Introduction

The modelling of flooding events, either in terms of historical reconstructions, hypothetical scenarios for improving resilience, and future projections, requires, besides accurate, fast and stable numerical models, the use of proper input data. If bathymetrical information can be obtained from high-resolution Digital Terrain Models (DTM), and roughness coefficients can be calculated via calibration processes, the definition of the boundary conditions, and particularly the inflow hydrographs, remains a crucial issue.

This challenge derives from the fact that direct discharge measurement is difficult to carry out during flood events in natural rivers; therefore, discharge hydrographs are usually inferred from observed stage hydrographs by means of rating curves. However, this procedure presents several limitations. Firstly, rating curves are seldom well calibrated for high discharge values. Moreover, the stage-discharge relationship is not

unique due to inertial and/or backwater effects. Furthermore, rating curves hold only for the considered gauging station and they are influenced by the river geometry, which actually changes from one season to the other and by natural (erosion/deposition) and/or artificial processes. Finally, water levels gauges can be damaged during extreme flooding events, consequently causing corruptions in the historical registrations.

Therefore, the knowledge of discharge hydrographs in ungauged river sections is still a relevant hydraulic problem not only, as mentioned before, for flood modelling purposes, but also for more practical issues related to flood protection measures, hydropower plants, water resource management, design of new structures, etc.

In the framework of numerical models, discharge hydrographs at a specific site can be evaluated by coupling rainfall-runoff and flood propagation schemes. However, rainfall-runoff models (Beven, 2011) present several uncertainties associated for example with the choice of the model for the basin schematization, with the evaluation of the effective rainfall, and with the calibration procedure.

A different method for defining the discharge hydrograph in an ungauged river section concerns the application of a reverse flow routing process: this numerical procedure allows evaluating the upstream flood wave starting from the knowledge of the downstream hydrograph and the hydraulic characteristics of the river reach.

It is noteworthy that the solution of an inverse problem presents three main challenges: the solution may not exist, or it may be non-unique, and during the computation instabilities in the solution may arise.

First attempts of solving the reverse flow routing problem for flood propagation were based on the solution of a reverse form of the Saint Venant equations. Following this approach, Eli et al. (1974) calculated the upstream discharge, once assumed as known quantities the discharge series at the end of the river and the initial conditions. Similar results were obtained by Szymkiewicz (1993) adopting an implicit scheme, under the hypothesis of subcritical flow conditions. However, in both works, the known information is assumed free of errors and instabilities in the discharge computation arise.

For a given single shape of inflow hydrograph, Dooge and Bruen (2005) investigated the

stability of the reverse routing problem for flood propagation and demonstrated that it is related to the adopted discretization parameters and to the channel bed slope.

Adopting a different approach, Das (2009) solved the reverse stream flow routing by adopting a reverse form of the Muskingum model and highlighted the importance of separately calibrating the Muskingum model parameters. However, in this work no errors corrupting the observations were considered. Furthermore, the Muskingum model provides a simplified flood routing based on the adoption of two parameters that supply a detailed description of the channel characteristics.

A different literature branch uses the water levels measured in two or three gauging stations of a river reach, in order to estimate the discharge hydrograph in one of these sections. For natural and artificial channels, Aricò et al. (2009; 2010) investigated and validated a hydraulic diffusive flow routing model, which requires the knowledge of synchronous water level measurements in two river sections a few kilometers apart, and estimates both the discharge hydrograph in the upstream section and the channel roughness. Perumal et al. (2007) adopted a Muskingum numerical scheme for estimating the discharge hydrograph at the upstream or downstream gauging station, from the knowledge of the stage hydrographs registered at the two gauging stations; recently, Barbetta et al. (2017) extended the method as to take into account later inflows.

Considering the presence of later contributions, and using the water level measurements at two gauged sections, Spada et al. (2017) estimated the discharge hydrograph at the downstream one and the channel roughness.

The cited works require the knowledge of the water levels in two/three river sections in order to estimate (considering or not later contributions) the discharge hydrograph in one of the section with register instruments. However, they do not manage to estimate the flow hydrograph in an ungauged section located upstream a gauging one.

Zucco et al. (2015) investigated the reverse flood routing process in natural channels, and estimated the discharge hydrograph in ungauged sections, by means of a Genetic algorithm. The discharge hydrograph is described by means of a Pearson type III distribution and thus the algorithm estimates three parameters. However, this assumption

limits the estimation of real flood waves with irregular shapes. Moreover, the equifinality problem may arise, since different set of parameters can produce the same downstream-observed hydrographs.

Recently, D’Oria and Tanda (2012) solved the reverse flow routing problem adopting a novel Bayesian Geostatistical Approach (BGA), which considers a flow hydrograph as a statistical continuous random function that presents autocorrelation and accounts for uncertainties. For 1D cases, the authors showed the capability of the BGA methodology to estimate the flow discharge in an upstream-ungauged section without having knowledge of the water levels in this section, and having information only in a downstream section: the procedure evidenced no instabilities also in presence of corrupted downstream discharge.

The BGA method was further extended in order to adopt as downstream observations stage hydrographs instead of the discharge ones (D’Oria et al., 2014). Focusing on a real river reach that presents also junctions, the BGA methodology allowed the inflow estimation of the tributary channel, having information about the inflow discharge on the main channel and the stage hydrograph recorded downstream the confluence (D’Oria et al., 2014). Additionally, the same methodology was adopted to estimate the flow through a levee breach assuming the knowledge of the water levels recorded downstream and/or upstream the river bank failure (D’Oria et al., 2015): all these test cases were performed using a 1D forward model.

However, in many real cases of rivers including large floodable areas, it is necessary to adopt a 2D Shallow Water Equation model to capture the complex hydrodynamic field, even if this poses the drawback of the high computation costs, with respect to the 1D scheme.

Therefore, this chapter extends the BGA methodology for reverse flow routing to 2D test cases in order to model natural rivers with complex geometry, including flood plains and meanders. Thus, the goal of the procedure is to estimate the discharge hydrograph in an upstream-ungauged river section, having water level information only in a downstream observation site. Since the inverse procedure requires also a stable, accurate and fast

forward numerical model, the 2D GPU finite volume model illustrated in Section 1.2 is thus adopted.