Reply on RC1

However, after reading the manuscript, one is left with the impression that on the assimilation approach the authors have come with a “preconceived” plan without seriously considering standard assimilation possibilities that are already out there. It is my belief that studies that defend alternatives methods should always consider existing approaches for benchmarking. Otherwise, there is the risk that we are left with a bunch of alternatives and without enough information to decide whether it is worth (or better) to try one or another. Also, one is left with the impression that the overall results in this study could be rather different using alternative assimilation approaches. This does not invalidate the study at all, but it should be augmented with results from applying (assimilation) techniques that are already on the table.

the joint assimilation of both traditional static sensors and satellite derived flood extent for improving the performances of a distributed unsteady bidimensional hydraulic model. The use of Quasi-2D hydraulic modelling is a challenge for real time DA frameworks and injecting distributed EO data (in addiction to single point static sensor observations) makes the challenger harder. To our knowledge, while there are a number of papers also using 2D hydraulic modelling (see ID: R1_06 comment) this is a quite challenging and still a much debated topic requiring further research to develop methods, procedure towards DA that include 2D models. Moreover, we also better specified why using some novel approaches for updating the state variables in the DA framework.

Authors'actions:
In the following lines we isolated every Referee comment assigning a specific ID with a progressive number (e.g. R1_XX) and our point-by-point reply.

Referee comments:
In the introduction, it would have been also nice also to refer, somehow, to the advances in Numerical Weather Prediction (NWP) to frame the context of the early warning systems.

Authors' reply:
We agree with the Referee' suggestion.

Actions:
We added some references on the NWP in the introduction: "In case of medium term forecast (i.e. days/weeks ahead), rainfall and runoff observations are not sufficient and Numerical WeatherPrediction (NWP) models are required, especially for basins whose concentration time is limited so that emergency measures, such as evacuation, cannot be properly applied on time (Hopson and Webster, 2010). In this regard, recent advances in NWP models in weather forecasting were developed adopting ensemble prediction systems (EPS) (Buizza et al., 2005) as inputs of hydrological and hydraulic models."

And:
"DA models are used both in NWP and hydrologic-hydraulic modelling. Advances in EPS approaches and increasing of computational power allowed to improve the accuracy of NWP models as inputs of flood forecasting systems (Yu et al., 2016). Successful examples of advanced EPS approaches in NWP models for flood forecasting services at large scale are the EPS-ECMWF -from the European Centre for Medium Range Weather Forecasts- (De Roo et al., 2003) and the COSMO-LEPS -from the Consortium for Small-Scale Modelling -Limited-area Ensemble Prediction System (Marsigli et al., 2005). Flood models can be updated in DA approaches by ingesting outputs of NWP models or direct rainfallrunoff observations."

Referee comments:
It is not easy to select a starting point as best reference on assimilation in this field along the chain of available papers, ranging from general methodological assimilation papers (outside from hydrology) to more applied ones in this specific field of flood forecast. Perhaps, on the assimilation of real satellite-based flood extent observations with a 2D model stretching over a number of rivers (some main rivers plus some tributaries), and using an upstream coupled hydrologic model for the inflow timeseries (as well as estimating online model parameters, as this study), the authors should look back to García-Pintado et al. (2015). From there, they could also further look into citing articles to go forward in time towards more recent studies.

Authors' reply:
We thank the Referee for the suggestion.

Actions:
In the introduction, we extended the analysis about the state of the art of the scientific literature on Data Assimilation applications with hydraulic modelling for flood forecasting including also García-Pintado et al. (2015) reference. We report here an extract of the introduction: "To tackle these issues, in the last ten years, Earth Observation (EO) data were used to inject water altimetry observations in DA frameworks for updating flood models, usually adopting radar Synthetic Aperture Radar (SAR) technologies and 1D (Matgen et al., 2007;Neal et al., 2009;Matgen et al., 2010;Giustarini et al., 2011) or 2D ( Andreadis et al., 2007Hostache et al.,2010;Mason et al., 2012;García-Pintado et al., 2013;Andreadis and Schumann, 2014) hydraulic routing algorithms. One of the critical issues of the model state updating is the persistence of the improvements of the model performances. Regardless the DA algorithm (e.g. Direct Insertion, PF, EnKF) the assimilation of the model states in real and synthetic scenarios brought to more accurate predictions immediately after the updating step, and they quickly decrease, depending on the specific case study, few hours or even few minutes after the state updating, going back to the same performances of the open-loop model realisation (Andreadis et al., 2007;Matgen et al., 2010;García-Pintado et al., 2013;Andreadis and Schumann, 2014). Some of these studies demonstrated that the updating of inflows boundaries can increase the persistence of the errors reductions between the observations in both 1D (Matgen et al., 2010) and 2D (Andreadis et al., 2007;García-Pintado et al., 2013) hydraulic models. Other studies investigated on the spatial weighting of remote sensing-derived water levels observations in DA approaches. For example Giustarini et al. (2011) found significant benefits in a local weighting procedure of assimilating unbiased very precise water levels observations, while a global weighting procedure is recommended for water level observations in ungauged basins. However, if the local weighting is combined with but poorly spatially distributed field data , the model updating can lead to an over-correction that could even decrease the overall model performances. In fact, the frequency of the model corrections seems to be effective mostly during the rising limb of the flow hydrograph, while it seems not to be significant efficient during the recession limb (Giustarini et al., 2011;García-Pintado et al., 2013). García-Pintado et al. (2015) proposed a novel methodology to test the performance of a global formulation, a traditional local formulation and their own novel local formulation of the EnKF model to improve the forecast of a 2D hydraulic model assimilating SAR derived water levels. Their novel local formulation of the EnKF was able to remove the unphysical relationships and spurious correlations that characterized the global filter. The authos also proved that the updating of the 2D hydraulic model friction and channel bathymetry seems to have second order effect, as respect to the inflow updating, in flood inundation models applied to gradually varied flow in large rivers. Andreadis and Schumann (2014) applied a local EnKF for assimilating synthetic SAR derived water levels, inundation width and flood extent in a 2D hydraulic model, partitioning the Ohio River (516 km) in reaches of equal lengths. The authors obtained similar results for reach lengths varying from 5 to 50 km. On the other hand, SAR-derived inundation extent mapping techniques were tested to provide spatially distributed information to support near real-time flood detection services (Martinis et al., 2015;Pierdicca et al., 2009) 2018) underlined opportunities of SAR images, overcoming visibility issues of optical sensors due to clouds, but also stressing some limitations of water altimetry approaches. In particular, the need of high resolution topographic data, challenging pre-processing and hydraulic modelling development make SAR-derived DA approaches hard to replicate and to be applied at varying scales (Mason et al., 2012;Wood et al., 2016). Dasgupta et al. (2021b) proposed a novel Mutual Information-based likelihood function for assimilating SAR derived flood extents in an high resolution 2D hydraulic model adopting a PF approach. Dasgupta et al.(2021a) investigated on the timing, the positioning and the frequency of the SAR-derived flood extents, on the performances of the PF assimilation of a 2D hydraulic model, finding that the optimal strategy for the image acquisition depends on the river morphology and flood wave arrival timing. Moreover, it was found that the number of observations to significantly improve the performances of the DA model increase with the with the narrowing of the floodplain valley."

Referee comments:
For a more general context about flood forecast considering assimilation, it is also advisable to read the review by Grimaldi et al. (2016). Although this paper is cited here, the manuscript indicates the authors have not actually gone through the paper details.

Authors' reply:
We agree that Grimaldi et al., (2016) provided a good starting point as framework on the state of the art of DA modelling for flood forecasting.
Actions: According to the referee suggestions, we extended the introduction considering Grimaldi et al. (2016) ID: R1_05

Referee comments:
Last but not least, it is surprising that despite the effort put into this study the authors do not show any spatial graphical output (despite they use a 2D model) for diagnosis of the assimilation. They could well show maps with, for example, assimilation increments at specific times, covariances, etc. These kind of 2D plots are very helpful to try to understand what is under the hood in the assimilation. This greatly eases the understanding and evaluation of what is working well or not. significant improvement of the revised work. What seems to indicate that the authors have actually not read these papers.

Authors' reply:
We thank the Referee for pointing out this error.

Referee comments:
L57: Clarify here in which sense is the DA framework novel? Specifically, the EnKF has now a long history.

Authors' reply:
We agree that the novelty of the proposed research is not the DA model itself (EnKF is a consolidated approach). Actions: we removed the "novel" word for the DA framework and we better clarified the novel aspects of the proposed research: "Despite the remarkable progresses in the integration of remotely sensed observations in DA frameworks, there are still majorchallenges that need to be faced (Grimaldi et al., 2016). For example, there is not still in scientific literature an approach able to assimilate heterogeneous observations from both local and distributed datasets coming from different sources (i.e. traditionalstage gauges and remotely sensed flood extents). Moreover, Quasi-2D and 2D hydraulic models can be sensitive to differentsimultaneous local state updating (i.e. water level corrections at specific time steps), because contiguous channel/floodplaincells can be characterized by different elevations, geometry and roughness, therefore instability issues can rise during themodel corrections. Another critical issue is that large scale flood forecasting models need to provide timely predictions but their spatial resolution can limit the effectiveness of the assimilation of satellite derived flood extents (Hostache et al., 2018).In this work, a DA framework supported by heterogeneous observations coming from both local water level observations(i.e. stage gauges) and spatially distributed information gathered from satellite images -is proposed and tested. This researchseeks to develop a more flexible DA scheme that may value all available sources of observations for distributed flood modellingupdates. The aim of this work is to mitigate flood prediction uncertainties by combining heterogeneous data and an integrated topographic-hydrologic-hydraulic modelling approach, while maintaining inundation forecasting robustness, scalability andnumerical stability. In achieving this goal, novel scientific advances and technical challenges of EO-driven DA approaches forflood prediction are investigated and in particular: A methodology for updating the state variable from multiple local stagegauges observations of a hydraulic model for distributed flood routing in floodplain domains; the gathering of spatially dis-tributed water level observations by means of flood extension processing and detection from satellite images, also adopting GIS algorithms for overcoming the issues of the different resolutions between the ensembles of the flood extents retrieved fromthe satellite derived images and the ones generated from the hydraulic model simulations. " ID: R1_08

Referee comments:
L77: The regular grid and simple IO formats do not make the model more "suitable" to DA than using an unstructured mesh or more complex (e.g. hierarchical) IO formats. This just allows for a simpler code.
Authors' reply: we agree with the Referee comment.

Referee comments:
L94 "assess maximum flood energy gradients". How is this relevant? How is energy coupled with the 2D flood model or used here? It is also unclear if the floodplain computational domain evolves with time along with model integrations or is preset, based on GFPLAIN.

Authors' reply:
We agree that the description of how we defined the computational domain (that however does not evolve with time) can move the reader's attention away from the main purpose of the manuscript and it is not relevant to this study.

Actions:
We removed this paragraph.

ID: R1_10
Referee comments: L114: No uncertainty is taken into account in the rainfall input. It is worth to a) discuss briefly the errors in rain gauge data [e.g. the possibility of generating quantitative precipitation ensembles via Sequential Gaussian Simulations, etc.] and how the uncertainty is propagated downstream in the forecasting chain, and b] some reference to coupling with [possibly ensemble] NWPs.

Authors' reply:
We agree that we did not directly take in to account the uncertainty in rainfall input. However, we expressed the inflow uncertainty including all the sources of uncertainty of the hydrologic modelling. In this revied version, we included the temporal correlation of the inflow errors and the standard deviation of the white noise component is derived considering the frequency distribution of its relative flow errors (observed versus simulated flow values) obtained by the calibration and validation of the hydrologic model.
We calibrated and validated the hydrologic model considering four small ungauged basins of the Tiber river basin in order to find the optimal values of the channel/hillslope flow velocity and infiltration parameters.
Since we directly compared the simulated and the observed streamflow values, the standard deviation error equal to 0.28, indirectly takes in to account all the sources of errors: from the rainfall inputs and its spatial distribution along the basin, the simplified modelling of the flow routing, the neglected physical processes, such as the groundwater flow, the mud and debris flow, the antecedent soil moisture conditions. We agree that more refined methods such as the generation of quantitative precipitation ensembles should be mentioned.
Actions: we added the following lines: "The hydrologic model is affected by different sources of uncertainties: the structural uncertainties, given by the simplification of the modelled physical processes (e.g. we adopted a simplified lumped WFIUH approach, neglecting groundwater flow,mud and debris flow), the input uncertainties (given by the rainfall values and antecedent soil moisture conditions), and para-metric uncertainties due to the inaccuracy of the model calibration). All of these sources of uncertainty should be considered. For example input rainfall uncertainty from rain gauges can be estimated considering quantitative precipitation ensembles (Clark and Slater, 2006), such as Sequential Gaussian Simulations (Goovaerts et al., 1997;Rakovec et al., 2012a). Precipitation ensemble generated with NWPs can than be coupled with hydrologic models to improve flood forecasting (Jaspe ret al., 2002;Sorooshian et al., 2008). In this work we decided adopted a simplified procedure taking in to account all the modeling uncertainties considering the frequency distribution of the errors between the observed and simulated flow values obtained by the calibration and validation of four small tributaries of the Tiber river basin in past flood events. "