Derivation of rating curve by the Tsallis entropy

- Rating curve is developed and verified by maximizing the Tsallis entropy.
- The entropy index is found to m = 3.
- The modified rating curve using the M parameter is equivalent to that determined using the Lagrange multipliers methods.
- The entropy method can predict the rating curves correctly for ungauged stations.
- The entropy method satisfactorily extends the rating curve beyond the range of values used in its construction.

SummaryThe stage-discharge relation, often called rating curve, is employed to determine discharge in natural and engineered channels. There are several methods for deriving a rating curve most of which are empirical. It is well recognized that rating curves are subjected to significant uncertainty, yet most of these methods do not have any provision to account for or do not quantify the uncertainty. This study employs the Tsallis entropy for deriving the rating curve, based on two simple constraints: (1) total probability and (2) mean discharge. Parameters of the derived curve are determined with the use of these two constraints. The rating curve is also determined by reparameterization with the use of an entropy parameter. The Tsallis entropy permits a probabilistic characterization of the rating curve and hence the probability density function of discharge underlying the curve. It also permits a quantitative assessment of the uncertainty of discharge obtained from the rating curve. The derived rating curve is found to be in agreement with field data and is also applied to ungaged watersheds. The rating curve is also extended beyond the range of discharge values used in its construction and its validity is then evaluated.