The central hypothesis of a nonlinear geophysical flood theory postulates that, given space‐time rainfall intensity for a rainfall‐runoff event, solutions of coupled mass and momentum conservation differential equations governing runoff generation and transport in a self‐similar river network produce spatial scaling, or a power law, relation between peak discharge and drainage area in the limit of large area. The excellent fit of a power law for the destructive flood event of June 2008 in the 32,400‐km² Iowa River basin over four orders of magnitude variation in drainage areas supports the central hypothesis. The challenge of predicting observed scaling exponent and intercept from physical processes is explained. We show scaling in mean annual peak discharges, and briefly discuss that it is physically connected with scaling in multiple rainfall‐runoff events. Scaling in peak discharges would hold in a non‐stationary climate due to global warming but its slope and intercept would change.

The goal of this study is to diagnose the manner in which radar‐rainfall input affects peak flow simulation uncertainties across scales. We used the distributed physically based hydrological model CUENCAS with parameters that are estimated from available data and without fitting the model output to discharge observations. We evaluated the model's performance using (1) observed streamflow at the outlet of nested basins ranging in scale from 20 to 16,000 km² and (2) streamflow simulated by a well‐established and extensively calibrated hydrological model used by the US National Weather Service (SAC‐SMA). To mimic radar‐rainfall uncertainty, we applied a recently proposed statistical model of radar‐rainfall error to produce rainfall ensembles based on different expected error scenarios. We used the generated ensembles as input for the hydrological model and summarized the effects on flow sensitivities using a relative measure of the ensemble peak flow dispersion for every link in the river network. Results show that peak flow simulation uncertainty is strongly dependent on the catchment scale. Uncertainty decreases with increasing catchment drainage area due to the aggregation effect of the river network that filters out small‐scale uncertainties. The rate at which uncertainty changes depends on the error structure of the input rainfall fields. We found that random errors that are uncorrelated in space produce high peak flow variability for small scale basins, but uncertainties decrease rapidly as scale increases. In contrast, spatially correlated errors produce less scatter in peak flows for small scales, but uncertainty decreases slowly with increasing catchment size. This study demonstrates the large impact of scale on uncertainty in hydrological simulations and demonstrates the need for a more robust characterization of the uncertainty structure in radar‐rainfall. Our results are diagnostic and illustrate the benefits of using the calibration‐free, multiscale framework to investigate uncertainty propagation with hydrological models.