed to assess its largest area. Thus, the preceding scores will only evaluate MHYST's ability to reproduce the maximum flood extent.

MHYST has two free parameters (Table ): Kch (the Strickler roughness coefficient for the channel) and Kfp (the Strickler roughness coefficient for the floodplains). Preliminary studies showed that, for the Loing catchment, a length of 1000m was a good trade-off between accuracy and computation time; consequently L was fixed at 1000m in the rest of this study. Kch and Kfp values were tested in the range 0.1;30 in order to explore a wide range of possibilities (121 combinations were tested).

To help make a decision on the optimal parametrisation of the model, we used the following graphs, on which each (Kch, Kfp) couple is characterised by one overall value:

two contour plots (Fig. ) showing the impact of Kch and Kfp for the two main scores (BIAS and CSI);

Last, to be able to analyse the variability of results between reaches (we have a total of 90 reaches affected by the inundation), we also computed the CSI and BIAS reach by reach, and produced two cumulative distribution plots showing these results (Fig. ). We found the following:

The fit criteria are very sensitive to the Kfp value and much less to the Kch value (Fig. ): this should not be a surprise given that we deal with the maximum flood extents for calibration, where Kch only plays a minor role. Remember also that (i) we are modelling a very extreme event (T>500 years) with substantial overflowing, and (ii) that we are working with a channel geometry derived from hydraulic geometry relationships. All this contributes to making the estimation of the channel roughness coefficient more difficult.

Last, Fig.  provides a good illustration of how parameter sets interact with the FAR, POD and CSI criteria: choosing from the parameter sets with the best CSI makes it possible to find a compromise between a high POD and a low FAR.

Figures  to provide a further illustration with a colour-coded classification of each reach depending on its CSI and BIAS value. A total of 11 regions are highlighted and numbered because of their poor performance. The reasons of why MHYST was not able to reproduce the inundation extent in these regions are explained below.

For the downstream-most part of the Loing (Fig. ), the reaches are red or orange because this area is only partially covered by the observation, which stops just after the confluence with the small tributary.

In order to complete our interpretation of MHYST behaviour, we conducted two sensitivity analyses, one with the Morris method and the other with the Sobol method details can be found in Appendix . We chose to assess the effect of six potential parameters, Kch, Kfp, α, β, δ and ω, that may play a major role in the computation of HT–Q relationships. In both analyses, we found that ω, which parameterises the regionalisation of bankfull heights, has the most substantial effect on the performance and that, surprisingly, Kfp has no influence at all. As a matter of fact, when we conducted the Sobol analysis with fixed hydraulic geometry parameters, we showed that Kfp is considerably more influential than Kch. We concluded that the previous results were due to the fact that these sensitivity analyses explore the parameter space in detail, and even with reasonable boundaries, they can reach values that may not be consistent with the characteristics of the catchment studied.

It is possible to assess the sensitivity to the DEM in two ways: first by aggregating our DEM from 5 m to various resolutions (10, 25, 50 and 100 m) and then by changing the source of the DEM. Figure  provides the CSI scores obtained by the model while changing the resolution. It shows that the resolution has relatively little effect on the optimal value, which varies between 0.65 and 0.69. However, the position of this optimal, i.e. the combination of parameters (Kch and Kfp) leading to it, changes. We can also see that for some resolutions, such as 25 or 50 m, the equifinality zone is much smaller than the one for the 100 m resolution, for example. If we also look at the “physical” set of parameters we previously identified (Kch=10 and Kfp=5m1/3s-1), we can see that the CSI reached by the model for this combination varies between the resolutions. Nevertheless, the result still seems satisfying, so it could be used as a “default” parameterisation, for instance for ungauged catchments. But this should be tested on other catchments with observed data to lead to a more comprehensive conclusion.

Before using the RGE 5 m DEM from IGN, we tried to use the 25 m EU-DEM from the European Environment Agency, and it showed poorer results, because it was not precise enough. Figure  shows the evolution of CSI for the same combinations of parameters as before. We see that the best combinations of parameters only lead to a 0.53 maximal CSI, which is more than 10 points below what we can obtain with the RGE DEM. There is also strictly no connection between the best values of BIAS and those of CSI, the latter being obtained for a clear overestimation of the flood extent (BIAS∼1.5). These results are due to the lack of precision of the EU-DEM, which does not distinguish the channel from the floodplain, leading to a 2 km wide channel in some parts of the river.