Theoretical link between rainfall and flood magnitude

The rainfall–run‐off convolution integral is analytically solved for several models for the elementary hydrograph. These solutions can be combined with available rainfall frequency analyses to predict flood flows along streams for different recurrence intervals, using no free parameters for gauged streams and one estimable parameter for ungauged streams. Extreme discharge magnitudes at gauged sites can be typically estimated within a factor of two of actual records, using no historical data on extreme flows. The flow predictions reproduce several important characteristics of the flood phenomenon, such as the slope of the regression line between observed extreme flows and basin area on the conventional logQ versus logA plot. Importantly, for the models and data sets investigated, the storm duration of greatest significance to flooding was found to approximate the intrinsic transport timescale of the particular watershed, which increases with basin size. Thus, storms that deliver extraordinary amounts of rainfall over a particular time interval will most greatly activate basins whose time constants approximately equal that interval. This theoretical finding is supported by examination of the regional hydrological response to the massive storms of September 14, 2008, and April 28–30, 2017, which caused extraordinary record flooding of basins of about 5–100 km² and 500–4,000 km², respectively, but produced few records in basins that were larger or smaller than those ranges.