# Using Optimisation Meta-Heuristics for the Roughness Estimation Problem in River Flow Analysis
## 1 Introduction  
To 'ensure availability and sustainable management of water and sanitation for all' is one of the Sustainable Development Goals (SDGs) targeted by the United Nations. Since its launch in 2015, this goal (i.e., SDG 6), assumed a wider focus and already in 2018 it was agreed that all aspects of the water cycle, being of paramount importance for development, should be considered and embedded directly and indirectly in all the remaining 17 SDGs [1]. In this context, hydraulic models are of pivotal importance to support the progress in achieving multiple SDGs. Indeed, the latter find their place in several fields of application having key implications on sustainable management of water sources [2], preserving the environment [3] and mitigating climate changes [4]. More in detail, hydraulic models can be used to predict the impact of floods, thus helping with insurance claims [5], to calculate the number of pollutants in rivers [6], estimate backwater [7] and simulate water free-surface flow [8]. Models also worth mentioning are those capable of characterising floods through sediment transport and analysis [9, 10], also used to develop flood protectionsystems [11], and those generating flooding maps, simulating drainage channels and water temperatures [12; 13; 14]. Note that in the vast majority of cases, including the aforementioned studies, good models of water flows are needed to produce meaningful results. However, even if established models are nowadays available, they are sensitive to parameters that have to be found either empirically or numerically via computational methods [15]. In hydraulic engineering, such parameters typically assume two different natures:  
* the 'physical' parameters are those describing the properties of materials--these are usually constant values;
* the empirical parameters are those added to deal with the complexity and the variability of specific aspects characterising hydraulic properties of a channel. Amongst these, the so-called 'roughness' coefficient, which indicates the resistance to flood flows in both channels and flood plains, is one of the most important. Other empirical parameters are the vegetation, the alignment of the channel and its irregularities, shape and size, stage and discharge, suspended sediment load and bed sediment loads. These have to be calculated mathematically through analytical models [16]. Using [PERSON]'s equation [17], which is expressed in terms of the roughness coefficients '\(n\)', is amongst the most popular empirical methods for estimating uniform flow in open channels. In the simplest case, suggested \(n\) values can be taken from established references, where they are tabulated [18], but they can also be calculated through a number of empirical formulas [19] or field measurements. It must be remarked that the estimation of \(n\), commonly also referred to as 'Manning's coefficients', can significantly affect the final results [20]. Hence, the use of precise numerical techniques to compute such coefficients is preferred to achieve better simulations, thus allowing for more informed prevention guidelines to avoid floods and related natural disasters. Still, finding the optimal \(n\) value is a challenging task. In 2007, [PERSON] et al. [21] implemented the GSA-GLUE methodology to determine the model's sensitivity on Manning coefficients. That analysis showed that the floodplain roughness coefficients have a small influence on flow, even though larger than that of the geometry. In the work by [PERSON] et al., in 2010 [22], a calibration model was applied to assess the Manning coefficients in solving the 1-D Saint Venant equations. In 2011 [PERSON] et al. [23], applied a calibration methodology of a steady-state two-dimensional flood inundation model using a Bayesian approach comparing their results with previous ones based upon the same data set and flood inundation model, where the uncertainty was analyzed using GLUE.