How to address velocity measurements during high floods by using the entropy theory: the Case Study of the gauged section along the Alzette River, Mersch, in Luxembourg

It is well known that the information entropy represents a measure of the uncertainty linked to a probability distribution and it is fundamental for solving several problems based on statistical models, where the absence of data requires general assumptions for parameters estimating. This is the case of the flow velocity distribution at river cross-sections and, in this context, Chiu (1987) was the pioneer in applying the entropy theory for the formulation of the velocity distribution in the probability domain. Based on this probabilistic formulation, the mean velocity, um, can be expressed as a linear function of the maximum velocity, , through a dimensionless entropy parameter M. The correlation among these two quantities was investigated for some gauged river sections across the world and a perfect agreement between the mean and maximum velocity was found. Therefore, the possibility to assess the velocity distribution only considering the maximum velocity and the entropic parameter M is of great interest in the context of discharge monitoring by standard techniques and, in particular, during high floods. Although there are a large number of studies on the velocity profiles analysis in natural channels, only a few have been addressed for estimating the spatial velocity distribution during high floods when it is not possible to measure the whole velocity field and in particular in the lower portion of the flow area. Indeed, the sampling procedure of velocity measurements in a river cross section, in this case, might be difficult and dangerous for cableway operators. On the other hand, the value of maximum flow velocity could be more easily sampled since its position is located in the upper portion of the flow area where velocity measurements can be carried out also during high flow conditions. Considering that the rating curve accuracy is strictly connected to experimental data availability which have to be referred both to low and high flow depths, we well know as a quick and accurate determination of high flow passing through a river section may be fundamental for the rating curve assessment. Therefore, a model based on the entropy theory and able to assess the velocity profiles, also when velocity data are not available in any portion of the flow area is welcome. The objective of the work is, then, to propose a procedure for addressing the velocity measurements during high floods by sampling the only. The procedure is based on the entropy velocity profile coupled with an elliptical/parabolic distribution of the maximum velocities, in the flow area. The elliptical/parabolic distribution of is derived by Chezy’s formula and by assuming a distribution of the verticals depth in the flow area depending on the depth of the vertical where is sampled and on the distance of each vertical from the side walls. The gauged section, named Mersch, along the Alzette River in Luxembourg is used as case study. A sample of fifteen velocity measurements for a total number of 200 sampled verticals is considered for the analysis. The entropic parameter value, M, is found equal to 2.32 with a determination coefficient R2=0.95. The velocity profiles uncertainty is also addressed by considering the error in estimating the observed velocity points. Results show that the procedure is able to estimate accurate velocity profiles using all velocity points sampled in the upper portion of the flow area and fairly accurate profiles if only is surmised sampled, regardless the distribution of maximum velocities adopted in the flow area. The main assets of the proposed approach are both to guarantee the safety conditions for the operators during the measurement and to drastically drop the time of sampling. It is worth noting that this aspect is fundamental for the practice hydrology because the monitoring of the maximum velocity, nowadays, can be done by using a portable radar unit, which makes possible a very quick measurement and, hence, for the same flood more gauged river sites can be monitored; condition that never can be accomplished by using traditional techniques as the one based on the use of current meter