# Using Optimisation Meta-Heuristics for the Roughness Estimation Problem in River Flow Analysis
## 3 Hydrodynamic River Flows Analysis
### Mathematical Formulation and Software Settings for Implementing the Model
The basic assumption is that the cross-section number 1 is located upstream of the cross-section number 2. To complete the description, the equation defining the distance weighted reach length \(L\) is made available below:  
\[L=\frac{L_{lob}\bar{Q}_{lob}+L_{ch}\bar{Q}_{ch}+L_{rob}\bar{Q}_{rob}}{\bar{Q}_ {lob}+\bar{Q}_{ch}+\bar{Q}_{rob}}, \tag{6}\]  
where the subscripts \(lob\), \(ch\), \(rob\) referred to the left overbank, main channel, and right overbank respectively. The quantity \(\left(\bar{Q}_{lob}+\bar{Q}_{ch}+\bar{Q}_{rob}\right)\) represents the arithmetic average of the flows between the sections of the left overbank, the main channel and the right overbank. It should be stressed out that the \(h_{e}\) term plays a fundamental role, as it includes the effects of the channel contraction and extension and the related friction losses caused by both the riverbed and banks on the flowing water. If the depth is known in one cross-section, it may be also determined in the second cross-section. This value must be available to calculate the depth distribution along the channel, with the cross-sections being the inlet or the outlet boundary of the channel reach. For this reason, in hydraulic jargon, this is also referred to as the 'boundary condition'. It has also to be noted that the kinematic energy terms and the friction losses depend on the magnitude of the flow quantified with the discharge term \(Q\). Moreover, this model takes also into consideration the influence of floodplains while calculating St. Venant coefficients and the weighted distance between cross-sections. The approached used in HEC-RAS to determine both the total conveyance and the velocity coefficient for a specific cross-section, requires separating the water flow into sections where velocities can be considered uniformly distributed. This is obtained by dividing the river's flow in the overbank areas by using the cross-section break-point values (locations where \(n\)-values change). Then, the conveyance is calculated within each subdivision through Equation (3). The empirical correlation between [PERSON]'s equation and [PERSON]'s coefficient is then given by the conveyance coefficient \(K\) of each ordained subdivision. This is expressed as follows  
\[K=\frac{1.486}{n}AR^{2/3}, \tag{7}\]  
where \(n\) is the [PERSON]'s roughness coefficient, \(A\) is the flow area and \(R\) is the hydraulic radius (area/wetted perimeter)--all terms refer to a specific subdivision and can significant vary in different subdivisions. The HEC-RAS software program sums up all the incremental conveyances in the overbanks to obtain a conveyance for the left overbank and the right overbank. Note that the conveyance o the main channel is commonly computed as a single conveyance component. Then, the three subdivision conveyances (i.e., 'left', 'channel' and 'right') are summed up to obtain the total conveyance for the cross-section. A single Manning's \(n\) can be used for multiple sections or calculated for each section (i.e., 'left', 'channel', 'right').This study uses a number of 3\(\varsigma\) Manning's roughness coefficients, as indicated in Section 3.1. On top of evaluating these Manning's \(n\)-values by solving the optimisation problem formulated in Section 2, for the sake of completeness and comparison, we also adopt the standard approach. This consists in considering tabulated values by [PERSON] (1988), which are displayed in Figure 2.