A Moment-Based, Calibration-Free Evaluation of Unit Hydrograph Shape Across UK Catchments

The choice of unit hydrograph (UH) shape remains a long-standing problem in rainfall–runoff modelling, with widely used forms such as the Gamma, triangular and other hydrographs often adopted without explicit validation. Here we present a calibration-free, moment-based framework to objectively test UH shape assumptions using observed rainfall–runoff events.

Treating time as a random variable and rainfall and runoff as probability weights, we exploit the additivity of central moments under convolution to directly estimate the first three temporal moments of the UH from data. For each event, UH moments are obtained as differences between runoff and rainfall moments, without fitting hydrographs or calibrating model parameters. For two-parameter unit hydrograph shape families, such as the Gamma (Nash) and triangular hydrographs, the first two temporal moments uniquely determine the shape parameters. The third central moment is therefore not a fitting target but an independent prediction, allowing systematic errors from the assumed shape to be identified.

The framework is applied to hourly data from 431 UK catchments from the CAMELS-GB dataset, using the 50 largest events per catchment. Event- and catchment-level diagnostics highlight systematic differences between alternative unit hydrograph shape families, including Gamma and triangular representations. By comparing moment-based consistency across shapes and response time scales, the analysis provides an objective basis for identifying the most appropriate unit hydrograph form for different hydrological conditions.

Overall, the proposed moment-based framework offers a physically interpretable and calibration-free approach for evaluating and comparing unit hydrograph shapes, with clear potential for application in catchment classification and regionalisation.