Dimensional analysis of literature formulas to estimate the characteristic flood response time in ungauged basins: A velocity-based approach

When investigating the wide literature dealing with the assessment of the critical time scale for basin hydrological response, several aspects still to be clarified can be acknowledged. Despite the high sensitivity of design flood peaks to the estimated time parameter value, there is still no agreement on the conceptual and operational definitions of the basin response time, resulting in several different approaches and formulations available. In our work, we suggest a conceptual approach to either reject or recommend formulas of the characteristic basin response time for ungauged basins, with the aim of devising some practical steps in the choice of a robust formulation to be used in hydrological modelling and flood hydrograph design. To this end, 29 empirical and semi-empirical formulas, all containing a basin length and slope, have been carefully selected and their structure compared in dimensional terms, using a simple hydraulic reasoning (e.g., the Chezy formula) as indicator of hydraulic consistency. 13 hydraulically consistent formulas have been identified. Starting from wave celerities and using the river network morphology of 135 watersheds in north-western Italy, we have then investigated and compared the variability of the average flow velocities estimated using all formulas whose input data are available within the study area. By comparing the magnitude and basin scale dependence of the inferred velocities with the values observed in the literature, which generally increase with basin size, some formulas are considered not reasonable, while 5 of them are identified as more robust, i.e. consistent with the observations. These are the formulas of Chow (1962), NERC (1975), SCS (1954), McEnroe and Zhao (1999) and Watt and Chow (1985). Our findings lead to identify analytically the relationships between the exponents of each formula and those of the scaling law linking the length and slope of the basin. These relationships, driving the increase or decrease in the velocity values with basin size, allow us to identify the range of length and slope exponents in the characteristic time formulas for which velocity increases with basin area, as literature suggests. Based on the same relationships, one of the 5 formulas above can be adopted in practical applications and a guideline for calibrating new formulations can be followed.