  Many simple hydrologic models are based on parametric statistical relations
between the river flow and catchment properties such as its area, precipitation
rates, soil properties, etc., fitted to the available data. The main objective
of this work is to explain how these statistical relations emerge from the
physical laws governing surface and subsurface flow at a catchment scale. The
main achievement of this work is the derivation of an analytic formula for
predicting peak monthly and annual river flows. It does not require any
parameter calibration, but requires a measurement or estimation of the mean
flow at the given catchment's outlet. We found that this model 1) has a simple
physical interpretation, 2) provides more precise estimates than the median
maximum annual flow (QMED) estimation method from the Flood Estimation Handbook
(FEH), commonly used to estimate flood risk in the ungauged catchments in the
UK, and 3) is highly accurate for all types of catchments, including the small
catchments, for which the standard FEH method is the least accurate.
