Assuming time-averaged velocity as a random variable, this study develops an entropy theory for deriving two-dimensional (2D) distribution of velocity in open channels. The theory comprises five parts: (1) Tsallis entropy; (2) principle of maximum entropy (POME); (3) specification of information on velocity in terms of constraints; (4) maximization of entropy; and (5) derivation of the probability distribution of velocity. The entropy theory is then combined with a hypothesis on the cumulative distribution function of velocity in terms of flow depth to derive a 2D velocity distribution. The derived distribution is tested using field as well as laboratory observations reported in the literature and is compared with known velocity distributions. Agreement between velocity values computed using the entropy-based distribution and observed values is found satisfactory. Also, the derived distribution compares favorably with known distributions.

Assuming time-averaged velocity as a random variable, this study develops an entropy theory for deriving the one-dimensional distribution of velocity in open channels. The theory includes five parts: (1) Tsallis entropy; (2) the principle of maximum entropy (POME); (3) the specification of information on velocity for constraints; (4) the maximization of entropy; and (5) the probability distribution of velocity and its entropy. An application of the entropy theory is illustrated by deriving a one-dimensional velocity distribution in open channels in which the dimension is vertical or the flow depth. The derived distribution is tested with field and laboratory observations and is compared to Chiu’s velocity distribution derived from Shannon entropy. The agreement between velocity values are computed with the entropy-based distribution.

Localized and severe storms can cause citywide flooding, leading drainage systems to surcharge and overflow to nearby water courses. Urban catchments feature high degrees of imperviousness and heterogeneity, often resulting in highly nonlinear hydrologic responses with shorter time of concentration, lag times, and sharper peak flows. Additionally, due to population and economic growth, urban drainage systems have attempted to evolve to more efficiently drain surface waters and reduce vulnerabilities. A critical outcome of this evolution is the need for finer spatio-temporal resolution rainfall measurements and hydrological modeling. As the major driving mechanism, the spatio-temporal variability in rainfall is acknowledged as a key source of uncertainty for urban hydrological modeling. The objective of this research is to revisit the impact of the temporal and spatial resolution of rainfall measurements on urban hydrological applications. We first provide a quantitative analysis of the spatiotemporal structure and variability of rainfall using both a 9-member hourly rain gauge network spaced ~10 km apart and a single WSR-88D dual-polarimetric weather radar with precipitation resolved every 5 minutes at ~500 m. Precipitation data from each observing system extracted at different time steps is aggregated within urban catchments and compared for three typical intense storms over a set of urban catchments located in Chicago Metropolitan area. Then the rain-runoff dynamics for 9 geographically-diverse (relative to the underneath sewer system) and differently-sized catchments are examined utilizing MetroFlow – a coupled hydrologic and hydraulic modeling system. Finally, city-wide flooding risks are simulated by routing the predicted surface runoff through the as-built sewer system. Additional mitigating storage capacity is also considered by numerical modeling the deep tunnel and reservoir in construction. The sensitivity of urban flood variables (i.e., mean and peak depth as well as duration) to rainfall spatio-temporal resolution is analyzed. Our results complement and advance the limited literature attempting to resolve the ideal resolution of rainfall data relevant for urban hydrology and stormwater management.